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% Macros for References
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\def\authorref#1{\textsc{#1:}}
\def\titleref#1{\textit{ #1,\ }}
\def\publisher#1#2#3{\textrm{#1, #2, (#3).}} %publisher, place, year
\def\journal#1#2#3#4{\textrm{#1\ #2\ (#3), pp.~#4.}} % jour, num, year, pages
\def\appearjournal#1{\textrm{to\ appear\ in:\ {#1}}}% jour
\def\preprint#1#2#3#4{\textrm{#1\ #2\ (#3), ftp: #4.}} % uni, num, year, ftp ad
%
%% Macros to define multi character operators or symbols.
\DeclareMathAlphabet{\mathrmsl}{OT1}{cmr}{m}{sl}
\def\operrm#1#2{\def#1{\mathop{\mathrm{#2}}\nolimits}}
\def\operit#1#2{\def#1{\mathop{\mathit{#2}}\nolimits}}
\newcommand{\rsoper}[3][n]{\newcommand{#2}{\mathop{\mathrmsl{#3}}\ifx
  n#1\nolimits\else\limits\fi}}
%% Macros of use to any budding analyst.
%
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\def\nrm#1{|\kern-.1em|#1|\kern-.1em|}
\def\restr#1{|_{\lower1pt\hbox{${}_{#1}$}}}
%
%% Macros for inner products
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\def\xgenip#1#2,#3??#4{#1#2,#3#4}
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\def\ipp#1{\genip{\langle\kern-.2em\langle}{#1}{\rangle\kern-.2em\rangle}}
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%
% Set theoretical variables
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\def\represent{\mathrm{repr}\,}
\def\support{\mathrm{supp}}
\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\def\H{\mathbb{H}}
\def\F{\mathbb{F}}
\def\A{\mathbf{A}}
\def\asso{\hat} %associated space
\def\implies{\Rightarrow}
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% Linear algebraic variables
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\def\binomiel#1#2{\hbox{${#1 \choose #2}$}\,}
\def\Tens{\mathop{\bigotimes}} %big tensor product
\def\tens{\otimes} %small tensor product
\def\cartan{\odot} %Cartan tensor product
\def\Cartan{\mathop{\bigodot}}
\def\Dsum{\mathop{\bigoplus}} %big direct sum
\def\dsum{\oplus} %small direct sum
\def\id{\mathit{id}} %identity map
\def\isomorph{\cong} %isomorphism sign
\def\prop{\sim} %proportionality sign
\def\sig{\hbox{sig}} %signature
\def\dual{*} %dual vector space
\def\dim{\mathrm{dim}\,}
\def\kernel{\mathrm{ker}\,}
\def\image{\mathrm{im}\,}
\def\inner{\mathop{\raise.1em\hbox{$\lrcorner$}}}
\def\innerH{\,\lrcorner_H\,}
\def\wedgeH{\wedge_H}
\def\cowedge{\triangle}
\def\coproduct{\triangle}
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\def\End{\mathrm{End}\,} %vector spaces related to vector space
\def\Hom{\mathrm{Hom}\,} %vector spaces related to vector space
\def\Skew{\mathrm{Skew}\,}
\def\Sym{\mathrm{Sym}\,}
\def\Weyl{\mathrm{Weyl}\,} %Weyl curvature tensors
\def\Cl{\mathrm{Cl}\,}
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\def\GL{\mathbf{GL}\,} %groups related to vector space
\def\Aut{\mathbf{Aut}\,}
\def\SL{\mathbf{SL}\,}
\def\CO{\mathbf{CO}\,}
\def\O{\mathbf{O}\,}
\def\SO{\mathbf{SO}\,}
\def\Pin{\mathbf{Pin}\,}
\def\Spin{\mathbf{Spin}\,}
%\def\U{\mathbf{U}\,}
\def\SU{\mathbf{SU}\,}
\def\Sp{\mathbf{Sp}\,}
\def\Usp{\mathbf{Usp}\,}
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\def\lie#1{\mathfrak{{#1}}} %Lie algebra notation
\def\g{\lie{g}}
\def\p{\lie{p}}
\def\l{\lie{l}}
\def\gl{\lie{gl}} 
\def\sll{\lie{sl}}
\def\co{\lie{co}}
\def\o {\lie{o}}
\def\so{\lie{so}}
\def\su{\lie{su}}
\def\sp{\lie{sp}}
\def\sp{\lie{sp}}
\def\der{\lie{der}}
\def\U{\mathfrak{U}} %universal enveloping algebra
\def\Centre{\mathfrak{Z}} %centre of algebra
\def\generatedby#1{{<}{#1}{>}}
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\def\rep{\theta} %Lie algebra representation
\def\Rep{\Theta} %Lie group representation
\def\ad{\mathrm{ad}} %adjoint representation
\def\coad{\mathrm{coad}}
\def\ind{\mathrm{ind}}
\def\Ind{\mathrm{Ind}}
\def\Ad{\mathrm{Ad}}
\def\Casimir{\mathrm{Cas}} %Casimir operator
\def\casimir{\mathrm{cas}} %Casimir operator
\def\Spinor{S} %spinor representation space
\def\Verma{\mathrm{Verma}\,}
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\def\Roots{\Phi} %set of roots
\def\tang{\top} %orthogonal projections
\def\normal{\perp}
\def\rad{\mathrm{rad}} %radial component
\def\spher{\mathrm{spher}} %spherical component
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\def\skew{\bigtriangleup} %skew symmetric endomorphism generated by two vectors
\def\Hodge{\ast} %Hodge star operator
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\operrm\alt{alt} %alternating part of a multilinear form
\operrm\sym{sym} %symmetric part
\operrm\trace{tr} %trace of an endomorphism
\operrm\deter{det} %determinant
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% Clifford algebraic variables
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\def\eve{^{\mathrm{ev}}}
\def\odd{^{\mathrm{od}}}
\def\twist#1{#1^\ast}
\def\rever#1{\widetilde{#1}}
\def\conju#1{\overline{#1}\,}
\def\invert{^\times}
\def\realpart#1{{\langle{#1}\rangle}_{\R}}
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% Differential topological variables
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\def\boundary{\partial} %boundary of a manifold
\def\homolog{\sim} %homological sub manifolds
\def\Lieb#1{\genip\lbrack{\,#1\,}\rbrack}
\def\immersion{\hookrightarrow} %immersion symbol
\def\orie{\mathrmsl{or}} %orientation
\def\coorie{\mathrmsl{coor}} %coorientation
\def\outernormal{\mathrmsl{out}}
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% Differential geometrical variables
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\def\com{\mathsf{c}} %conformal metric
\def\cip#1{\letterip{\com}{#1}}   %conformal inner product
\def\del{\partial} %Euclidian derivative
\def\delbydel#1{{\del\over\del #1}} %d by d t
\def\Lie{{\mathcal L}} %Lie derivative
\def\LieWeyl{\partial}%
\def\christo{\gamma} %Christoffel symbols
\def\Christo{\Gamma}
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% Differential operators
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\operrm\Tor{Tor} %Torsion tensor
\operrm\Ricci{Ric} %Ricci endomorphism
\operrm\ricci{ric} %Ricci bilinear form
\operrm\nric{r} %normalised ricci
\operrm\scal{scal} %scalar curvature
\operrm\nscal{s} %normalised scalar curvature
\operrm\Weylcurv{W^\com} %Weyl curvature of conformal structure
\operrm\Bachtensor{B^\com} %Bach tensor of n=4 conformal structure
\def\Cotton{\mathrm{C}} %Cotton York tensor
\operrm\Wein{Wein} %Weingarten map
\operrm\Koszul{Koszul} %Koszul operator
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\rsoper\Fermi{Fermi} %Fermi derivative
\rsoper\CGeod{Geod} %Conformal third derivative
\rsoper\divR{div}
\def\dR{\mathrmsl{d}}
\rsoper\divH{div_H}
\rsoper\dH{d_H}
\def\deltadR{\delta^{\mathrmsl{dR}}}
\def\evaluation{\mathrmsl{ev}}
\rsoper\grad{grad}
\def\Laplace{\Delta}
\def\Quabla{\square}
\operrm\Nabla{\nabla}
\rsoper\Killing{Kill}
\rsoper\Codazzi{Coda}
\rsoper\Lanczos{Lanc}
\rsoper\Lanczosdiv{div}
\rsoper\UltraLanczos{UltraLanc}
\rsoper\Hesse{Hesse}
\rsoper\divdiv{divdiv}
\rsoper\Fierz{Fierz}
\rsoper\Fierzdiv{Fierzdiv}
\rsoper\Bianchi{Bianchi}
\rsoper\LinWeyl{LinWeyl}
\rsoper\Bach{Bach}
\rsoper\Symdiv{Symdiv}
\rsoper\Skewdiv{Skewdiv}
\rsoper\Twistor{Twist}
\rsoper\Branson{Bran}
%
%% Analytic shorthands
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\def\Lebesque{{\cal L}}
\def\Sobolev{{\cal W}}
\def\Hoelder{{\cal C}}
\def\jet{\mathit{jet}}
\def\Jet{\mathit{Jet}}
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\def\other#1{{\widetilde #1}}
%\def\smallfrac#1#2{\hbox{${#1 \over #2}{}$}}
\def\smallfrac#1#2{{#1 \over #2}}
\def\oneover#1{\smallfrac{1}{#1}\,}
\def\twoover#1{\smallfrac{2}{#1}}
\def\const{\mathrm{const.\ }}
\def\sign#1{\mathrm{sign}(#1)}
\def\sky{{\mathrm{Sky}}}
\def\Cone{\mathrm{Cone}}
\def\Cano{\mathrm{Can}}
\def\Eot{\mathrm{E\"ot}}
\def\Cor{\mathrm{Cor}}
%
%\def\Dimof#1{\hbox{dim}\left(#1\right)}
%\def\Dim#1{\hbox{({\it #1\/})}}
\def\Dimof#1{\mathrm{dim}\left(#1\right)}
\def\Dim#1{(\mathit{#1}\,)}
\def\cyclic{\hbox{cyc\,}}
\def\swap{\hbox{sym\,}}
\def\stackyrel#1#2{\!\!\stackrel{#1}{#2}\!\!}
%% Variables and fundamental constants
%
\def\Sum{\sum}
\def\smallsum{\Sigma}
\def\eps{\varepsilon}
\def\rho{\varrho}
\def\light{c}
\def\coulomb{\hbox{$\,{1\over {4\pi\eps_0}}\,$}{}}
\def\oneovercoulomb{\,4\pi\eps_0{}\,}
\def\electron{{e}}
\def\newton{\gamma}
\def\galilei{g}
\def\boltzmann{k}
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\begin{document}
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\begin{center}
\ 
\vskip 3cm
{\Huge Conformal Geometry,}\\
\vskip 2mm
{\Huge Representation Theory}\\
\vskip 2mm
{\Huge and Linear Fields}\\
\vskip 3cm
{\bf Dissertation}\
\vskip 3mm
zur\\
\vskip 3mm
Erlangung des Doktorgrades (Dr.~rer.~nat)\\
\vskip 3mm
der\\
\vskip 3mm
Mathematisch-Naturwissenschaftlichen Fakult\"at\\
\vskip 3mm
der\\
\vskip 3mm
Rheinischen Friedrich-Wilhelms-Universit\"at Bonn\\
\vskip 3cm
vorgelegt von\\
\vskip 3mm
Tammo Diemer\\
\vskip 3mm
aus\\
\vskip 3mm
Essen\\
\vskip 15mm
Bonn 1999
\end{center}
\newpage
\thispagestyle{empty}
\
\newpage
\thispagestyle{empty}
\ 
\vskip 2cm
{\flushleft Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen
Fakult\"at\\
der Rheinischen Friedrich-Wilhelms-Universit\"at Bonn}
\vskip 14cm
{\flushleft 1.~Referent: Prof.~Dr.~Hermann Karcher}
\vskip 2mm
{\flushleft 2.~Referent: Prof.~Dr.~Jens Franke}
\vskip 12mm
{\flushleft Datum der Promotion:}
\newpage
\thispagestyle{empty}
\ 
\newpage
\pagenumbering{roman}
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\tableofcontents
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\newpage
\section*{Introduction}

The aim of this dissertation is to uncover and explore the intimate
relation between three different branches of mathematics: (A) local
differential geometry on conformal manifolds, in particular
conformally invariant linear and bilinear differential operators, (B)
homomorphisms and coproducts between induced modules in representation
theory, and (C) interactions of particles and fields in classical
relativistic linear field theories for electromagnetism and gravity in
conformal Minkowski space.

The M\"obius group $G=\O(p+1,n-p+1)$ of conformal diffeomorphisms acts
transitively upon the conformal sphere $S^{n-p,p}$ of signature
$(n-p,p)$; hence the sphere occurs as homogeneous space
$S^{n-p,p}=G/P$ with a (parabolic) stabilizer group
$P$. Representations $W$ of the M\"obius group enter into all three
contexts in a fundamental way. In conformal geometry (A), the vector
bundle associated to a $G$-representation $W$ carries a canonical
covariant derivative (induced by the normal Cartan connection); on the
conformal sphere, the covariant derivative is flat and the space of
parallel sections is isomorphic to $W$. This is interpreted in
representation theory (B) as a canonical surjection from the
$G$-module induced by $W^\dual$ as a $P$-representation onto
$W^\dual$ using Frobenius reciprocity. The physical significance (C)
of such representations $W$ was discovered by Penrose, see
\cite{PenroseRindler}, who called elements in $W$ \emph{twistors}. He
identified $W$ as the kernel of an overdetermined conformally
invariant operator, called a \emph{twistor operator}. In linear field
theories twistors play the role of generalizations of electric charges
such as gravitational masses, and they play a key role in this
dissertation.

A second manifestation of the above relationship between conformal
geometry, representation theory and field theory, is the fact that
variations of the deRham complex of exterior derivatives between
alternating forms occur in all three of them: (A) associated to any
$G$-representation $W$ is a locally exact complex of conformally
invariant differential operators between sections of homogeneous
bundles over $S^{n-p,p}=G/P$, see Eastwood Rice \cite{EastwoodRice} or
theorem \ref{ConformalBGGcomplex}, (B) any $G$-representation $W$ is
subject to the parabolic Bernstein Gelfand Gelfand resolution in terms
of modules induced by $P$-representations, see Lepowsky
\cite{LepowskyBGG} or theorem \ref{ParabolicBGGresolution}, and (C)
in $n=4$ dimensions with Lorentzian signature $(3,1)$ any such space
$W$ characterizes a linear field theory based upon gauges, potentials
and fields, see \cite{PenroseRindler} or section
\ref{Ingredientsofageneralfieldtheory}. For the trivial representation
$W=\R$ we recover electromagnetism since Maxwell's equations are based
upon the deRham complex. The complexes occurring in (A), (B) and (C)
are named after Bernstein, Gelfand and Gelfand, and are called BGG
complexes for short. The first operator in such a complex is the
twistor operator.

The main result of this dissertation is a new construction of the BGG
complexes using a projection $S$ defined on twisted deRham complexes
by an explicit finite power series. This projection induces a homotopy
equivalence between the twisted deRham complex and the corresponding
BGG complex, and has the additional benefit of revealing a new
feature: not only linear operators, but bilinear operators,
generalizing the wedge product on the deRham complex, are manifest at
the interface of these three topics. In conformal geometry (A), the
projection $S$ translates the wedge product on alternating forms into
bilinear differential pairings on the BGG complexes. These pairings
and operators in the BGG complex are subject to a general Leibniz rule
and hence induce a cup product on the BGG cohomology, see theorem
\ref{Conformalexteriorpairings}. In algebraic terms (B), if $\g$ is a
semisimple Lie algebra and $\p\subset\g$ a parabolic subalgebra, then
$S$ is an explicit $\g$-projection on the $\g$-modules induced by
$\Lambda^k(\g/\p)\tens W^\dual$, see corollary
\ref{Projectionmap}. The image of the projection $S$ is isomorphic to
the $\g$-module induced by the relative cohomology space
$H^k((\g/\p)^\dual,W^\dual)$. The map $S$ defines a homotopy
equivalence between the deRham resolution twisted by $W^\dual$ and the
parabolic BGG resolution of $W^\dual$, see theorem
\ref{ParabolicBGGresolution}. The wedge coproduct induces a new
coproduct on the BGG resolution which satisfies a general co-Leibniz
rule, see theorem \ref{CoLeibnizrule}. In linear field theories (C)
the \emph{helicity} of a field theory is a number reflecting a
particular \emph{polarization property} of plane wave solutions of
that theory. For example electromagnetic waves have helicity one and
gravitational waves are expected to have helicity two. The bilinear
differential pairings provide a wide-reaching generalization of
helicity raising and lowering between solutions in different
conformally invariant linear field theories, which leads, for example,
to general constructions of conserved quantities, see the applications
in section \ref{Generalfieldtheory}.

The motivation to study linear field theories other than the
electromagnetic theory stems from the quest for a better understanding
of gravitational waves. This phenomenon is the gravitational analogue
of electromagnetic waves---the latter have a satisfactory description
in terms of Maxwell's equations. One way to study gravitational
interactions is to set up a linear field theory with fields subject to
equations analogous to Maxwell's equations and with a force law which
predicts the motion of gravitational particles within these
fields. Einstein's theory of gravity is nonlinear and linearizations
of the Einstein equation are obvious candidates for a linear theory of
gravity. The naive linearization of the Einstein equation on the level
of the metric, with Minkowski background, fails to be conformally
invariant. Instead we take the electromagnetic theory as a starting
point for a development of general linear field theories. The guiding
principle will be conformal invariance of the theory: gravitational
waves are expected to travel with the speed of light and the light
cone geometry only determines a conformal structure.

A linear field theory describing gravitational interaction deals with
the following physical phenomenon: motion of gravitational masses
produces fields which travel with the speed of light and these fields
influence the motion of gravitational masses. In particular
accelerated sources will produce gravitational wave solutions. Any
$G$-representation $W$ leads to such a field theory, but the expected
polarization properties of plane gravitational waves rule out most $W$
leaving only three theories due to Fierz, Bach and Penrose, which we
will recall in some detail, see sections \ref{Fierzstheoryofgravity},
\ref{SectionBachstheoryofgravity}, \ref{Penrosestheoryofgravity}. The
mentioned generalizations of helicity raising and lowering allow us to
suggest the beginning of a classical theory of motion i.e.~a theory
describing the interaction of particles and fields in arbitrary field
theories with conformally flat background: the elements in $W$ play
the role of generalized charges or gravitational masses in the
corresponding linear field theory. We will demonstrate how an
arbitrary worldline with an associated gravitational mass (i.e.~an
element in $W$, a twistor) gives rise to a conserved distributional
source, see paragraph \ref{Chargedparticlesassources}. Moreover we can
solve the general field equations with that right hand side: the
Lienard Wiechert fields in electromagnetism are solutions of Maxwell's
equations with a distributional source coming from an electrically
charged pointlike worldline. We will construct general Lienard
Wiechert fields solving the general field equations with a
distributional source representing a gravitational mass associated to
an arbitrary worldline, see paragraph
\ref{DistributionalsourcesandgeneralLienardWiechertfields}. In
electromagnetism a straight worldline induces the static Coulomb
field, see paragraph \ref{Coulombfield} and accelerated
worldlines induce radiation fields, i.e.~fields which fall of like
$1/r$ where $r$ is the (luminosity) distance to the source, see
paragraph \ref{ElectromagneticLienardWiechertfields}. Hence the
general Lienard Wiechert fields provide solutions which model the
phenomenon that motion of gravitational masses produces gravitational
waves that travel with the speed of light. On the other hand the
Lorentz force law in electromagnetism predicts the force which acts on
an electrically charged test particle moving in an electromagnetic
field, see paragraph \ref{Lorentzforcelaw}. This force is proportional
to the electric test charge. Similarly the gravitational mass
(i.e.~the element in $W$) of a test particle allows to define a force
from a general field which we suggest to be the force of the field
acting on the test particle, see paragraph
\ref{Lorentzforcelawingeneralfieldtheories}. This force law models the
phenomenon that gravitational fields influence the motion of
gravitational test particles. We claim that the above is the beginning
of a theory of motion for gravitational particles influencing each
other through their gravitational fields. Notice that pointlike
particles as sources in the theory obtained by a linearization of the
Einstein equation are only conserved, if they follow a straight line,
see Stephani \cite{Stephani} p.~89 or remark
\ref{PointlikesourcesinEinsteinstheory}. This prevents a theory of
motion of various particles influencing each other in a linearized
Einstein theory, since already their individual conservation laws
forces them to follow straight lines.

Invariant differential operators are of interest in differential
geometry: the Bianchi identity of the curvature tensor or the Codazzi
equation of the second fundamental form of a hypersurface are examples
where differential operators occur in integrability conditions. The
first analytic question on a manifold with conformal structure is
which differential operators are intrinsically defined. A
classification of linear conformally invariant first order operators
has been given by Fegan, Hitchin, Gauduchon \cite{Fegan,
HitchinLinear, Gauduchon}, see theorem
\ref{Fegansfirstorderconformalinvariants}. Second order operators were
classified by Branson \cite{BransonNonlinear, BransonSecond}, see the
examples \ref{ConformalLaplacian}, \ref{Bransonsoperator} and
proposition \ref{Simplesecondorderdifferentialoperators}. Some of
these operators are twistor operators, i.e.~a section in the kernel of
such an operator is the curved analogue of an element in a
$G$-representation $W$, see propositions \ref{ConformalHessian},
\ref{Conformaltwistoroperatorforweightone}. Motivated by conservation
of energy momentum in relativity, we will explain, in a geometric
context, how twistors give rise to new conserved properties along
conformal geodesics using bilinear differential pairings along curves,
see propositions \ref{Fierzconservation}, \ref{Killingconservation},
\ref{Twistorconservation}. We extend the theory of first and second
order conformally invariant operators by characterizing conformally
invariant bilinear differential pairings by algebraic data and
construct some new classes of simple examples, see
\ref{Algebraicfirstorderpairings},
\ref{Simpleconformallyinvariantfirstorderpairings},
\ref{Laplacepairing} ff.
%\ref{Bransonpairing},
%\ref{Hessepairing},
%\ref{Simpleconformallyinvariantsecondorderpairings},
%\ref{Algebraicsecondorderpairings}.
First and second order operators and pairings are sufficient to do
helicity raising and lowering between electromagnetism and the three
linear theories of gravity as indicated in sections
\ref{Fierzstheoryofgravity}, \ref{SectionBachstheoryofgravity},
\ref{Penrosestheoryofgravity}. Indeed it was these examples of
conformally invariant bilinear pairings and Leibniz rules which led us
to conjecture the general existence of differential pairings imitating
the wedge product between forms.

In this dissertation we will explore the new product structure in BGG
complexes on homogeneous parabolic geometries, i.e.~on homogeneous
spaces $G/P$ where $G$ is a semisimple Lie group and $P$ is a
parabolic stabilizer group. For an extension of our result to curved
parabolic geometries using Cartan connections see the article by
Calderbank and the author \cite{CalderbankDiemer}. Parabolic geometry
includes for example conformal geometry in $n\ge 3$ dimensions,
M\"obius geometry in $n=2$ dimensions, projective geometry,
Gra{\ss}mannian geometry and Cauchy Riemann geometry. The programme of
parabolic invariant theory was initiated by Fefferman and Graham
\cite{Fefferman,FeffermanGraham}. We will focus on $G$-equivariant
differential operators and pairings on $G/P$ acting on sections of
homogeneous bundles. Let $\p\subset \g$ denote the Lie algebras of
$P\subset G$. The intimate relation between geometry and algebra is
given in this context by the fact that equivariant differential
operators on $G/P$ correspond to $\g$-equivariant homomorphisms
between Verma modules induced by $\p$-representations. First versions
of this observation go back to Bernstein Gelfand Gelfand
\cite{BGGDifferential} and Eastwood Rice \cite{EastwoodRice} and were
fully discovered independently by Baston Eastwood
\cite{BastonEastwood}, Collingwood Shelton \cite{CollingwoodShelton}
and Soergel \cite{Soergel}. The study of homomorphisms between Verma
modules was initiated by Verma \cite{Verma} and continued by Bernstein
Gelfand Gelfand \cite{BGGStructure} and Lepowsky
\cite{LepowskyBGG}. In interesting special cases such as the conformal
case all Verma module homomorphisms are known due to Boe Collingwood
\cite{BoeCollingwood}, but in general a complete classification has
not been achieved. The resolution of a $\g$-representation in terms of
induced parabolic Verma modules due to Bernstein Gelfand Gelfand
\cite{BGGStructure} and Lepowsky \cite{LepowskyBGG} provides us with a
complex of homomorphisms between Verma modules, which are called
standard homomorphisms. The corresponding sequence of invariant
differential operators is a locally exact complex. The problem with
these standard homomorphisms constructed by Verma and Bernstein
Gelfand Gelfand is that they were only explicit in special cases, for
instance if the corresponding highest weights are related by a simple
root reflection. In a sequence of pioneering papers
\cite{BastonVerma,BastonAHS}, Baston introduced a number of general
methods to construct invariant differential operators in conformal
geometry and a related class of parabolic geometries, which he called
almost hermitian symmetric structures. In the general parabolic
setting {\v Cap, Slov\'ak and Sou\v cek} constructed in
\cite{CapSlovakSoucekIV} the standard operators by an inductive
process. In this context our main result is a new construction of the
Bernstein Gelfand Gelfand resolution in terms of an explicit finite
power series of simple endomorphisms. The applied method permits us to
define the additional structure of a coproduct on the resolution. The
corresponding bilinear differential product on the sequence of
differential operators satisfies a Leibniz rule.

The structure of this dissertation is as follows. The first chapter
brings together dimensional analysis in physics and geometry. The
following scalars: a length scale, an electric charge, an inertial
mass, or a volume density have different distinctive
\emph{dimensions}. Therefore we will not treat all of them as real
numbers, but as elements in different one dimensional vector spaces,
see definition \ref{Density}. A choice of a \emph{unit} corresponds to
the choice of a basis vector in such a one dimensional space. The
dimensional analysis, well known to physicists, carries over to
geometry when a preferred length scale is not available. It allows to
define conformal structures invariantly, see definition
\ref{Conformalvectorspaces}, it determines the order of differential
operators and pairings, and it provides a first hint towards the
physical interpretation of a given tensor---see the examples in
section \ref{Physicaldimensionsandfundamentalconstants}. In section
\ref{Densitybundlesanddivergenceoperators} we recall the deRham
complex its wedge product and Leibniz rule. The structure of this
complex lies at the heart of this dissertation since our main result
proved in the last chapter \ref{BGGtheory} is an application and
generalization of it. The last section \ref{SectionWeylderivatives}
applies densities to differential geometry, i.e.~defines a covariant
derivative on these line bundles. These so called \emph{Weyl
derivatives} not only provide a geometric interpretation of the first
Maxwell equation, but also become fundamental in conformal geometry on
manifolds.

Chapter \ref{Elementaryconformalgeometry} defines conformal structures
on affine spaces and smooth manifolds and relates it to
electromagnetism and general relativity. It contains in section
\ref{Specialrelativityandelectromagnetism} an introduction to
Minkowski's relativistic formulation of electromagnetism emphasizing
its conformally invariant aspects such as the electromagnetic
constitutive relation in vacuum. In section
\ref{Conformalstructureonmanifolds} we describe how Weyl derivatives
of the density bundle parameterize the covariant derivatives of the
tangent bundle on conformal manifolds. A differential geometric
construction involving a choice of Weyl derivative is called
conformally invariant, if it does not depend upon that choice. In
particular for constructions involving higher derivatives, like the
curvature tensor, we present, in section
\ref{ChangingtheWeylderivative}, a convenient way to study the
dependence on the Weyl derivative. In section
\ref{DifferentialBianchiidentities} we recall the first few
integrability conditions of the conformal curvature tensor. These
Bianchi identities motivate the field equations for geometric theories
of gravity, such as Einstein's or Bach's theory.

Chapter \ref{Algebraofconformalgeometry} provides a link between the
change of Weyl derivative and the Lie algebra $\g$ of vector fields
leaving the conformal affine structure invariant. In section
\ref{Tensorialtwistorsinconformalgeometry} we identify this M\"obius
algebra as a linear Lie algebra $\g=\so(p+1,n-p+1)$. This allows to
use subspaces of the tensor algebra of $\R^{p+1,n-p+1}$ as
representations $W$ for $\g$. Elements of $W$ are called
\emph{twistors} and they induce \emph{twistor fields} on the affine
conformal space, i.e.~functions with values in the space of
\emph{coinvariants} of $W$. This inclusion, twistors $\mapsto$ twistor
fields, is the beginning of the conformal Bernstein Gelfand Gelfand
complex. In chapter \ref{Differentialconformalinvariants} we will
construct invariant differential operators annihilating these twistor
fields for $W=\Lambda^{k+1}\R^{p+1,n-p+1}$. Hence such a twistor
operator is the next map in the BGG complex.

Chapter \ref{Differentialconformalinvariants} splits into two
parts. The first three sections
\ref{Conformalinvariantsalonglightlikecurves},
\ref{Conformalinvariantsalongnonlightlikecurves} and
\ref{Weightonetwistorsandconservationlaws} deal with conformal
invariants along curves. The invariance of the lightlike acceleration
suggests a classical law of the interaction of an electromagnetic
field with a light ray. This gives a classical model for the
light-light interactions predicted by quantum electrodynamics,
although these interactions are expected to be very weak and so it
remains a great challenge to observe them in the laboratory. Motivated
by energy momentum conservation in general relativity we present in
section \ref{Weightonetwistorsandconservationlaws} a geometric
interpretation of twistor fields: they lead to conserved properties
along conformal geodesics (circles) on arbitrary conformal
manifolds. In conformal affine space this observation has the
reinterpretation that a conformal geodesic (perhaps with an additional
parallel tensor attached to it) induces a natural twistor up to
scale. The second part, sections \ref{Firstorderconformalinvariants}
and \ref{Secondorderconformalinvariants}, focuses on differential
invariants up to order two for sections of natural bundles on
conformal manifolds. The main definition is a $2$-jet operator
\ref{Secondjetoperator} in terms of a Weyl derivative. This operator
transforms under a linear change of Weyl derivative in a purely
algebraic way, hence any second order conformal invariant known from
the affine space translates into an invariant on an arbitrary
conformal manifold. This allows to characterize linear and bilinear
conformal invariants algebraically. We will give some general examples
of operators and pairings which were motivated from the study of
linear field theories for gravity, which are special cases of BGG
complexes including their wedge product structure.

Chapter \ref{Conformallinearfieldtheories} begins with a general
analysis of the ingredients and principles a linear field theory like
electromagnetism should follow. In sections
\ref{Chargesinfieldtheories} and \ref{LienardWiechertfields} we
demonstrate how differential pairings and Leibniz rules support
Penrose's idea to view twistors as generalized charges. In section
\ref{Generalfieldtheory} we use our main result (which we will prove
in chapter \ref{BGGtheory}) to define the notion of a conformally
invariant linear field theory. In the following three sections
\ref{Fierzstheoryofgravity}, \ref{SectionBachstheoryofgravity} and
\ref{Penrosestheoryofgravity} we investigate three particular field
theories in connection with linear gravity. We will calculate the
relevant pairings here in terms of a Weyl derivative. This happens in
advance of the main result to motivate and to engender a better
understanding of pairings.

Chapter \ref{BGGtheory} develops the Bernstein Gelfand Gelfand
resolution in the context of homogeneous parabolic geometry. In
section \ref{Projectivelightcone} we already observed that the
conformal sphere is an example of parabolic geometry. In the first
section \ref{Vermamodulesandtheholonomicjetoperator} we define a jet
operator encoding the information of all derivatives of a section of a
homogeneous bundle at a point into a linear form on an induced module,
also called a \emph{Verma module}. Invariant linear differential
operators and bilinear pairings are then in one to one correspondence
with homomorphisms and coproducts between Verma modules. In section
\ref{Homogeneousparabolicgeometry} we fix the notion of a parabolic
subalgebra in the way we will use it in the next section
\ref{EilenbergsrelativeLiealgebrahomology} on finite dimensional
relative Lie algebra homology. The operators of the BGG complex act
between sections of homogeneous bundles associated to these relative
Lie algebra homology spaces. In section \ref{SectionTwisteddeRhamresolution}
we rewrite the deRham complex in terms of homomorphisms between Verma
modules. The key observations are made in section
\ref{ProjectiononVermacochains} where we present a variation of
Kostant's Hodge theory of Lie algebra cohomology. This allows to
define a projection $S$ on the Verma modules induced by the cochains
which projects onto a submodule isomorphic to the Verma module induced
by the cohomology. This projection $S$ translates the deRham
homomorphism to the BGG homomorphisms and the wedge coproduct to a BGG
coproduct.

\section*{Acknowledgments}\noindent
First of all I would like to thank my advisor Hermann Karcher for his
help and support, his constant encouragement and his inspiring insight
in relativity. It is also a great pleasure for me to thank David
Calderbank with whom I have been discussing conformal geometry and
linear field theories for several years. My discussions with him have
greatly influenced the approach to differential conformal geometry
taken here and it was him who conjectured a special case of my main
result two years ago. I am equally indebted to Gregor Weingart who
over the last two years explained various aspects of representation
theory to me, in particular Eilenberg's relative Lie algebra homology
and the original approach to the Bernstein Gelfand Gelfand
resolution. I have also benefitted from his understanding of geometric
structures and invariant operators. I would also like to thank Jens
Franke who gave me the opportunity to talk and discuss in his algebra
seminar and Andreas Kewenig for useful information and references on
algebraic matters.

I am very grateful to the Deutsche Forschungsgemeinschaft,
Sonderforschungsbereich 256, ``Nichtlineare partielle
Differentialgleichungen'', for providing financial support throughout
the duration of this research and also to the Mathematisches Institut
Bonn for a stimulating environment in which to carry out my work. This
manuscript was typeset in LaTeX2$\eps$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Densities and dimensional analysis}
\label{Densitiesanddimensionalanalysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagenumbering{arabic}
%
This chapter, although elementary and simple, illuminates the role
played by scalar densities: we will bring together scale invariance in
analysis, physics and geometry. To each number $w$ there is a one
dimensional vector space $L^w$ of elements with geometric dimension
$\Dim{length}^{w}$. These spaces will allow us to define conformal
metrics intrinsically (see definition \ref{Conformalvectorspace}) and
to specify domain and target of conformally invariant differential
operators properly (see chapter \ref{Differentialconformalinvariants}
on differential conformal invariants). Here densities enable us to
develop a simple geometric dimensional analysis of arbitrary
tensors. One novel feature is a discussion of the exemplary
correspondence between physical dimensions and geometric dimensions:
scalars of some particular physical dimension will be identified with
elements in a $L^w$. We will indicate how the Sobolev number in
functional analysis also relates to $L^w$. Volume densities $L^{-n}$
allow to treat integration on manifolds globally without orientability
assumptions. Particular applications are natural adjoints of
differential operators, distributions and integral theorems, see
paragraphs \ref{Adjoints}, \ref{Distributions} and
\ref{Divergencetheorem}. Adjoints, distributions and integral theorems
clarify fundamental constructions in physical field theories such as
Maxwell's equations \ref{SummaryElectromagneticfieldtheory} and
sources from pointlike charges, see paragraph
\ref{Chargedparticlesassources}. We clarify the distinction between
integration over oriented and cooriented submanifolds in section
\ref{Orientationsandintegrationoversubmanifolds}, which allows to
distinguish between electric and magnetic charge, see definitions
\ref{Electriccharge} and \ref{Magneticcharge}. We will recall the
divergence theorem \ref{Divergencetheorem} on manifolds with
boundaries since in full generality it is less widely appreciated than
the Stokes' theorem, but is more natural since it avoids orientability
assumptions. On manifolds the density bundles can be equipped with a
Weyl derivative which illustrates the differential geometric aspect of
densities and will be pursued later in the context of conformal
geometry. We view the mentioned concepts of densities, dimensional
analysis and Weyl derivatives as being as fundamental as tangent
vectors or linear connections.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Linear algebra of densities}
\label{Linearalgebraofdensities}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We begin this chapter with the linear algebra of densities and define
a simple geometric dimensional analysis for tensors. From a
representation theoretic view point the geometric dimension is
characterized by the weight of the tensor representation restricted to
the centre of the general linear group. The following explicit
definition is taken from \cite{CalderbankFaraday}:

\begin{definition}\label{Density}
Let $V$ be a real $n$-dimensional vector space and $w\in\R$ a real
number. A homogeneous map $\mu \from \Lambda^nV \minuszero \to \R$
with the property $\mu(\lambda\omega) = |\lambda|^{- w/n} \mu(\omega)$
for all $\lambda \in \R\minuszero$ and $\omega\in\Lambda^nV\minuszero$
is called a \emph{density of weight $w$}.
\end{definition}

The set of all densities of weight $w$ forms a one dimensional vector
space $L^w(V)$ or simply $L^w$. This vector space is oriented, since a
nontrivial density takes either positive or negative values. The
various vector spaces of densities satisfy canonical isomorphisms:
$L^{w_1} \tens L^{w_2} = L^{w_1+w_2}$, ${L^w}^\dual=L^{-w}$ and
$L^0=\R$.

\begin{remark}\label{densityasrep}
The vector space $L^w$ can equally well be characterized as the
representation $\GL(V) \to \GL(L^w) = \R\minuszero$ given by $A
\mapsto |\deter A|^{w/n}$. Dilations (positive multiples of the
identity) $A=\lambda \id$ act by multiplication with $\lambda^w$ on
$L^w$. The multiples of the identity form the centre of $\GL(V)$ and
$w$ is also called the \emph{central weight} of $L^w$.
\end{remark}

Elements of $L^1$ may be thought of as scalars with dimensions of
$\Dim{length}$. General tensors of $V$, i.e.~elements of $L^k \tens
V^i \tens {V^\dual}^j$ will be said to have central weight $w:=k+i-j$
or dimensions of $\Dim{length}^{k+i-j}$, since dilations $\lambda\id$
naturally act upon them by multiplication with $\lambda^{k+i-j}$
(hence $w=k+i-j$ is sometimes called the \emph{degree of
homogeneity}). The central weight of the tensor product of two tensors
is given by the sum of individual central weights.

\begin{example}\label{pseudoscalars}
Real numbers $\R=L^0$ are weightless and dimensionless.  Vectors of
$V$ have central weight $+1$ and describe translations of dimensions
$\Dim{length}$. A positive density of weight $-n$ assigns naturally a
real number as volume to an $n$-dimensional parallelepiped and has
dimension $\Dim{length}^{-n}$. The tensor product $L^{-n} \tens
\Lambda^nV$ is the weightless space of \emph{pseudoscalars}. This one
dimensional space naturally carries a norm given by $|\mu \tens
\omega|:=|\mu(\omega)|$.  The two orientations of $V$ are in one to
one correspondence with the two unit elements of $L^{-n} \tens
\Lambda^nV$.
\end{example}

\begin{remark}
This definition of weight is in agreement with Gauduchon's convention
in \cite{Gauduchon}. In the fundamental articles by Fegan \cite{Fegan}
and Hitchin \cite{HitchinLinear} the definition of weight differ from
ours by a minus sign. However, in parts of the literature the
definition of the weight is very different and those definitions of
weight do not allow the satisfactory dimensional analysis as discussed
in the next section. Eastwood in \cite{EastwoodNotes} assigns $k$ as
weight to elements in $L^k \tens V^i \tens {V^\dual}^j$. Similarly
Bergmann \cite{Bergmann}, Stephani \cite{Stephani} and Weyl
\cite{WeylRZM} prefer to assign $-k/n$ as weight for $L^k \tens V^i
\tens {V^\dual}^j$.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Physical dimensions and fundamental constants}
\label{Physicaldimensionsandfundamentalconstants}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Physical quantities are measurable aspects of physical objects. The
value of a scalar quantity is the product of a \emph{real number} and
a chosen \emph{unit} (e.g.~a ruler has length
$30\mathrm{cm}$). Independent of the unit is the \emph{dimension}
characterizing the type of the scalar quantity in question. From a
basic set of dimensions like $\Dim{length}$, $\Dim{time}$,
$\Dim{inertial\ mass}$, $\Dim{electric\ charge}$, $\Dim{gravitational\
mass}$ and $\Dim{temperature}$, one can generate derived dimensions as
commutative products and powers of these basic dimensions like
$\Dim{acceleration} := \Dim{length}/\Dim{time}^2$.

Mathematically, different one dimensional real vector spaces can
represent different basic dimensions. Scalar quantities of a given
(derived) dimension are elements of the appropriate tensor product of
these one dimensional vector spaces (and their duals). The choice of a
basis in such a one dimensional space corresponds to a choice of a
unit of that dimension and this is called a \emph{gauge fixing}. The
one dimensional vector spaces $L^w$ from definition \ref{Density} are
geometric examples of spaces carrying scalars of dimension
$\Dim{length}^w$.

For theoretical purposes one can reduce the number of basic dimensions
by one, if one postulates that a certain proportionality factor which
appears in a (fundamental) law of nature is a (fundamental) constant
of nature. The first example is Newton's law of motion, which says
that the force (e.g.~of a spring) acting on a body is proportional to
the inertial mass of the body times its acceleration: $F=\lambda
ma$. The proportionality factor $\lambda$ is universal. This factor is
always set to unity, which turns Newton's law in a combination of a
law of motion and a definition for the dimension of force (see
\cite{Parrot} p.~239) $\Dim{force} = \Dim{inertial\
mass}\Dim{acceleration}$.

Further examples of these laws are Einstein's relation between energy
and inertial mass of massive particles $E=m\light^2$, Planck's
relation between energy and frequency of massless particles
$E=\hbar\omega$, Coulomb's force law between two electric charges
$F=\coulomb{q_1q_2 \over r^2}$, Newton's force law between two 
gravitational masses $F=\newton {m_1m_2\over r^2}$ and Boltzmann's law 
between thermical energy per degree of freedom and temperature of a 
molecule $E=\oneover{2}\boltzmann T$. The dimensions of the mentioned 
constants of nature are as follows:
\begin{eqnarray*}
\hbox {Velocity of light: }
 \Dimof{\light} &=& 
 \Dim{length}/\Dim{time},\\ 
\hbox{Planck's constant: } 
 \Dimof{\hbar} &=& 
 \Dim{energy}\Dim{time} =\Dim{force}\Dim{length}\Dim{time},\\
\hbox{Coulomb's constant: } 
 \Dimof{1/(\oneovercoulomb)} &=& 
 \Dim{force}\Dim{length}^2/\Dim{electric\ charge}^2,\\ 
\hbox{Newton's constant: } 
 \Dimof{\newton} &=&
 \Dim{force}\Dim{length}^2/\Dim{gravitational\ mass}^2,\\
\hbox{Boltzmann's constant: } 
 \Dimof{\boltzmann} &=&
 \Dim{energy}/\Dim{temperature}.
\end{eqnarray*}
One way to deal with fundamental constants is to choose appropriate 
\emph{units} in all basic dimensions such that these constants have the 
\emph{value} $1$. Having done this, all constants disappear in formulas. 
We don't want to choose units in basic dimensions here. Instead we will 
use fundamental constants as factors in front of physical quantities such 
that the product has a dimension which is simply a power of 
$\Dim{length}$:

\begin{example}\label{Kinematicdimensions}
\summary{Kinematic dimensions} Using $\light$ one reduces the basic
\emph{kinematic} dimensions $\Dim{length}$ and $\Dim{time}$ to powers
of $\Dim{length}$,
e.g.~$\Dim{frequency}=\Dimof{\light^{-1}}\Dim{length}^{-1}$.
\end{example}

\begin{example}\label{Mechanicaldimensions}
\summary{Mechanical dimensions} Using $\light$ and $\hbar$ one reduces
the basic mechanical dimensions $\Dim{length}$, $\Dim{time}$ and
$\Dim{inertial\ mass}$ to powers of $\Dim{length}$: note,
$\Dimof{\hbar\over\light} =\Dim{length}\Dim{inertial\ mass}$ and
$\Dimof{\hbar} =\Dim{length}\Dim{linear\ momentum}$. In classical
mechanics the linear momentum is an element of the dual tangent space
of the configuration space, hence has geometric dimension
$\Dim{length}^{-1}$ in agreement with this choice of fundamental
constants.
\end{example}

\begin{example}\label{Electromagneticdimensions}
\summary{Electromagnetic dimensions} In the same way one reduces the
basic electromagnetic dimensions $\Dim{length}$, $\Dim{time}$,
$\Dim{inertial\ mass}$ and $\Dim{electric\ charge}$ using $\light$,
$\hbar$ and $\coulomb$ to powers of $\Dim{length}$. Notice that
$\Dimof{\hbar\light} =\Dim{force}\Dim{length}^2$ and hence there is a
\emph{natural unit} for electric charges since $\Dim{electric\
charge}^2 ={\hbar\light\oneovercoulomb}$ and electric charge $q$
becomes dimensionless (i.e.~a real number) when divided by
$\sqrt{\hbar\light\oneovercoulomb}$.

In electrostatics there are two basic quantities. The \emph{electric
field strength} $E$ has dimension $\Dimof{E} =\Dim{force}
/\Dim{electric\ charge}$. The \emph{dielectric displacement} $D$ has
dimension $\Dimof{D} = \Dim{electric\ charge}/\Dim{area}$. Notice,
$$ \Dimof{\sqrt{{\oneovercoulomb\over\hbar\light}}\, E } 
   = \Dimof{ \sqrt{1 \over {\oneovercoulomb\hbar\light}}\, D} 
   = \Dim{length}^{-2}. $$

In magnetostatics there are also two basic quantities. It is
convenient to introduce the dimensions $\Dim{current} :=\Dim{electric\,
charge}/\Dim{time}$ and $\Dim{voltage} :=\Dim{energy}/\Dim{electric\,
charge}$. The \emph{magnetic flux} $B$ has dimension $\Dimof{B}
={\Dim{voltage}\Dim{time}/\Dim{area}}$ and the \emph{magnetic loop
tension} $H$ has dimension $\Dimof{H}
=\Dim{current}/\Dim{length}$. Notice once more,
$$ \Dimof{\sqrt{{\oneovercoulomb\light\over\hbar}}\, B } 
   = \Dimof{ \sqrt{1 \over {\oneovercoulomb\hbar\light^3}}\, H} 
   = \Dim{length}^{-2}. $$

In relativistic electromagnetism, the electric field $E$ and the
magnetic flux $B$ are summarized into a $2$-form $F$, the so called
\emph{Faraday form} $F$. The first Maxwell equation $\dR F=0$ is
expressed in terms of the natural exterior derivative see paragraphs
\ref{Exteriorderivative} and \ref{Electromagneticfieldtheory}. The
Faraday form therefore carries the geometric dimension
$\Dim{length}^{-2}$. Similarly, the dielectric displacement $D$ and
the magnetic loop tension $H$ are summarized into a bivector density
(denoted by $G$). The second Maxwell equation is expressed in terms of
the natural exterior divergence see paragraphs \ref{Exteriordivergence}
and \ref{Electromagneticfieldtheory}. The bivector density $G$
therefore carries the geometric dimension $\Dim{length}^{2-n}$, where
$n$ is the spacetime dimension. Hence with the above choice of
fundamental constants and in $n=4$ spacetime dimensions, the natural
geometric dimensions and the physical dimensions are in full
correspondence.
\end{example}

\begin{example}\label{Gravitationaldimensions}
\summary{Gravitational dimensions}
Analogously, using $\light$, $\hbar$ and $\newton$ one reduces the
basic gravitational dimensions $\Dim{length}$, $\Dim{time}$,
$\Dim{inertial\ mass}$ and $\Dim{gravitational\ mass}$ to powers of
$\Dim{length}$. A gravitational mass becomes dimensionless, when
divided by $\sqrt{\hbar\light/\newton}$.
\end{example}

\begin{example}\label{thermodynamicaldimensions}
\summary{Thermodynamical dimensions} Using $\light$, $\hbar$ and
$\boltzmann$ one reduces the basic thermodynamical dimensions
$\Dim{length}$, $\Dim{time}$, $\Dim{inertial\ mass}$ and
$\Dim{temperature}$ to powers of $\Dim{length}$.
\end{example}

Fundamental constants as factors allow scalars of any physical
dimension to be viewed as elements in one of the geometrically defined
spaces $L^w$ of the previous chapter. Similarly, a tensorial physical
quantity is mathematically described by an element in the
corresponding tensor space of the central weight according to its dimension.

\begin{remark}
The elementary electric charge $\electron$ of the electron defines
another natural unit of dimension $\Dim{electric\ charge}$. Hence the
quotient ${\electron^2/(\hbar\light\oneovercoulomb)}$ determines a
(dimensionless) real number, the so called \emph{electric fine
structure constant}. A gravitational analogue to an elementary
electric charge has not been observed. We will not use the elementary
electric charge of the electron as a unit.
\end{remark}

\begin{remark}
The equivalence principle formulated as the proportionality between
inertial and gravitational mass of a massive particle leads to another
constant which one could call \emph{Galilei's constant} $\galilei$:
$$\hbox{Galilei's constant: }
   \Dimof{\galilei}
   =\Dim{gravitational\ mass}/\Dim{inertial\ mass}.
$$
For historical reasons, this constant is always set to unity (the
common units for inertial mass are also used for gravitational
mass). Using Galilei's constant $\galilei$ together with $\light$,
$\hbar$ and $\newton$ Planck found natural units for all dimensions
involved, in particular $\Dimof{\galilei^2{\hbar\newton{\light^{-3}}}}
= \Dim{length}^2$.  We will not make use of the equivalence principle
in that way, hence we will not use Galilei's constant and we will not
use the resulting units of Planck. (We believe that Galilei's constant
is not fundamental, instead it we would like to view it as a
consequence of a proper understanding of Mach's principle.)
\end{remark}

\begin{remark}\label{Einsteinconstants} 
In Einstein's theory of gravity one usually uses $\light$, $\newton$
and Galilei's constant $\galilei$ from above without Planck's constant
$\hbar$ to reduce the basic gravitational dimensions to powers of
$\Dim{length}$ (see \cite{MisnerWheeler}, p. 596). With this choice of
constants we have $\Dimof{{\galilei\newton}{\light^{-2}}}
=\Dim{length}/\Dim{gravitational\ mass}$ instead of a natural unit for
gravitational mass. Taking also Coulomb's electric constant
$1/(\oneovercoulomb)$ into account, one finds for electric charge in
analogy to gravitational mass
$\Dimof{{\galilei\light^{-2}}\sqrt{\newton/(\oneovercoulomb)}}
=\Dim{length}/\Dim{electric\ charge}$. Note that in this case the
electric field strength has dimension
$\Dimof{\galilei\light^{-2}\sqrt{\newton\oneovercoulomb}E}
=\Dim{length}^{-1}$, so that we would lose the correspondence between
physical dimensions and natural geometric dimensions explained in
paragraph \ref{Electromagneticdimensions}.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Density bundles and divergence operators}
\label{Densitybundlesanddivergenceoperators}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Applying the construction in definition \ref{Density} of densities of
weight $w$ pointwise to the tangent space of a $n$-dimensional
manifold $M$ leads to a family of natural real line bundles
$L^w=L^w(TM)$ over $M$.  Alternatively one could use the (first order)
frame bundle $\GL(M)$ of $M$ and the associated bundle construction
for the representation $|\deter{}|^{w/n}$ of $\GL(\R^n)$ to obtain:
$L^w(TM)=\GL(M) \cross_{|\deter{}|^{w/n}} \R$. These line bundles are
oriented and hence trivializable, but except for $w=0$ do not carry a
natural trivialization. The sections of $L^{-n}$ over $M$ play the
role of natural integrands. We will follow Calderbank
\cite{CalderbankClifford} to define natural adjoints of differential
operators and distributions. We apply this definition to the deRham
complex to obtain the complex of exterior divergences.

\begin{discussion}\label{Discussionintegration}
\summary{Integration} We fix a notion of integration of smooth
functions defined on a unit cube $[0,1]^n$ in Euclidean $n$-space
which satisfies linearity and the transformation formula under
diffeomorphisms of the cube to itself. If $F\from [0,1]^n \to M$ is
smooth and $\mu\in C^\infty(M,L^{-n})$ is a density of weight $-n$ then
$x\in[0,1]^n$ and $(F^\dual\mu)_x := \mu_{F(x)}(\del
F_x(.)\wedge\ldots \wedge\del F_x(.))$ define a density on $\R^n$. We
say a real valued $\R$-linear functional $\int_M\from
C^\infty_o(M,L^{-n}) \to \R$ from densities of weight $-n$ with
compact support \emph{locally agrees with the Euclidean integral} if
for any $F\from [0,1]^n \to M$ and any $\mu\in C^\infty_o(M,L^{-n})$
with $\support(\mu)\subset\image(F)$ we have:
$$
\int_M \mu = \int_{[0,1]^n} F^\dual\mu(e_1\wedge\ldots\wedge e_n),
$$
where $e_i$ is the standard basis in $\R^n$. A partition of unity
argument shows existence:
\end{discussion}

\begin{proposition}\label{integration}
There is a linear functional $\int_M \from C^\infty_o(M,L^{-n}) \to
\R$ which locally agrees with the Euclidean integral and which
therefore is unique and invariant under diffeomorphisms of $M$.
\end{proposition}

\begin{discussion}\label{Divergenceofvectorfields}
\summary{Divergence of vector fields} Let $X\in C^\infty(M,TM)$ be a
vector field and $\mu\in C^\infty(M,L^{-n})$ a positive density of
weight $-n$. The Lie derivative $\Lie_X\mu$ is defined by
differentiating the pull back of $\mu$ by the local flow of $X$. The
\emph{divergence of $X$ with respect to $\mu$} is a function on $M$
defined by $(\divR_\mu X)\mu = \Lie_X\mu$. The divergence can be
interpreted as the rate of volume expansion of the flow of $X$ with
respect to $\mu$. Notice that for any function $f\in C^\infty(M,\R)$
we have $\Lie_{(fX)}\mu = \Lie_X(f\mu)$, such that the above formula
defines a linear first order differential operator on vector field
densities like $\mu\tens X$:
$$
\divR\from C^\infty(M,L^{-n}\tens TM)\to C^\infty(M,L^{-n}).
$$
\end{discussion}

A vector field $X\in C^\infty_o(M,TM)$ with compact support on the
interior of $M$ has a complete flow. The diffeomorphism-invariance of
the integral leads to the following vanishing result:

\begin{proposition}\label{Integrationbyparts}
\summary{Integration by parts} If $X\in C^\infty_o(M,TM)$ has compact
support and $\mu\in C^\infty(M,L^{-n})$ is positive then $\int_M
\divR(\mu\tens X) = 0$.
\end{proposition}

\begin{definitions}\label{Jetsdifferentialoperatorsandpairings}
\summary{Jets, differential operators and pairings} Let $EM\to M$ be a
vector bundle. At a point $x\in M$ we define for any $k\in\N$ the
vector space of sections which vanish up to order $k$ at $x$ to be
$Z^k_x(EM):=\setof{e\in C^\infty(M,EM)\property e(x)=0,
de(X)=0,\ldots,d^ke(x)=0}$. The projection $\jet^k(e)_x$ of a section
$e\in C^\infty(M,EM)$ onto the quotient $\Jet^k(EM)_x:=
C^\infty(M,EM)/Z^k_x(EM)$ is called the \emph{$k$-jet of $e$ at
$x$}. The vector bundle $\Jet^k(EM)\to M$ is called the $k$-jet bundle
of $EM$. We have a bundle inclusion $\Sym^kT^\dual \tens EM\to
\Jet^k(EM)$ defined at $x\in M$ by a function $f\in C^\infty(M,\R)$
with $f(x)=0$ as $(\del f)^k\tens e \mapsto \jet^k(f^ke)$. This is the
beginning of a short exact sequence of bundle maps:
$$
0\to \Sym^kT^\dual \tens EM\to \Jet^k(EM)\to \Jet^{k-1}(EM)\to 0.
$$
Let $FM\to M$ denote another vector bundle. A \emph{linear
differential operator of order $k$} is a $\R$-linear map
$$
\Nabla\from C^\infty(M,EM)\to C^\infty(M,FM)
$$
which factors through a bundle map $\pi\from \Jet^k(EM)\to FM$ as
$\Nabla=\pi\circ\jet^k$ with nonzero \emph{symbol} defined to be the
composite: $\Sym^kT^\dual\tens EM\to \Jet^k(EM)\to FM$. The
\emph{universal $k$-jet operator} $\jet^k\from C^\infty(M,EM)\to
C^\infty(M,\Jet^k(EM))$ is a particular example. Similarly we call a
$\R$-bilinear map
$$
X\from C^\infty(M,E_1M)\cross C^\infty(M,E_2M)\to
C^\infty(M,FM)
$$
which factors through a bundle map
$Q\from\Jet^k(E_1M)\tens\Jet^k(E_2M)\to FM$ as
$X(e_1,e_2)=Q(\jet^k(e_1)\tens\jet^k(e_2))$ a \emph{bilinear
differential pairing of order $k$}.
\end{definitions}

\begin{discussion}\label{Adjoints}
\summary{Adjoints} Let $EM\to M$ and $FM\to M$ be two vector bundles
over $M$ and $\Nabla\from C^\infty(M,EM)\to C^\infty(M,FM)$ be a
linear differential operator. A differential operator
$\Nabla^\dual\from C^\infty(M, L^{-n}\tens F^\dual M)\to
C^\infty(M,L^{-n}\tens E^\dual M)$ is called \emph{adjoint} to
$\Nabla$, if for all sections with compact support in the interior of
$M$ we have $\int_M\ip{\Nabla e,\phi} =
\int_M\ip{e,\Nabla^\dual\phi}$. If $\Nabla^\dual$ exists, it is
unique. One way to proof adjointness of two given operators $\Nabla$
and $\Nabla^\dual$ is to construct a \emph{bilinear differential
pairing}
$$
X_{\Nabla} \from C^\infty(M,EM) \cross C^\infty(M,L^{-n}\tens F^\dual M) \to
C^\infty(M,L^{-n}\tens TM),
$$
such that a divergence formula holds:
$$
\divR (X_{\Nabla}(e,\phi)) = \ip{\Nabla e,\phi} - \ip{e,\Nabla^\dual\phi}.
$$
The differential order of the pairing is one less then the order of the 
operator. In particular for first order operators the pairing is 
tensorial.
\end{discussion}

\begin{definition}\label{Covariantderivative}
\summary{Covariant derivative} A linear first order operator $D\from
C^\infty(M,EM)\to C^\infty(M,T^\dual\tens EM)$ with symbol given by
the identity on $T^\dual\tens EM$ is called \emph{covariant derivative
on $EM$}.
\end{definition}

Associated to a covariant derivative $D$ is the \emph{Riemann
curvature tensor $R^D$}. It defines the local obstruction against the
existence of parallel sections:

\begin{proposition}\label{Curvature}
\summary{Curvature} Let $D$ be a covariant derivative of a vector
bundle $EM$. The following trilinear differential pairing $R^D\from
C^\infty(M,TM)\tens C^\infty(M,TM)\tens C^\infty(M,EM)\to
C^\infty(M,EM)$ defined by $R^D(X,Y)e
:=D_X(D_Ye)-D_Y(D_Xe)-D_{\Lieb{X,Y}}e$ (where $X$, $Y$ are vector
fields, $e$ a section and the Lie bracket on functions is defined by
$\del_{\Lieb{X,Y}} = \del_X\del_Y-\del_Y\del_X$) is indeed zero order
and defines the \emph{curvature} of $D$ to be an endomorphism valued
$2$-form $R^D\in C^\infty(M,\Lambda^2T^\dual\tens E^\dual\tens EM)$.
\end{proposition}

\begin{remark}
A covariant derivative on $EM$ induces a covariant derivative on the
dual bundle $E^\dual M$ by the product rule: $\del_X(\ip{\eta,e})=
\ip{D_X\eta,e} + \ip{\eta,D_Xe}$, with a vector field $X$ and sections
$e\in C^\infty(M,EM)$ and $\eta\in C^\infty(M,E^\dual M)$. Consider
the tensorial pairing $EM \tens (L^{-n}\tens T\tens E^\dual M) \to
L^{-n}\tens TM$ with $e\tens Y\tens\eta\mapsto Y\ip{e,\eta}$. It
satisfies the divergence formula:
$$
\divR( \ip{e,(Y\tens\eta)} )
= \ip{D_Ye,\eta} + \ip{e,D_Y\eta} + \ip{e,\eta}\divR(Y),
$$
with $Y\in C^\infty(M,L^{-n}\tens TM)$. This shows that a covariant
derivative has an adjoint $D^\dual\from C^\infty(M,L^{-n}\tens T\tens
E^\dual M)\to C^\infty(M,L^{-n}\tens E^\dual M)$ given by
$D^\dual(Y\tens\eta)=-D_Y\eta - \divR(Y)\tens\eta$.
\end{remark}

\begin{discussion}\label{Distributions}
\summary{Distributions} Let $M$ be a $n$-dimensional manifold and
$EM\to M$ a vector bundle. The space of \emph{test functions}
$C^\infty_o(M,L^{- n}\tens E^\dual M)$ consists of smooth sections
with compact support in the interior of $M$. The space of sections of
the $k$-jet bundle $C^\infty_o(M,\Jet^k(L^{-n}\tens E^\dual M))$ is
equipped with the \emph{compact open topology} and this pulls back
under $\jet^k\from C^\infty_o(M,L^{-n}\tens E^\dual M)\to
C^\infty(M,\Jet^k(L^{-n}\tens E^\dual M))$ to induce the
\emph{$C^k$-topology} on $C^\infty_o(M,L^{-n}\tens E^\dual M)$. A
\emph{distributional section} of $EM$ is a linear map
$C^\infty_o(M,L^{-n}\tens E^\dual M)\to\R$ which is continuous with
respect to the $C^k$-topology for all $k\in\N$. The set of all such
distributions is denoted by ${\cal D}(M,EM) :=
C^\infty_o(M,L^{-n}\tens E^\dual M)^\dual$. Locally integrable
sections of $EM$, i.e.~elements of
$\Lebesque^1_{\mathrm{local}}(M,EM)$ define distributions via
pointwise contraction and integration over $M$.

Let $\Nabla\from C^\infty(M,EM)\to C^\infty(M,FM)$ be a linear
differential operator. Its \emph{transpose} in terms of distributions
always exists: $\Nabla^{\mathrm{trans}}\from {\cal D}(M,L^{-n}\tens
F^\dual M)\to {\cal D}(M,L^{-n}\tens E^\dual M)$. If the restriction
of $\Nabla^{\mathrm{trans}}$ to smooth sections gives smooth sections,
then this restriction is the adjoint $\Nabla^\dual\from
C^\infty(M,L^{-n}\tens F^\dual M)\to C^\infty(M,L^{-n}\tens E^\dual
M)$. If this adjoint exists the operator $\Nabla$ can be extended to
distributional sections $\Nabla\from {\cal D}(M,EM)\to {\cal D}(M,FM)$
via $\ipp{\Nabla e,\phi}=\ipp{e,\Nabla^\dual\phi}$ with a distribution
$e\in {\cal D}(M,EM)$ and a test function $\phi\in
C^\infty_o(M,L^{-n}\tens F^\dual M)$.

Since a covariant derivative $D$ of $EM$ always has an adjoint, one can 
say that distribution can infinitely often be differentiated.
\end{discussion}

\begin{discussion}\label{Exteriorderivative}
\summary{Exterior derivative} On any manifold the exterior derivative
on alternating forms 
$$
\dR \from C^\infty(M,\Lambda^kT^\dual M)
  \to  C^\infty(M,\Lambda^{k+1}T^\dual M)
$$
defines a linear first order differential operator. The sequence of
exterior derivatives builds the \emph{deRham complex}:
$\dR \circ\dR=0$. The resulting cohomology is called \emph{deRham
cohomology}. The deRham complex is \emph{locally exact} in the sense
that it becomes exact when restricted to a contractible open subset of
$M$. The wedge product is a zero order bilinear pairing:
$$
\wedge\from
C^\infty(M,\Lambda^kT^\dual M)\cross C^\infty(M,\Lambda^lT^\dual M)
\to  C^\infty(M,\Lambda^{k+l}T^\dual M),
$$
subject to the following \emph{Leibniz rule}:
$$
\dR (\alpha\wedge\beta)
= \dR \alpha \wedge \beta + (-1)^k \alpha\wedge \dR \beta.
$$
Hence the wedge product induces a \emph{cup product} in the deRham 
cohomology.
\end{discussion}

For a vector space $V$ we have a $\GL(V)$-equivariant pairing
$\ip{{\,},}\from \Lambda^kV^\dual\tens\Lambda^kV\to\R$. We define
\emph{interior multiplication by forms} to be a linear map
$\inner\from \Lambda^kV^\dual\tens \Lambda^{k+l}V\to \Lambda^l V$
which is adjoint to the wedge product in the sense $ \ip{\alpha \inner
a,\beta} =\ip{a,\alpha\wedge\beta} $, with $\alpha\in
\Lambda^kV^\dual$, $a\in \Lambda^{k+l}V$ and $\beta\in\Lambda^l
V$. Similarly \emph{interior multiplication by multivectors} is a
linear map $\inner\from \Lambda^kV\tens \Lambda^{k+l}V^\dual\to
\Lambda^l V^\dual$. On a $n$-dimensional manifold $M$ the bundle of
alternating forms can be tensored with the bundle of pseudoscalars
$L^{-n}\tens\Lambda^nTM$ and the above interior multiplication
provides an isomorphism
\begin{eqnarray*}
\Lambda^kT^\dual\tens(L^{-n}\tens\Lambda^n T)
&\stackrel{\isomorph}{\longrightarrow}&
L^{-n}\tens\Lambda^{n-k} T\\
\alpha\tens\orie&\mapsto& \alpha\inner \orie
\end{eqnarray*}
to obtain the bundle of multivector densities. Likewise we can tensor
the bundle of multivector densities by the (dual) pseudoscalars
$L^n\tens\Lambda^nT^\dual M$ to recover the bundle of alternating forms:
\begin{eqnarray*}
L^{-n}\tens \Lambda^kT \tens(L^{n}\tens\Lambda^n T^\dual)
&\stackrel{\isomorph}{\longrightarrow}&
\Lambda^{n-k} T^\dual\\
a\tens\orie^\dual&\mapsto& a\inner \orie^\dual.
\end{eqnarray*}

\begin{discussion}\label{Exteriordivergence}
\summary{Exterior divergence} The bundle of dual pseudoscalar
$L^{n}\tens\Lambda^nT^\dual M$ over $M$ has rank one, comes with a
natural norm and hence with local sections $\orie^\dual$ which are
parallel. The \emph{exterior divergence} on multivector densities
$$
\divR\from C^\infty(M,L^{-n}\tens \Lambda^{k}TM)
    \to C^\infty(M,L^{-n}\tens\Lambda^{k-1} TM),
$$
is then defined in terms of the exterior derivative by
$$
(\divR a)\inner\orie^\dual := (-1)^{k+1} \dR(a\inner\orie^\dual).
$$
The sequence of divergences defines a complex $\divR\circ\divR=0$. The
resulting homology is called \emph{deRham homology}. Interior
multiplication $\inner$ is a zero order pairing
$$
\inner \from
C^\infty(M,\Lambda^kT^\dual M)
\cross C^\infty(M,L^{-n}\tens\Lambda^{k+l}TM)
\to  C^\infty(M,L^{-n}\Lambda^lTM).
$$
For $\alpha\in C^\infty(M,\Lambda^kT^\dual M)$ and $a\in
C^\infty(M,L^{-n}\tens \Lambda^{k+l}TM)$ we get from the Leibniz rule
the following divergence formula:
$$
(-1)^k \divR (\alpha\inner a) ={\dR \alpha}\inner a + \alpha \inner \divR a.
$$
For that note $(\alpha\inner a)\inner\orie^\dual
=(-1)^{k(k+l+1)}\alpha\wedge(a\inner\orie^\dual)$. Hence the interior
multiplication defines a \emph{cap product} between deRham cohomology
and homology. For $l=1$ the above divergence formula shows that $\dR$
and $-\divR$ are adjoints. 
\end{discussion}

\begin{remark}
For $k=0$ we recover the divergence on vector fields from paragraph
\ref{Divergenceofvectorfields}, since for functions $f\in
C^\infty(M,\R)$ and vector fields $X\in C^\infty(M,L^{- n}\tens TM)$
we clearly have the divergence formula $\divR(fX) = \ip{\dR f, X} + f
\divR X$.
\end{remark}

\begin{remark}
Let $D$ be a torsion-free covariant derivative of $TM$ (see definition
\ref{Covariantderivative} and proposition \ref{Torsion}). (Local
coordinates induce torsion-free derivatives.)  With a dual basis
$t_i$, $\theta^i$ of $TM$ we like the exterior derivative on a form
$\alpha$ to be $\dR \alpha=\Sum_i \theta^i\wedge D_{t_i}\alpha$. With
the above sign convention we obtain for a multivector $a$ applied to
the divergence $\divR a=\Sum_i \theta^i\inner D_{t_i}a$.
\end{remark}

\begin{remark}
The divergence operator can also be defined invariantly on
decomposable multivector density fields using the Lie derivative: for
$k=1$ let $\mu$ denote a nonvanishing density of weight $-n$ and $X$,
$Y$ vector fields then $\divR (\mu \tens X \wedge Y)=(\Lie_X \mu) \tens
Y -(\Lie_Y \mu) \tens X + \mu \tens \Lieb{X,Y}$.
\end{remark}

\begin{application}\label{Covariantexteriorderivative}
\summary{Covariant exterior derivative} Let $EM$ be again a vector
bundle over $M$ and $D$ a covariant derivative on $EM$. Since the
exterior derivative on forms is first order it can be twisted by $D$
to obtain a sequence of first order differential operators
$\dR^D\from C^\infty(M,\Lambda^kT^\dual\tens EM)\to
C^\infty(M,\Lambda^{k+1}T^\dual\tens EM)$ between forms with values in
$EM$. As an example, if $X$, $Y$ denote vector fields and $\alpha$ a
$1$-form with values in $EM$ then $\dR^D\alpha(X,Y) := D_X(\alpha(Y))
-D_Y(\alpha(X)) -\alpha(\Lieb{X,Y})$. The obstruction for the twisted
deRham sequence to be a complex is the Riemann curvature: $R^D\in
C^\infty(\Lambda^2T^\dual\tens\gl(EM))$ which acts on a $k$-form to
give a $(k+2)$-form as
$$
\dR^D\circ \dR^D\alpha 
= \oneover{2}\Sum_{i,j} \theta^i\wedge\theta^j\wedge R^D(t_i,t_j)\alpha
=: R^D\wedge \alpha,
$$
where $t_i$, $\theta^i$ is a dual basis of $TM$. Clearly $D$ induces
also a derivative on $\gl(EM)$ and the Leibniz rule gives
$$
\dR^D\circ (\dR^D\circ \dR^D)\alpha 
= \dR^D R^D\wedge \alpha + (-1)^2 R^D\wedge \dR^D\alpha
= \dR^D R^D\wedge \alpha + (\dR^D\circ \dR^D)\circ \dR^D \alpha.
$$
This observation proves the following (see
e.g.~\cite{BerlineGetzlerVergne}):
\end{application}

\begin{proposition}\label{Bianchiidentity}
\summary{Bianchi identity} The curvature tensor $R^D$ of a covariant
derivative $D$ on $EM$ satisfies a first order integrability
condition: $0=\dR^DR^D$.
\end{proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Orientations and integration over submanifolds}
\label{Orientationsandintegrationoversubmanifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We recall that the one dimensional \emph{space of pseudoscalars} $L^{-
n}(V)\tens\Lambda^nV$ of a $n$-dimensional real vector space $V$ comes
with a natural norm $|\mu\tens v|=|\mu(v)|$. The two unit elements
correspond to the two orientations of $V$: if
$v:=t_1\wedge\ldots\wedge t_n$ with $t_i\in V$ is a positive basis of
$V$ then this induces $\orie=\mu\tens v$ with $0<\mu$ determined by
$\mu(v)=1$. A unit element $\orie\in L^{-n}(V)\tens\Lambda^nV$ induces
a unique element $\orie^\dual\in L^n(V)\tens\Lambda^nV^\dual$ in the
dual space by $\ip{\orie^\dual,\orie}=1$. Alternatively, the above
norm in the one dimensional space $L^{-n}(V)\tens\Lambda^nV$ induces
an inner product, which identifies it with its dual space:
$L^{-n}(V)\tens\Lambda^nV =L^n(V)\tens\Lambda^nV^\dual$

\begin{definition}\label{Orientation}
\summary{Orientation} A vector bundle $EM\to M$ of rank $k$ over a
$n$-dimensional manifold $M$ is called \emph{orientable} if the bundle
of pseudoscalars $L^{-k}(EM)\tens\Lambda^kEM$ is trivializable,
i.e.~if there is a global section $\orie\in
C^\infty(M,L^{-k}(EM)\tens\Lambda^kEM)$ of norm one $|\orie|=1$. A
choice of such a trivialization on an orientable bundle is called an
\emph{orientation of $EM$} and $EM$ with such a choice is called
\emph{oriented}.
\end{definition}

\begin{remark}
In general, if $M$ is connected and $x\in M$ a point then there is a
group homomorphism $\pi_1(M,x)\to \setof{+1,-1}$ between the
fundamental group and $\O(\R)=\setof{+1,-1}$ induced by parallel
transport in $L^{-k}(EM)\tens\Lambda^kEM$. This homomorphism factors
through the maximal Abelian quotient $\pi_1(M,x)_{Ab}$, hence the
homomorphism corresponds to a cohomology class $w_1(EM)\in
H^1(M,\Z/2\Z)$. The vector bundle is orientable, iff this class is
trivial. In particular if $M$ is simply connected, $\pi_1(M,x)=1$, any
vector bundle over $M$ is orientable.
\end{remark}

\begin{definition}\label{Orientationandcoorientation}
\summary{Orientation and coorientation} Let $S\immersion M$ denote a
$k$-dimensional immersed submanifold inside a $n$-dimensional manifold
$M$. The submanifold $S\immersion M$ is called \emph{orientable} iff
its tangent bundle $TS\to S$ is orientable. The submanifold
$S\immersion M$ is called \emph{coorientable} iff the normal bundle
(quotient bundle) $TM/TS \to S$ is orientable.
\end{definition}

\begin{remark}
Points in $M$ are always oriented and the manifold $M$ itself is
always cooriented. The conventional (co)orientations are often denoted
by $+1$, and the opposite orientation is denoted by $-1$.
\end{remark}

\begin{application}\label{integrationoverorsubmfds}
\summary{Integration over oriented submanifolds} Let $S\immersion M$
by an immersion as in definition \ref{Orientationandcoorientation}. If
$S$ is orientable and $\orie$ a choice of orientation then any
$k$-form $F$ on $M$ can be turned into a density $\ip{F,\orie}\in
C^\infty(S,L^{-k}(TS))$ over $S$ by means of the following
contraction:
$$
L^{-k}(TS)\tens \Lambda^k TS\tens \Lambda^k T^\dual M \to L^{-k}(TS).
$$
Hence $\ip{F,\orie}$ can be integrated naturally over $S$.
\end{application}

\begin{proposition}\label{sesandorientations}
If $0\to U \to V \to V/U \to 0$ is a short exact sequence of real vector 
spaces $U$, $V$ and $V/U$, then orientations of two of them induces an 
orientation of the third. Indeed if $k=\dim(U)$ denote the dimension we 
have the following canonical isomorphisms between one dimensional spaces:
\begin{center}
$L^{-k}(U)\tens\Lambda^{k}U \tens
L^{-(n-k)}(V/U)\tens\Lambda^{n-k}(V/U)
\stackrel{\isomorph}{\longrightarrow}
L^{-n}(V)\tens\Lambda^{n}V$,\\
$\Lambda^{k}U \tens
\Lambda^{n-k}(V/U)
\stackrel{\isomorph}{\longrightarrow}
\Lambda^{n}V$,\\
$L^{-k}(U) \tens
L^{-(n-k)}(V/U)
\stackrel{\isomorph}{\longrightarrow}
L^{-n}(V)$.
\end{center}
\end{proposition}

\begin{application}\label{integrationovercoorsubmfds}
\summary{Integration over cooriented submanifolds} Let $S\immersion M$
denote an immersion as in definition
\ref{Orientationandcoorientation}. If $S$ is coorientable and
$\coorie\in C^\infty(M,L^{n-k}(TM/TS) \tens
\Lambda^{n-k}(TM/TS)^\dual)$ a choice of coorientation then any
$(n-k)$-multivector density $G$ on $M$ can be turned into a density
$\ip{G,\coorie}\in C^\infty(S,L^{-k}(TS))$ over $S$, by means of the
following map (we will use the notation $TS^\perp=(TM/TS)^\dual$):
\begin{eqnarray*}
L^{-n}(TM) \tens \Lambda^{n-k}TM \tens
L^{n-k}(TM/TS)\tens \Lambda^{n-k} TS^\perp
&\to& L^{-n}(TM) \tens L^{n-k}(TM/TS)\\
&\isomorph& L^{-k}(TS).
\end{eqnarray*}
Hence $\ip{G,\coorie}$ can be integrated naturally over $S$.
\end{application}

\begin{application}\label{Outernormal}
\summary{Outer normal} If $S$ is a manifold with boundary $\boundary
S$ then the outer normal induces an orientation of the normal bundle
of the boundary $\outernormal\in L^{-1}(TS/T\boundary S)\tens
\Lambda^1(TS/T\boundary S)$ with $|\outernormal|=1$ along $\boundary
S$. Applying the proposition \ref{sesandorientations} for $0\to
T\boundary S\to TS\to TS/T\boundary S\to 0$ and for $0\to
TS/T\boundary S\to TM/T\boundary S\to TM/TS \to 0$ we have two induced
isomorphisms:
\begin{eqnarray*}
L^{-k}(TS)\tens \Lambda^k(TS) &\to&
L^{1-k}(T\boundary S)\tens\Lambda^{k-1}(T\boundary S),\\ 
L^{n-k}(TM/TS)\tens \Lambda^{n-k}(TS)^\perp &\to&
L^{n+1-k}(TM/T\boundary S)\tens\Lambda^{n+1-k}(T\boundary S)^\perp.
\end{eqnarray*}
Hence a (co-)orientation of $S$ induces a (co-)orientation of
$\boundary S$.
\end{application}

\begin{proposition}\label{Stokestheorem}
\summary{Stoke's theorem} If $F\in C^\infty(M,\Lambda^kT^\dual M)$ is
a smooth $k$-form on a $n$-manifold $M$, and $\Sigma \immersion M$
an (immersed) compact oriented $(k+1)$-dimensional submanifold with
boundary $\boundary\Sigma$ then 
$$
\int_{\boundary\Sigma} \ip{F,\orie} 
= \int_{\Sigma} \ip{\dR F,\orie}.
$$
\end{proposition}

\begin{proposition}\label{Divergencetheorem}
\summary{Divergence theorem} If $G\in C^\infty(M,L^{-n}\tens
\Lambda^{n-k}TM)$ is a smooth $(n-k)$-multivector density on an
$n$-manifold $M$, and $\Sigma \immersion M$ an (immersed) compact
cooriented $(k+1)$-dimensional submanifold with boundary
$\boundary\Sigma$ then
$$
\int_{\boundary\Sigma} \ip{G,\coorie}
= \int_{\Sigma} \ip{\divR G,\coorie}.
$$
\end{proposition}

\begin{remark}
In the case of $M$ being a manifold with boundary $\boundary M$ the outer
normal always induces a coorientation of $\boundary M$ and $M$ itself
is always cooriented. Hence the above divergence theorem applies to
the whole of $M$ without any orientability assumptions which is called
\emph{Gau{\ss} theorem}, whereas Stoke's theorem needs an orientable
$M$.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Scale invariance in functional analysis}
\label{Scaleinvarianceinfunctionalanalysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%

In real functional analysis the weight in definition \ref{Density} is
related to the so called \emph{Sobolev number}. To demonstrate this
relation we will briefly recall Lebesgue and Sobolev spaces using
density valued functions. The dimensional analysis then gives an
explanation of the H\"older and Sobolev inequalities. Recall that the
integral of a sections of the line bundle $L^{-n}$ over a
$n$-dimensional compact domain is invariant under diffeomorphisms of
the domain.

\begin{discussion}\label{lebesquespaces}
\summary{Lebesgue spaces} A scalar function lives in the Lebesgue
space $\Lebesque^p$ for some $p\in\R$ if the $p$th power of its
modulus is integrable. Therefore such a scalar function should be
considered as a section of $L^{-n/p}$, i.e.~one assigns $-n/p$ as
the weight of that function. Hence $\Lebesque^p$ can be called the
\emph{space of integrable functions of weight $w=-n/p$}. For $0<w$
(i.e.~$p<0$) this space of functions fails to be a vector space (local
problem) and for $w<-n$ (i.e.~$0\leq p< 1$) it fails to satisfy the
triangle inequality. Therefore a natural range for $w$ is $-n\leq
w\leq 0$. We may denote the norm of an integrable function $u$ of
weight $w$ by
$$
||u||_{(w)}:=\int |u|^{-n/w}.
$$
H\"older's inequality implies that the product of two integrable
functions of weight $w_1$ and $w_2$ respectively is integrable of
weight $w_1+w_2$.
\end{discussion}

\begin{discussion}\label{sobolevspaces}
\summary{Sobolev spaces} If a function has weight $w$ then its $k$th
derivative will have weight $w-k$. If a function lives in the Sobolev
space $\Sobolev^{k,p}$ then in particular its $k$th derivative is in
$\Lebesque^p$. Hence it is natural to assign $w$ with $w-k=-n/p$ as
the weight of that function; $w$ the equals the Sobolev number
$w=k-n/p$. For a compact domain with smooth boundary and in the case
of the first derivative $k=1$ we reformulate Sobolev's and H\"older's
inequality in terms of weights as follows: for $(1-n)\leq w < 0$ we
have Sobolev's inequality:
$$
||u||_{(w)}\leq C(n,w) ||\del u||_{(w-1)},
$$
with a constant $C(n,w)$ depending on the domain and the weight $w$,
similarly for $0 < w \leq 1$ we have the H\"older estimate:
$$
|u(x)-u(y)| \leq C(n,w) |x-y|^w ||\del u||_{(w-1)}.
$$
Note how the dimensional analysis takes care of the correct
exponents.
%Let us call $\Sobolev^{k,p}$ the \emph{space of $k$-times
%differentiable integrable functions of weight $w=k-n/p$} with $k-n\leq
%w \leq k$. As it is well known, in the case $k-n<w<l<k$ the Sobolev
%inequality implies that $k$-times differentiable integrable functions
%of some weight are also $l$-times differentiable integrable of the
%same weight.
\end{discussion}

\begin{example}
If a differential operator $\Nabla$ of order $k$ is selfadjoint then
the pointwise inner product $\ip{\Nabla\phi,\psi}$ should be
considered as a density of weight $-n$ (since these are the natural
integrands) leaving $w=(k-n)/2$ as natural weight for $\phi$ and
$\psi$. This weight coincides with the Sobolev number of the space
$\Sobolev^{k/2,2}$ which is the usual domain of $\Nabla$ in terms of
its weak formulation.
\end{example}

If a partial differential equation has a geometric origin, the
involved unknown function may have a natural geometric dimension
$\Dim{length}^w$. Such a natural or geometric weight $w$ usually
equals the \emph{critical Sobolev number}, i.e.~natural sections just
fail to be integrable. (For example the Euclidean distance
$\mathrm{dist}(\Sigma,.)$ to a submanifold $\Sigma$ is geometrically
given by a section of $L^1$ which is always critical.)

\begin{example}\label{Yamabeproblem}
The Yamabe problem of finding a metric within a given conformal class
which has constant scalar curvature (with respect to its own length
scale) is a typical example of such a situation. The problem reduces
to a nonlinear eigenvalue problem for the conformal Laplace operator
(see example \ref{ConformalLaplacian}) $\Laplace f^{(2-n)/2} = -
f^{(-2-n)/2}$ with normalization $\int f^{-n}=1$, where $f$ is a
positive section of $L^1$ playing the role of a length scale. The
geometric context has already determined the Laplace operator to act
on densities of weight $w=(2-n)/2$, which agrees with the Sobolev
number in the usual analytic setting in $\Sobolev^{1,2}$, see
\cite{LeeParker}.
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weyl derivatives}\label{SectionWeylderivatives}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Since the density line bundles $L^w$ over a manifold $M$ have no
preferred trivialization (except for $w=0$) covariant derivatives on
these line bundles are of differential geometric interest.  We will
indicate how densities and Weyl derivatives provide a way to
geometrize electromagnetism. Any first order differential operator can
be twisted by a Weyl derivative to define an operator on sections of
arbitrary central weight. We apply this to the deRham complex and its
adjoint.

\begin{definition}\label{Weylderivative}
\summary{Weyl derivatives} A covariant derivative $D$ of $L^1$ is
called a \emph{Weyl derivative} (see definition
\ref{Covariantderivative}). It naturally induces covariant derivatives
on all density bundles $L^w$ (e.g.~by means of parallel transport).
\end{definition}

\begin{definition}\label{Faradaycurvature}
\summary{Faraday curvature} The curvature (see proposition
\ref{Curvature}) of a Weyl derivative $D$ in $L^1$ is an endomorphism
valued $2$-form which for a line bundle is determined by a real valued
$2$-form $F^D\in C^\infty(M,\Lambda^2T^\dual M)$.
\end{definition}

The differential Bianchi identity, see proposition
\ref{Bianchiidentity}, applied to a given Weyl derivative $D$ means
that $F^D$ is a closed $2$-form, $\dR F^D=0$. Originally Weyl
interpreted a Weyl derivative as electromagnetic potential and its
curvature as electromagnetic field, which then automatically satisfies
the first Maxwell equation. Therefore we like to call the closed two
form $F^D$ the \emph{Faraday curvature}. The induced curvature of
$L^w$ is given by $wF^D$.

The Faraday curvature is the local obstruction for finding a parallel
trivialization of $L^1$. The Weyl derivative $D$ is called
\emph{closed} if $F^D=0$. For a closed Weyl derivative one can find
locally positive sections of $L^1$, which are $D$ parallel. If there
is a global positive $D$ parallel section $\mu$ of $L^1$ then $D$ is
called \emph{exact}. Conversely a positive
section of $L^1$ determines an exact Weyl derivative $D^{\mu}$ via
$D^{\mu}\mu = 0$.

The space of all Weyl derivatives forms an affine space modelled on
the linear space of smooth $1$-forms $C^\infty(M,T^\dual M)$: let $D$
and $\other{D}$ be two covariant derivatives in the line bundle
$L^1$. They differ by an endomorphism valued $1$-form $\other{D}-D =
\christo$ which for a line bundle is just a real valued $1$-form
$\christo\in C^\infty(M,T^\dual M)$. The induced derivatives in $L^k$
differ by $\other{D}-D = k\christo$. Closed and exact Weyl derivatives
are affine subspaces modelled on closed and exact $1$-forms
respectively.

\begin{application}\label{Twistedexteriorderivative}
\summary{Twisted exterior derivative} With the help of a Weyl
derivative $D$ the exterior derivative $\dR $ can be extended to forms
of arbitrary central weight (see paragraph
\ref{Covariantexteriorderivative}):
$$
\dR^D\from C^\infty(M,L^w \tens \Lambda^kT^\dual M) 
\to C^\infty(M,L^w \tens \Lambda^{k+1}T^\dual M).
$$
The obstruction for the twisted deRham sequence to be a complex is the
Faraday curvature $F^D$.
\end{application}

\begin{application}\label{Twistedexteriordivergence}
\summary{Twisted exterior divergence} Similarly a Weyl derivative $D$
defines divergences on multivector field densities of arbitrary
central weight
$$
\divR^D\from C^\infty(M,L^{w-n}\tens \Lambda^kTM) 
   \to C^\infty(M,L^{w-n} \tens \Lambda^{k-1}TM).
$$
For example if $\mu$ denotes a density of weight $w$ and $X$ a vector
field of central weight $1-n$, then $\divR^D (\mu \tens X) = \trace (D\mu
\tens X) + \mu \tens \divR X$.
\end{application}

\begin{remark}
Lie derivative, Weyl derivative and twisted divergence are related as
follows: for a density $\mu$ of weight $w$ and a vector field $K$ we
have $\Lie_K\mu = D_K\mu - \smallfrac{w}{n} \mu \divR^D K$.
\end{remark}

The vector field $K$ generates diffeomorphisms which leave a given
Weyl derivative $D$ invariant, iff it satisfies the following linear
partial differential equation of second order: $0=\Lie_KD$. Here
$\Lie_KD$ denotes a real valued $1$-form defined with the help of
vector field $X$ and a length scale $\mu\in L^1$ by
\begin{eqnarray*}
(\Lie_KD)_X\mu
&=& \Lie_K(D_X\mu) - D_{\Lie_KX}\mu - D_X(\Lie_K\mu)\\
%&=& D_K(D_X\mu) - \oneover{n} D_X\mu \divR^D K
%  - D_{\Lieb{K,X}}\mu - D_X(D_K\mu - \oneover{n} \mu \divR^D K)\\
&=& F^D(K,X)\mu + \oneover{n} \mu \,\del_X (\divR^D K).
\end{eqnarray*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Elementary conformal geometry}
\label{Elementaryconformalgeometry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This chapter provides the elementary background in relativity,
representation theory and differential geometry which we will use in
the sequel. In the first two sections we define a conformal structure
on a vector space and apply this to relativistic kinematics and
electromagnetism in Minkowski space. The freedom of choosing a
particular inner product in the conformal class corresponds to the
free choice of a unit for the dimension $\Dim{length}$. Light rays,
relative velocity of observers, conservation of momentum in collision
experiments and the electromagnetic constitutive relation in vacuum
only depend upon a conformal structure. Perihelion advance illustrates
the impact of special relativity on the motion in a central force
field. In section \ref{Conformalstructureonmanifolds} we recall how
manifolds can be equipped with a conformal structure. There is no
preferred covariant derivative on a conformal manifold: instead the
space of Weyl derivatives parameterizes compatible derivatives of the
tangent bundle. In sections
\ref{Elementarynotionsofrepresentationtheory} and
\ref{Bundlesandcurvatureonconformalmanifolds} we recall how vector
bundles are associated to the orthogonal frame bundle and determine
their curvature properties. In section \ref{ChangingtheWeylderivative}
we adapt Branson's method to Weyl derivatives, which reduces the
question of conformal invariance to a linear question. In the last
section we recall the first few integrability conditions of the
conformal curvature tensors. These Bianchi identities motivate the
field equations for Einstein's and Bach's theories of gravity.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conformal vector spaces}
\label{Conformalvectorspaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Conformal geometry concerns the concept of angle without that of
absolute length. A conformal structure of a vector space is an equivalence
class of inner products with the same notion of orthogonality. Hence
two such inner products differ by a multiplicative factor in
front. Since there is no preferred inner product in the class we give
the following invariant definition which is taken from
\cite{CalderbankFaraday}:

\begin{definition}\label{Conformalvectorspace}
\summary{Conformal vector space} A \emph{conformal structure} $\com$
of a real vector space $V$ is a normalized inner product on the
weightless space $L^{-1}\tens V$.  Normalized means that on $\Lambda^n
(L^{-1} \tens V) = L^{-n} \tens \Lambda^n V$ the canonical inner
product and the inner product induced by $\com$ coincide:
$|\deter\com|=1$. Alternatively $\com$ can be viewed as a normalized
$L^2$ valued inner product on $V$. If $\A$ is an affine space over the
conformal vector space $V$ then $\A$ carries a conformal structure.
\end{definition}

\begin{remark}
A conformal structure can also be defined as an equivalence class of
inner products: a positive element $\mu \in L^1$ is called a
\emph{length scale} and it determines by $\mu^{-2}\com$ a (real
valued) inner product on $V$ with $\mu^{-n}$ as volume density. On the
other hand, given a (real valued) inner product $g$ on $V$ then $g$
represents a conformal structure given by $\com := \mu(g)^2 g$, where
$\mu(g)$ is the length scale of $g$ (equivalently $\mu^{-n}(g)$ is the
volume density of $g$).
\end{remark}

\begin{notation}
We will use the conformal metric to identify $V$ with its dual,
precisely, we will use the isomorphisms $\sharp\from L^k\tens V^\dual
\to L^{k-2}\tens V$ and $\flat \from L^k\tens V\to L^{k+2}\tens
V^\dual$ sometimes without denoting them.
\end{notation}

The following remark is a bit technical. We will use it to integrate
sections in $L^{-1}$ along nonlightlike curves (application
\ref{Associateddistribution}) and to recover the electric field and
the magnetic flux from a Faraday $2$-form, see remark
\ref{EandBfield}.
 
\begin{remark}\label{Pullbackofdensities}
\summary{Pull back of densities} If $\iota\from U\to V$ denotes a
subspace of dimension $k$ on which the induced conformal inner product
stays nondegenerated, then we have an induced map $\iota^\dual:
L^{-k}(V)\to L^{-k}(U)$: for $\mu\in L^{-k}(V)$, $u\in\Lambda^kU$,
$\sign{\mu}=\pm 1$ we define
$\iota^\dual(\mu)(u):=\sign{\mu}\sqrt{\cip{\mu\tens
\iota(u),\mu\tens\iota(u)}}$, where $\com$ is the induced inner
product on $\Lambda^k(V^0)$. More generally for all weights $w$ we
have $\iota^\dual\from L^{w}(V)\to L^{w}(U)$ with
$\iota^\dual(\mu)(u)=\sign{\mu}(|(|\mu|)^{-k/w}\tens u|_\com)^{-w/k}$.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Special relativity and electromagnetism}
\label{Specialrelativityandelectromagnetism}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The simplest geometrical model of physical spacetime is an affine space 
$\A$ over a real $n=4$ dimensional vector space $V$. Points in $\A$ are 
called \emph{events} and an observer living for some time traces out a 
\emph{worldline} in $\A$. In principle an (infinitesimal) observer $v\in 
V$ along his worldline can measure the angle between two direction and 
can decide whether two rulers have equal length (at the same time). Hence 
(infinitesimal) spacelike measurements take place in a conformal class of 
(Euclidean) metrics (of the three dimensional quotient $V/\R v$). Note, 
that the focus on conformal metrics also corresponds to the following 
physical principle:

\begin{discussion}\label{constantvelocityoflight}
\summary{Constant velocity of light}
Lightrays travel through spacetime in a way which is independent of the 
motion of the emitter. In the geometrical model these naturally given 
lightrays form double cones through each (emitting) event. In special 
relativity these light cones come from a conformal class $\com$ of inner 
products in $V$ with Lorentzian signature $(3,1)$. Note that such a 
lightcone $\setof{v\in V\property \cip{v,v}=0}$ characterizes its 
conformal metric.
\end{discussion}

The geometry of lightcones describes the causal relation between
events: 

\begin{definition}\label{Causality}
\summary{Causality} Vectors $v\in V$ inside the lightcone
$\cip{v,v}<0$ of an event $x\in \A$ are called \emph{timelike},
vectors $v$ with $\cip{v,v}>0$ are called \emph{spacelike} and vectors
$v$ with $\cip{v,v}=0$ are called \emph{lightlike}.
\end{definition}

Similarly a (piecewise) differentiable path is called timelike
(respectively spacelike or lightlike), if all velocity vectors are
timelike (respectively spacelike or lightlike). We say that two events
$x$ and $x+v$ with $v$ timelike are \emph{in causal relation}. A
lightlike or timelike path joining two events $x$ and $x+v$ with $v$
spacelike needs to change its time direction.

Observers are smooth curves $c(t)\in \A$ with timelike velocity
vectors $\dot c$. The points in the intersection of the forward
lightcone of $c(t_0-\tau)$ with the backward lightcone of
$c(t_0+\tau)$ are called \emph{simultaneous events relative to
$c$}. The three dimensional linear space ${\dot c(t_0)}^\perp :=
\setof{v\in V\property \cip{v,\dot c(t_0)}=0}$ perpendicular to $\dot
c(t_0)$ describes the space of simultaneous events infinitesimally.

\begin{remark}\label{relativevelocity}
\summary{Relative velocity} Relativistic kinematics of particles is
also based upon the conformal Lorentzian metric: let $U\in
V^0:=L^{-1}\tens V$ be a unit tangent vector to a worldline of an
observer, hence $U$ is timelike and normalized $\cip{U,U}=-1$, see
also \ref{Geometryofcurves}. A second observer $N$ with $\cip{N,N}=-1$
meeting $U$ at an event $x$ \emph{points into the same time direction}
as $U$, if $\cip{U,N}<0$. In that case $N$ measures a \emph{relative
velocity} $\vec u\in N^\perp \subset V^0$ (as multiple of the velocity
of light, example \ref{Kinematicdimensions}) in his infinitesimal
space of simultaneous events $\cip{\vec u,N}=0$ given by the condition
$N + \vec u \prop U$, hence $N + \vec u = -U/\cip{U,N}$. Squaring this
equation gives $|\vec u|^2 = 1 - 1/{\cip{U,N}^2}$.
\end{remark}

\begin{remark}\label{Collisionexperiments}
\summary{Collision experiments} If two (pointlike) bodies collide at
an event in spacetime then their initial velocities $N_1$ and $N_2$
(weightless and normalized) and inertial masses $m_1$ and $m_2$ are
related to their final velocities $N_1'$ and $N_2'$ (weightless and
normalized) and final inertial masses $m_1'$ and $m_2'$ as $m_1N_1 +
m_2N_2 = m_1'N_1' + m_2'N_2'$ (see \cite{Parrot} p.~18ff). Note that
this conservation law at the (idealized) event of collision only
depends upon the conformal structure. From the dimensional analysis of
example \ref{Mechanicaldimensions} we think of inertial masses as
elements in $L^{-1}$ and $m_1N_1\in V^\dual$ is called the
\emph{linear momentum} of the body.
\end{remark}

Electromagnetism deals with the following physical phenomenon:
\emph{motion of electric charges produces fields which travel with the
speed of light and these fields influence the motion of charged
particles}. The theory describing electromagnetic interactions is
based upon Maxwell's equations. In their relativistic form due to
Minkowski the field equations are part of the deRham sequence of the
exterior derivative and its adjoint the sequence of exterior
divergences:

\begin{definition}\label{Electromagneticfieldtheory}
\summary{Electromagnetic field theory} A vector field $j$ of central
weight $1-n$ represents the \emph{charge current density}. It plays
the role of the \emph{source} and has to satisfy a \emph{conservation
law}: $\divR j = 0$. We have to distinguish between two realizations
of the field itself, the dynamic and the kinematic field.  The
\emph{dynamic field} $G$ is represented by a bivector density field of
central weight $2-n$ and it is coupled to the source by the second
Maxwell equation $j = \divR G$. The \emph{kinematic field} $F$ is
represented by a $2$-form of central weight $-2$ the so called
\emph{Faraday $2$-form} and satisfies an \emph{integrability
condition} $\dR F = 0$ the first Maxwell equation.
\end{definition}

\begin{remark}\label{EandBfield}
\summary{$E$ and $B$ field} Relative to an observer $\cip{N,N}=-1$
with its space of simultaneous events $N^\perp\subset V^0$ the
kinematic field $F$ splits into two tensors of central weight $-2$ in
$N^\perp$. We make secret use of the remark \ref{Pullbackofdensities}
and define a $1$-form the \emph{electric field strength} $E(\vec u) :=
F(N,\vec u)$ and a $2$-form the \emph{magnetic flux} $B(\vec u,\vec v)
:= F(\vec u,\vec v)$ with $\vec u, \vec v\in N^\perp$. Similarly the
dynamic field $G$ splits into the \emph{dielectric displacement}
$D(\vec u) := G(N,\vec u)$, a vector on $N^\perp$ of central weight
$-2$ and the \emph{magnetic loop tension} $H(\vec u,\vec v):=G(\vec
u,\vec v)$ a bivector on $N^\perp$ of central weight $-2$.  For the
physical dimensions see example \ref{Electromagneticdimensions}.
\end{remark}

\begin{discussion}\label{constitutiverelation}
\summary{Constitutive relation} Dynamic and kinematic fields are
related by the so called \emph{constitutive relation} which depends upon
the matter in which the field propagates. In vacuum (and in $n=4$
dimensions) the constitutive relation between $F$ and $G$ is simply
given by a conformal structure $\com$ of Lorentzian signature
$(n-1,1)$, i.e.~$F(X,Y) = \sum_{i,j} \cip{X,t_i}\cip{Y,t_j}
G(\theta^i,\theta^j)$, where $X$, $Y$ are vectors and $t_i$,
$\theta^i$ is a dual basis of $V$.
\end{discussion}

Since the kinematic field $F$ is closed there (locally) exists a
$1$-form $A$ of central weight $-1$, called \emph{potential}, such
that $F=\dR A$. This potential is not uniquely determined by $F$,
instead it is subject to (local) transformations $A\mapsto A + \dR f$,
where $f$ is a function, called a \emph{gauge function}.

\begin{summy}\label{SummaryElectromagneticfieldtheory}
To summarize, the electromagnetic field theory in vacuum takes place in 
an (affine) space modelled on a four dimensional vector space $V$ with 
conformal metric of Lorentzian signature. The \emph{kinematic 
sequence} is given by the beginning of the deRham sequence of the 
exterior derivative:
\[
\begin{array}{*{7}{c}}
\mathrm{gauge\ } &\ &
\mathrm{potential\ } &\ &
\mathrm{kinematic\ field} &\ &
\mathrm{} \\
C^\infty(\A,\R) &\stackyrel{\dR }{\to}&
C^\infty(\A, V^\dual) &\stackyrel{\dR }{\to}&
C^\infty(\A, \Lambda^2 V^\dual) &\stackyrel{\dR }{\to}&
C^\infty(\A, \Lambda^3 V^\dual) \\
f &\ &
A &\ &
F &\ &
0 
\end{array}
\]
The \emph{dynamic sequence} is given by the end of the sequence of the 
exterior divergence:
\[
\begin{array}{*{7}{c}}
0 &\ &
j &\ &
G &\ &
\ \\
C^\infty(\A, L^{-n}) &\stackyrel{\divR}{\ot}&
C^\infty(\A, L^{-n}\tens V) &\stackyrel{\divR}{\ot}&
C^\infty(\A, L^{-n}\tens \Lambda^2 V)&\stackyrel{\divR}{\ot}&
C^\infty(\A, L^{-n}\tens \Lambda^3 V) \\
\mathrm{}  &\ &
\mathrm{source\ } &\ &
\mathrm{dynamic\ field} &\ &
\mathrm{}
\end{array}
\]
The conformal metric is the constitutive relation between kinematic and 
dynamic field. Hence the Maxwell equations in vacuum are linear and 
conformally invariant given by:
\begin{center}
$F= G$ via the conformal structure $\com$,\\
$0 = \divR j$ and $j = \divR G$,\\
$0 = \dR F$ and $F = \dR A$ and $A \mapsto A + \dR f$.
\end{center}
\end{summy}

\begin{remark}
Note that the second Maxwell equation $\divR G=j$ between the dynamic
field and the source already determines all other differential
operators which are involved: the integrability condition for $\divR
G=j$ gives the conservation of the source $0=\divR\divR G =\divR j$,
adjoint to this conservation law is the operator for the gauge
transformation $A+\dR f$, these transformations are in the kernel of
the equation between potential and kinematic field $F=\dR A=\dR(A+\dR f)$
(which is also adjoint to $\divR G=j$), finally the integrability of
$\dR A=F$ gives the first Maxwell equation $0=\dR\dR A=\dR F$.
\end{remark}

\begin{discussion}\label{Lorentzforcelaw}
\summary{Lorentz force law} Galilei's principle of inertia means that
a free massive particle with initial velocity $N$ in spacetime has the
tendency to stick to that spacetime direction, i.e.~in the affine
model of spacetime a free massive particle moves along straight
timelike lines: $D_NN=0$, where $D$ denotes the affine
derivative. Obviously the principle of inertia isn't conformally
invariant, since it involves the affine derivative. If an
electromagnetic field $F$ is present, an electrically charged particle
$q\in\R$ is no longer free, instead it is influenced by $F$: a simple
relativistic law of motion is then given by 
$$
mD_NN = q \sharp (F(N)),
$$
where $m\in L^{-1}$ is the inertial mass of the charged particle and
$N$ is viewed as a test particle ($q/m$ is small compared to $F$).
\end{discussion}

\begin{example}\label{Coulombfield}
\summary{Coulomb field} Coulomb's (nonrelativistic) law determines the
force between two charges $q$ and $Q$ which are at rest relative to
the laboratory to be $qQ/r^2$. Viewing one charge $q$ as test particle
and the other $Q$ as source, the Lorentz force law $mD_NN = q F(N)$
determines the corresponding Faraday $2$-form $F$ which solves
Maxwell's equations - the Coulomb field: let $N$ be the unit timelike
vector of the two charges and identify the conformal affine spacetime
$\A$ with $V$ by taking an event $o\in \A$ along the straight
worldline of the source as origin. If $P\from V\to N^\perp\subset V$
denotes the projection $v \mapsto v - \ip{v,N}/\ip{N,N} N$ the radial
distance $r\from \A \to L^1$ to the source is given by $r(c) :=|x|$,
where $x=x(c):=P(c-o)\in N^\perp$ is the spacelike location of the
test particle. Coulomb's force law on the one hand and Lorentz force
law on the other hand gives $qQ/r^2\ {\vec r} = q F(N)$, where ${\vec
r} = x/|x| = \grad^D r$ denotes the unit radial field, which is the
gradient of the distance function with respect to the (usual) affine
derivative $D$. Therefore if $F\from\A\to \Lambda^2 V^\dual$ is a pure
electric field for $N$ then it is given by
$$
F = F_{\mathrm{Coul}} := Q / r^2 \ (\ip{\vec r,.}N -\ip{N,.}\vec r).
$$
\end{example}

\begin{discussion}\label{Motioninacentralfield}
\summary{Motion in a central field}
Next we like to study the relativistic motion of a test particle $c\from 
\R \to \A$ subject to the Lorentz force in the Coulomb field. For that we 
solve the ordinary differential equation
$$
\ddot c = -a/r^2 \ (\ip{\vec r,\dot c}N -\ip{N,\dot c}\vec r),
$$
where $a\in \R$ is a number. Clearly $\ip{\dot c,\dot c}$ is a constant 
of motion. For the projection $x:=P(c)$ we have
$$
\ddot x = P(\ddot c) = -a/r^2 (-\ip{\dot c,N}){\vec r}.
$$
The function $-\ip{\dot c,N}$ satisfies $\delbydel{t}(-\ip{\dot c,N})
= a\delbydel{t}(1/r)$, hence
$$
-\ip{\dot c,N}=b + a/r,
$$
where $b=-\ip{\dot c(0),N}- a/r(0)$ depends on the initial
conditions only. Hence we can introduce the central potential
$U\from\R^+\to \R$ defined by
$$
U(r):= - ab/r - (\oneover{2} a^2)/r^2,
$$
and (with $U(x)=U(|x|)$) the differential equation becomes
\begin{eqnarray*}
\ddot x &=& -\grad U_x = - U'(|x|) x/|x| \\
r(t) &=& |x(t)| \\
\dot c(t) &=& \dot x(t) + (b + a/r(t))N.
\end{eqnarray*}
We have the usual conserved quantities in a central potential: angular
momentum $\vec l:=\dot x \wedge x$ with length squared $l^2 = \ip{\dot
x,\dot x} r^2 - (r\dot r)^2$ and energy $E:=\oneover{2}\ip{\dot x,\dot
x} + U(|x|)= \oneover{2}{\dot r}^2 + \oneover{2} l^2/r^2 + U(r)$. The
conservation of the bivector $\vec l$ means that the solution curve
for $x(t)$ lies in a plane $x=r(\cos(\phi),\sin(\phi))$ and for the
polar angle $\phi$ we have $\dot\phi=l/r^2$. Note $\ip{\dot c,\dot c}=
2E - b^2$.
%The differential equation for the distance $r(t)$ is given by $\ddot r= -
%U_{\mathrm{eff}}'(r)$, where the \emph{effective potential} is defined 
%to be
%\begin{eqnarray*}
%U_{\mathrm{eff}}(r) &:=& \oneover{2} l^2/r^2 + U(r)\\
%&=& - ab/r + \oneover{2} (l^2-a^2)/r^2.
%\end{eqnarray*}
\end{discussion}

\begin{discussion}\label{Perihelionadvance}
\summary{Perihelion advance} In the case of a Coulomb field of a
charge $Q$ and an opposite test charge $q$ with inertial mass $m$ we
have $a=- Qq/m>0$, i.e.~an attractive potential. A timelike solution
curve $\ip{\dot c,\dot c}=-1=2E-b^2$ which directs into the same time
$0<- \ip{\dot c,N}$ is guaranteed if $b>0$. If in addition $-1/2<E<0$
and $a^2<l^2<a^2/(-2E)$ we have bounded orbits $0\leq(\dot r)^2 = 2E
-2U(r) - l^2/r^2$ hence $r_P\leq r\leq r_A$ with $r_{A/P}=r_0\pm q$,
$(-2E)r_0:=ab$ and $(-2E)q^2:={a^2/(-2E) -l^2}$. Since $(\dot r)^2
= (- 2E)(r_A-r)(r-r_P)/r^2$ the polar angle satisfies
$$
\phi'(r)
= {\dot\phi\over \dot r}
= {l\over r^2\dot r}
= { l\over \sqrt{-2E}}\ {1 \over r\sqrt{(r_A-r)(r-r_P)} },
$$
hence with $\phi(r_P)=0$ we find
$$
\phi(r)
={ l\over \sqrt{(-2E)r_Ar_P} }\arccos\left({r_Pr_A-r_0r \over qr}\right).
$$
Finally $\phi(r_A)=\pi l/\sqrt{(-2E)r_Ar_P}$ and the total angle
between two perihelion constellations (moment of nearest approach to
the centre) is given by:
$$
2\pi {l \over \sqrt{(-2E)r_Ar_P}}
= 2\pi {1\over\sqrt{1 - (a/l)^2}}
\approx 2\pi( 1 + \oneover{2}(a/l)^2 + 
\smallfrac{3}{8} (a/l)^4 + \ldots).
$$
(We could compare the above special relativistic perihelion advance
with the perihelion advance in the Schwarzschild model (see
e.g.~\cite{ONeill} p.~379): the qualitative behavior is similar,
since the leading term is in both cases given by $(a/l)^2$, but the
special relativistic factor in front is $\oneover{2}$ whereas in the
Schwarzschild model it is $3$.)
\end{discussion}

\begin{discussion}
\summary{Charge} In physical situations where a smooth source $j$
models the charge distribution with a dynamic field $G$ subject to
$\divR G=j$ we deduce from the divergence theorem that the integral of
$j$ over a cooriented $(n-1)$-dimensional ball $\int_{B}\ip{j,\coorie}
= \int_{B}\ip{\divR G,\coorie} =\int_{\boundary B}\ip{G,\coorie}$ only
depends upon the boundary $\boundary B$ rather then the spanning
$B$. This leads to the definition \ref{Electriccharge} of charge
represented by $G$ for the case of $G$ being smooth in a region away
from $\support(j)$ even if $j$ is distributional. We remark that the
integral $\int_{\boundary B} \ip{G,\coorie}$ only depends upon the
homology class of $\boundary B$ as will be explained in proposition
\ref{Vanishingelectriccharge}. In what follows we let $M\subset\A$ be
an open region away from $\support(j)$ where the field $G$ is smooth.
\end{discussion}

\begin{definition}\label{Electriccharge}
\summary{Electric charge} If $G$ denotes a smooth divergence-free
bivector density we define the \emph{electric charge} (represented by
$G$) contained in an (immersed) cooriented $(n-2)$-dimensional sphere
$S^{n-2} \immersion M$ by:
$$Q_e:=\int_{S^{n-2}}\ip{G,\coorie}.$$
\end{definition}

As is well known, this quantity satisfies a conservation law, which 
follows from the divergence theorem:

\begin{proposition}\label{Vanishingelectriccharge}
If $S^{n-2}=\boundary\Sigma$ is the boundary of an (immersed) compact
cooriented $(n-1)$-dimensional submanifold $\Sigma\immersion M$, then
$Q_e = 0 $. Consequently, if $S^{n-2},\coorie$ and
${S^{n-2}}',\coorie'$ are two immersed spheres, such that there is an
(immersed) compact cooriented $(n-1)$-dimensional submanifold
$\Sigma,\coorie$ with these two spheres as boundary:
$\partial\Sigma,\coorie=S^{n-2},\coorie\, \cup\,
{S^{n-2}}',-\coorie'$, then $S^{n-2}$ and ${S^{n-2}}'$ contain the
same amount of electric charge.
\end{proposition}

For completeness we will also define \emph{magnetic charge}
represented by a Faraday $2$-form:

\begin{definition}\label{Magneticcharge}
\summary{Magnetic charge} If $F$ denotes a smooth closed $2$-form we
define the \emph{magnetic charge} (represented by $F$) contained in an
(immersed) oriented $2$-dimensional sphere $S^{2} \immersion
M$ by:
$$Q_m:=\int_{S^{2}}\ip{F,\orie}.$$
\end{definition}

This quantity satisfies a conservation law, which 
follows from Stoke's theorem:

\begin{proposition}\label{Vanishingmagneticcharge}
If $S^{2}=\boundary\Sigma$ is the boundary of an (immersed) compact
oriented $3$-dimensional submanifold $\Sigma\immersion M$, then
$Q_m = 0 $. Consequently, if $S^{2},\orie$ and
${S^{2}}',\orie'$ are two immersed spheres, such that there is an
(immersed) compact oriented $3$-dimensional submanifold
$\Sigma,\orie$ with these two spheres as boundary:
$\partial\Sigma,\orie=S^{2},\orie\, \cup\,
{S^{2}}',-\orie'$, then $S^{2}$ and ${S^{2}}'$ contain the
same amount of magnetic charge.
\end{proposition}

\begin{remark}
The differential equations of electromagnetism make sense on any 
conformal manifold. Source-free electromagnetic fields can then give rise 
to charge due to nontrivial topology of the underlying manifold. See the 
joint article by Hadley and the author \cite{DiemerHadley} for a review 
and extension of this idea.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conformal structure on manifolds}
\label{Conformalstructureonmanifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we follow Weyl \cite{WeylRZM}, Bergmann and Einstein 
\cite{Bergmann} to define conformal structures and related geometries on 
arbitrary smooth manifolds. A Riemannian metric defines a conformal class 
of metrics, but there is no preferred metric within such a class. 
Consequently we define a conformal structure invariantly as a single 
Riemannian metric of the weightless tangent bundle. There is no preferred 
covariant derivative on a conformal manifold instead we deal with an 
affine space of derivatives modelled on the space of smooth $1$-forms.

\begin{definition}\label{Conformalstructure}
A conformal structure on a manifold $M$ is a (smooth) normalized $L^2$
valued inner product $\com$ on $TM$ (see definition
\ref{Conformalvectorspace}). It can be viewed as a nondegenerate
normalized (real valued) inner product on the weightless tangent
bundle $L^{-1} \tens TM$. This inner product between vector fields $X$
and $Y$ will also be denoted by $\ip{X,Y}=\cip{X,Y}$.
\end{definition}

\begin{definition}\label{Conformalgeometry}
\summary{Conformal geometry} A conformal structure $\com$ defines a
conformal geometry on $M$. Naively, a construction is called
conformally invariant, if it only depends upon $\com$, i.e.~if the
construction is defined using an additional choice one needs to show
that the construction is indeed independent of that choice.
\end{definition}

\begin{definition}\label{ConformalKillingfields}
\summary{Conformal Killing fields} A vector field $K$ on a conformal
manifold $M$ generates diffeomorphisms which leave the conformal
structure $\com$ invariant iff $0=\Lie_K\com$. A solution to this
linear first order partial differential equation is called a
\emph{conformal Killing field}.
\end{definition}

\begin{proposition}\label{Torsion}
\summary{Torsion} Let $D$ be a covariant derivative of the tangent
bundle $TM$. The following bilinear differential pairing $T^D\from
C^\infty(M,TM)\tens C^\infty(M,TM)\to C^\infty(M,TM)$ defined by
$T^D(X,Y):=D_XY - D_YX -{\Lieb{X,Y}}$ (where $X$, $Y$ are vector
fields and the Lie bracket on functions is defined by
$\del_{\Lieb{X,Y}} = \del_X\del_Y-\del_Y\del_X$) is indeed zero order
and defines the \emph{torsion} of $D$ to be a vector valued $2$-form
$T^D\in C^\infty(M,\Lambda^2T^\dual\tens TM)$.
\end{proposition}

A covariant derivative $D$ on $TM$ is called \emph{torsion-free} if
$T^D=0$. A covariant derivative $D$ on $TM$ induces covariant
derivatives on all tensor bundles including $L^w$ (for instance by
means of parallel transport or via the induced connection on the
first order frame bundle $\GL(M)$). The covariant derivative on
$L^1$ is called a Weyl derivative, see definition
\ref{Weylderivative}. The derivative $D$ is called \emph{compatible
with the conformal structure} if $D\com=0$.

\begin{theorem}\label{Fundamentaltheoremofconformalgeometry}
\summary{Fundamental theorem of conformal geometry} On a conformal
manifold $M$ there is a one to one correspondence between covariant
derivatives on $L^1$ and derivatives on $TM$, which are torsion-free
and compatible with the conformal structure $\com$. Such derivatives
will be called \emph{Weyl derivatives} and for vector fields
$X$,$Y$,$Z$ the \emph{Koszul formula} holds:
\begin{eqnarray*}
2\cip{D_XY,Z} 
  &=& D_X(\cip{Y,Z}) + D_Y(\cip{Z,X}) - D_Z(\cip{X,Y})\\
  && +\cip{\Lieb{X,Y},Z}-\cip{\Lieb{Y,Z},X}+\cip{\Lieb{Z,X},Y}.
\end{eqnarray*}
\end{theorem}

\proof The arguments needed here are the same as those used in
Riemannian geometry: first of all if a covariant derivative $D$ is
torsion-free and $D\com=0$ then the above Koszul formula
holds. Furthermore, if $f\in C^\infty(M,\R)$ is a function then $D$
defined by the Koszul formula satisfies $D_{fX}Y=fD_XY$ and
$D_X(fY)=(\del_Xf)Y+fD_XY$ since
$\Lieb{fX,Y}=f\Lieb{X,Y}-(\del_Yf)X$.\qed

\begin{definition}\label{Riemanniangeometry}
\summary{Riemannian geometry} A length scale, i.e.~a positive section
$\mu$ of $L^1$ plus a conformal structure $\com$ defines a Riemannian
geometry on $M$. Let $D$ be the (exact) Weyl derivative which leaves
the length scale parallel. The induced derivative on $TM$ is then the
Levi-Civita derivative of the metric $\mu^{-2}\com$.
\end{definition}

A vector field $K$ on a $n$-dimensional manifold $M$ with Riemannian
geometry, generates diffeomorphisms which leave the geometry
invariant, iff $K$ is a conformal Killing field which leaves the
length scale invariant i.e.~$0=\Lie_K\com$ and $0 = \Lie_K\mu$. The
second condition says that the vector field is divergence-free (with
respect to the length scale).  Using the Levi-Civita derivative, the
two equations can be summarized by saying that $DK$ is a skew
symmetric endomorphism field.

\begin{definition}\label{Weylgeometry}
\summary{Weyl geometry} A Weyl derivative $D$ in $L^1$ plus a
conformal structure $\com$ defines a Weyl geometry on $M$. 
\end{definition}

A vector field $K$ on a $n$-dimensional manifold $M$ with Weyl
geometry, generates diffeomorphisms which leave the geometry
invariant, iff $K$ is a conformal Killing field which leaves the Weyl
derivative invariant i.e.~$0=\Lie_K\com$ and $0 = \Lie_KD = F^D(K,\ )
+ \oneover{n} \del \divR^DK$. The conformal Killing equation can
be rewritten in terms of the Weyl derivative $D$:
\begin{eqnarray*}
(\Lie_K\com)(X,Y)
&=& \Lie_K(\cip{X,Y}) 
    - \cip{{\Lie_KX},Y} 
    - \cip{X,{\Lie_KY}}\\
%&=& D_K(\cip{X,Y}) 
%    - \twoover{n} \divR^DK \cip{X,Y}
%    - \cip{{D_KX-D_XK},Y} 
%    - \cip{X,D_KY-D_YK}\\
&=& \cip{D_XK,Y} 
    + \cip{X,D_YK} 
    - \twoover{n}(\divR^DK) \cip{X,Y}.\\
\end{eqnarray*}
A vector field solving both equations is called a \emph{Weyl Killing
field}. Its $2$-jet is already determined from its $1$-jet, since for
the induced derivative $D$ on $TM$ we clearly have $0=\Lie_KD$ and
\begin{eqnarray*}
(\Lie_KD)_XY
&=& \Lie_K(D_XY) -D_{\Lie_KX}Y -D_X{\Lie_KY}\\
&=& D_KD_XY -D_{D_XY}K -D_{\Lieb{K,X}}Y -D_XD_KY +D_XD_YK\\
&=& R^D_{K,X}Y +D^2_{X,Y}K.
\end{eqnarray*}

\begin{remark}\label{Weylsunifiedtheory}
\summary{Weyl's unified theory of electromagnetism and gravity}
Einstein's theory of gravity models the spacetime by a four
dimensional manifold $M$ with a Lorentzian metric $g$. Its
divergence-free Ricci curvature, see remark \ref{Einsteinequation}, is
coupled to the energy density of the matter in
question. Electromagnetism is then modelled by an additional field $F$
on $M$ (which might contribute to the matter energy). In
contradistinction Weyl geometry provides a geometric model of
spacetime where the electromagnetic field is intrinsically represented
by the Faraday curvature $F^D$, see \cite{WeylRZM} and the next remark
\ref{Twinparadox}.
\end{remark}

\begin{remark}\label{Twinparadox}
\summary{Twin paradox} In case of Weyl geometry with Lorentzian
signature we say that the parameterization of a timelike worldline
$c\from\R\to M$ is \emph{proportional to the proper time} of the
observer $c$ if $\ip{D_{\dot c}\dot c,\dot c}=0$. If two different
worldlines $c_1$ and $c_2$ start at the same point $x=c_1(0)=c_2(0)$
with the same unit of proper time $\ip{\dot c_1,\dot c_1}(0)=\ip{\dot
c_2,\dot c_2}(0)$ and both are parameterized proportional to their
proper time, it is clear that if they meet again $y=c_1(t_1)=c_2(t_2)$
the two proper times $t_1$, $t_2$ will be different. This is called
the twin paradox. In the context of Weyl geometry with nontrivial
Faraday curvature $F^D$ not only are the two parameters $t_1$, $t_2$
different, but also the two time units $\ip{\dot c_1,\dot c_1}(t_1)$
and $\ip{\dot c_2,\dot c_2}(t_2)$ are different. Similar, a parallel
transported length scale along the two worldlines will be different at
$y$ even if it is the same at $x$.  The two twins starting at $x$ and
meeting at $y$ not only have different ages, but also different sizes,
and they will age at different rates. This dependence on past history
was the main reason why Weyl geometry as a unified theory of gravity
and electromagnetism was rejected. This does not rule out the
possibility of a very small coupling constant $\lambda$ between the
Faraday curvature $F^D$ and the corresponding electromagnetic field
$\lambda F = F^D$.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Elementary notions of representation theory}
\label{Elementarynotionsofrepresentationtheory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we recall that Lie groups and their Lie algebras can act 
on vector spaces by linear maps. This is called a representation and we 
will use the basic definitions in the next section only to give a unified 
treatment of all relevant vector bundles on a conformal manifold.
 
\begin{notation}
The group of linear isomorphisms preserving the conformal structure 
$\com$ will be denoted by 
$\CO(V):=\setof{A\in\GL(V)\property\com.A=\com}$. It is a direct product 
$\CO(V)=\R^+\id_V \cross \O(V)$ with one dimensional centre. Its Lie 
algebra is given by $\co(V)=\R\id_V \dsum \so(V)$.
\end{notation}

$\CO(V)$ and $\co(V)$ is an example of a Lie group with its Lie
algebra. In general let $G$ be a real Lie group and $\g$ be its real
(or complexified) Lie algebra.

\begin{definition}\label{Grouprepresentations}
\summary{Group representations} A real vector space $W$ is called a
\emph{right representation space} for $G$ if there is a group action
from the right $\Rep\from W\cross G\to W$ by linear maps. It will also
be denoted by $\Rep(w,A):=w.A$ and satisfies $w.A.B =w.(AB)$ for all
$A,B\in G$ and $w\in W$. A left action of $G$ on $W$ would be denoted
by $G\cross W\to W$ satisfying $A.B.w=(AB).w$. Every left action has
an associated right action defined by $w.A = A^{-1}.w$ and vice
versa. If $W$ is a right representation of $G$ then the dual space
$W^\dual$ carries a left $G$-representation (defined without inverting
group elements): $\ip{A.\omega,w}=\ip{\omega,w.A}$ for $\omega\in
W^\dual$ and all $w\in W$.
\end{definition}

\begin{definition}\label{Liealgebras}
\summary{Lie algebras} Let $\F$ be a field (we are interested in the
cases $\F=\R$ and $\F=\C$). A finite dimensional vector space $\g$
over $\F$ with a skew symmetric bilinear multiplication $\g \tens \g
\to \g$, denoted on $X,Y \in \g$ by $X \tens Y\mapsto \Lieb{X,Y}$ is
called a \emph{Lie algebra}, if it satisfies the \emph{Jacobi
identity}: for all $X,Y,Z \in \g$ we have $\Lieb{X,\Lieb{Y,Z}} +
\Lieb{Y,\Lieb{Z,X}} + \Lieb{Z,\Lieb{X,Y}} = 0$.
\end{definition}

\begin{remark}
If $G$ denotes a real Lie group and $\g$ the space of left invariant
vector fields on $G$, then $\g$ carries a real Lie algebra structure
induced by the Lie bracket of vector fields.
\end{remark}

If $\p \subset \g$ is a subspace, such that $X,Y \in \p$ implies
$\Lieb{X,Y} \in \p$, then $\p$ is called a \emph{subalgebra} of
$\g$. A linear map $\phi \from \g \to \lie{h}$ between two Lie
algebras is called a \emph{Lie algebra homomorphism}, if
$\phi(\Lieb{X,Y})=\Lieb{\phi(X),\phi(Y)}$ for all $X,Y \in \g$ (this
condition is nonlinear in $\phi$ similar to the orthogonality
condition for a linear transformation).

\begin{example}
The space of endomorphisms $\End(V)=\gl(V)$ of a vector space $V$ over
the field $\F$ carries a natural Lie algebra structure by $\Lieb{A,B}
:= A \circ B - B \circ A$ and is called the \emph{general linear Lie
algebra}.
\end{example}

\begin{definition}\label{Modules}
\summary{Modules} If $\g$ denotes a Lie algebra over $\F$, a vector
space $W$ is said to carry a \emph{right $\g$-module structure} or $W$
is called a \emph{right $\g$-representation space}, if there is a
linear map $\rep\from W\tens\g\to W$ denoted by $\rep(w\tens X)=:w.X$
such that for $Y\in \g$ we have $w.\Lieb{X,Y} = w.X.Y-w.Y.X$.
\end{definition}

A $\g$-representation from the left $\rep\from\g\tens W\to W$ will be
denoted by $\rep(X\tens w)=X.w$ and satisfies $\Lieb{X,Y}.w = X.Y.w -
Y.X.w$. Every left action has an associated right action defined by
$w.X=-X.w$ and vice versa. If $W$ is a right module then $W^\dual$ is
a left module defined (without signs) by $\ip{X.\omega,w}
=\ip{\omega,w.X}$ for $\omega\in W^\dual$ and all $w\in W$.

\begin{remark}
If $G$ denotes a real Lie group with Lie algebra $\g$. Let $W\cross
G\to W$ be a right group representation, then this induces a right
representation of the Lie algebra $W\tens \g\to W$.
\end{remark}

\begin{examples}\label{Adjointrepresentation}
\summary{Adjoint representation} For any Lie algebra $\g$ the Jacobi
identity shows, that the Lie bracket itself induces a left
representation on $\g$ itself, called \emph{adjoint representation}
$\ad\from\g\tens \g \to \g$, $\ad(X)Y:=\Lieb{X,Y}$. The kernel of the
adjoint representation is given by the centre $\kernel\ad =
\Centre(\g)$. The \emph{coadjoint representation} $\coad \from
\g\tens\g^\dual\to\g^\dual$ is given by: $\coad := -\ad^\dual$.
\end{examples}

\begin{constructions}
If $W\tens\g\to W$ is a finite dimensional right representation and
$X\in\g$ then the action $.X$ is an endomorphism of $W$. Any $\F$
multiple of $\trace(\ .X) \from \F \to \F$ defines a $1$-dimensional
representation. If $W_1\tens\g \to W_1$ and $W_2\tens\g \to W_2$ are
representations, then $\ .X\dsum\ .X$ induces a representation on the
direct sum $W_1 \dsum W_2$. Similarly $\ .X \tens \id_{W_2} +
\id_{W_1} \tens \ .X$ induces a representation on the tensor product
$W_1 \tens W_2$.
\end{constructions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bundles and curvature on conformal manifolds}
\label{Bundlesandcurvatureonconformalmanifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we construct all relevant vector bundles on a conformal 
manifold. Riemann's curvature tensor can be defined in Weyl geometry as 
in Riemannian geometry and it defines an obstruction against local 
flatness. 

\begin{construction}\label{associatedvectorbundles}
\summary{Associated vector bundles} Let $M$ be a real $n$-dimensional
conformal manifold (modelled on $V$). Denote by $\CO(M)\to M$ the
bundle of conformal frames, i.e.~the set of all orthogonal frames of
equal length: for $x\in M$ we have $\CO(M)_x=\setof{f\from V\to
T_xM\property f\ \mathrm{linear}, f^\dual\com_x=\com}$. The group
$\CO(V)$ acts naturally on $\CO(M)$ from the right. A right
representation $\Rep\from E\cross \CO(V)\to E$ leads to a vector
bundle over $M$ by the \emph{associated bundle construction}:
$EM:=\CO(M)\cross_\Rep E:=\setof{\setof{(fA,e.A) \property A\in\CO(V)}
\property f\in\CO(M), e\in E}$.
\end{construction}

If $E$ and $F$ are $\CO(V)$-representation with associated bundles
$EM$ and $FM$ and $\phi\from E\to F$ is a $\CO(V)$-equivariant map,
then $\phi$ induces a bundle map $\phi\from EM\to FM$. The action of
the Lie algebra $E\tens \co(V)\to E$ is an example leading to the
bundle map $EM\tens \co(TM) \to EM$.

\begin{discussion}
Let $M$ be a conformal manifold with an associated vector bundle
$EM\to M$ coming from a $\CO(V)$-representation space $E$. Any Weyl
derivative $D$ on the line bundle $L^1$ induces a covariant derivative
on the tangent bundle $TM$ (which is torsion-free and compatible with
the conformal structure). Hence $D$ induces a connection on the
conformal frame bundle $\CO(M)$. Therefore $D$ also induces covariant
derivatives on all associated bundles $EM$. These derivatives are also
denoted by $D\from C^\infty(M,EM) \to C^\infty(M,T^\dual M\tens EM)$.
The target bundle $T^\dual M\tens EM$ is associated to $V^\dual\tens
E$.
\end{discussion}

\begin{discussion}\label{CurvatureinWeylgeometry}
\summary{Curvature in Weyl geometry} Let $M$ be a conformal manifold
and $D$ a Weyl derivative. Let $EM$ be an associated vector bundle
with induced derivative $D$. The curvature of $EM$ is a section
$R^{E,D}\in C^\infty(M,\Lambda^2 T^\dual\tens\gl(EM))$ defined on
vector fields $X$, $Y$ and $e\in C^\infty(M,EM)$ by
$$
R^{E,D}_{X,Y}e := D_XD_Ye - D_YD_Xe - D_{\Lieb{X,Y}}e.
$$
It is given by the Lie algebra action of the curvature
$R^D:=R^{T,D}\in C^\infty(M,\Lambda^2 T^\dual\tens\co(TM))$ of the
tangent bundle as $R^{E,D}_{X,Y}e = R^D_{X,Y}.e$.

Since $D$ is torsion-free $R^D$ satisfies the first Bianchi identity:
$0=R^D_{X,Y}Z +R^D_{Y,Z}X +R^D_{Z,X}Y$. Two tensorial contractions of
$R^D$ are possible. The contraction of the last two indices gives back
the \emph{Faraday curvature}:
$$
F^D(X,Y)=\oneover{n} \trace R^D_{X,Y}.
$$
The \emph{Ricci curvature} $\ricci\in C^\infty(M,T^\dual M\tens
T^\dual M)$ is defined to be the (natural) trace of the curvature of
the weightless tangent bundle $R^{L^{-1}\tens T,D}=
R^{D}-F^D\tens\id$:
$$
\ricci^D(Y,Z):= \trace(R^{L^{-1}\tens T,D}_{\ ,Y}Z)
             =  \trace( R^D_{\ ,Y}Z - F^D(\ ,Y)Z ).
$$
Since $D$ is torsion-free, the skew part of $\ricci^D$ is determined
by $F^D$ as follows:
$$
\ricci^D(Y,Z)-\ricci^D(Z,Y) = -(n-2)F^D(Y,Z).
$$
\end{discussion}

\begin{remark}
In addition to the curvature terms in Riemannian geometry, the Faraday
curvature $F^D$ occurs as a new ingredient. It is the local
obstruction of finding a parallel metric in the conformal class.
\end{remark}

In the linear algebraic context of conformal vector spaces we will
define a $\CO(V)$-equivariant projection from $\gl(V)\to\co(V)$. In
the rest of this chapter it will occur as a useful short hand
notation. The reason why we denote this projection by a Lie bracket
will become clear in section \ref{SectionConformalKillingfields}:

\begin{definition}
For a conformal vector space $V$ we define a $\co(V)$-equivariant
map:
$$
\Lieb{,}\from V^\dual\tens V\to\co(V);\quad
\Lieb{\alpha,v}
:=\alpha(v)\id_V + \alpha\tens v - \flat v \tens \sharp \alpha.
$$
\end{definition}

This projection satisfies the following properties: since it has values
in $\co(V)$ it acts on any representation, moreover it has the
following symmetries: $v,w\in V$ and $\alpha,\beta\in V^\dual$
\begin{eqnarray*}
\Lieb{\alpha,v}.w &=&\Lieb{\alpha,w}.v\\
\Lieb{\alpha,v}.\beta &=&\Lieb{\beta,v}.\alpha.
\end{eqnarray*}

\begin{proposition}\label{Splittingthecurvature}
\summary{Splitting the curvature} There is a natural splitting of
$R^{E,D}$ into two parts:
$$
R^{E,D}_{X,Y}e
= (\Weylcurv_{X,Y}-\Lieb{\nric^D(X),Y}+\Lieb{\nric^D(Y),X}).e,
$$
where
$$
\nric^D\in C^\infty(M,T^\dual M\tens T^\dual M)
$$
is the \emph{normalized Ricci bilinear form}, and
$$
\Weylcurv\in C^\infty(M,\Lambda^2T^\dual M\cartan \so(TM))
$$
is the \emph{Weyl curvature tensor}. The fact, that $\Weylcurv$ takes 
values in the Cartan tensor product means, that it satisfies the first 
Bianchi identity (separately) and all traces applied to $\Weylcurv$
vanish. Hence, in two and three dimensions the Weyl curvature is zero.
\end{proposition}

\proof The canonical Ricci curvature mostly used in Riemannian
geometry was defined above by $\ricci^D(Y,Z):= \trace(R^D_{\
,Y}Z-F^D(\ ,Y)Z)$. The so called \emph{normalized Ricci curvature}
$\nric^D$ is much more appropriate in conformal geometry and related
to the canonical Ricci as
$$
\ricci^D = (n-2)\nric^D + \nscal^D\com.
$$
Here we use the splitting of $V^\dual\tens V^\dual$ into symmetric
trace-free, trace and skew part:
$$
\nric^D = \nric_0^D + \oneover{n}\nscal^D \com - \oneover{2} F^D,
$$
where 
$$
2(n-1) \nscal^D
:= \trace_{\com} \ricci^D
$$
denotes the scalar curvature and
$$
(n-2) \nric_0^D(Y,Z)
:= \oneover{2} \ricci^D(Y,Z) + \oneover{2}\ricci^D(Z,Y)
   -\oneover{n}\trace_{\com}(\ricci^D)\ip{Y,Z}
$$
denotes the symmetric and trace-free Ricci curvature. In $n=2$
dimensions scalar and Faraday curvature together determine the whole
curvature tensor, hence in that case $\nric^D = \oneover{2}\nscal^D
\com - \oneover{2} F^D$. We refer to Calderbank
\cite{CalderbankMoebius} for more details. In $n=3$ dimensions the
Ricci curvature determines the whole curvature tensor. The Ricci
curvature written as curvature tensor $-\Lieb{\nric^D(X),Y}.Z +
\Lieb{\nric^D(Y),X}.Z$ satisfies the first Bianchi identity
(separately), hence so does $\Weylcurv$.  \qed
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Changing the Weyl derivative}
\label{ChangingtheWeylderivative}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Recall that a Weyl derivative is a covariant derivative of the line
bundle $L^1$. The set of all Weyl derivatives forms an affine space
modelled on the space of $1$-forms. On a conformal manifold any Weyl
derivative induces a covariant derivative on the tangent bundle. Let
$\Nabla^D$ be a linear differential operator defined using a Weyl
derivative $D$ and the conformal structure $\com$. The curvature
tensors $R^D$ and $\nric^D$ are particular (zero order) examples of
this situation. The aim of this section is to study the behavior of
such differential operators under a change of the Weyl
derivative. Such operators which are independent of the choice of Weyl
derivative are \emph{conformally invariant}. We adapt Branson's method
to Weyl derivatives, which reduces the question of conformal
invariance to a linear question, and more generally, provides a method
for computing explicitly the dependence on $D$ of an operator which is
polynomial in $D$ and its jets. We will recall that the Weyl part of
the curvature tensor only depends upon the conformal class.

\begin{proposition}\label{LinearizedKoszulformula}
\summary{Linearized Koszul formula} Let $D$ and $\other{D}$ be two
covariant derivatives in $L^1$ and $\christo\in C^\infty(M,T^\dual M)$
the $1$-form such that $\other{D}-D = \christo$. For $X,Y\in
C^\infty(M,TM)$ the Koszul formula
\ref{Fundamentaltheoremofconformalgeometry} determines the change of
the induced covariant derivatives in the tangent bundle to be:
$$
\other{D}_{X}Y-D_{X}Y
= \Lieb{\christo,X}.Y
:=  \christo(X)Y + \christo(Y)X -\cip{X,Y}\sharp\christo.
$$
\end{proposition}

\begin{corollary}
Let $EM$ be a vector bundle associated to the conformal frame bundle
and $e\in C^\infty(M,EM)$ a section, then the induced change of
covariant derivatives induced by $\christo\in C^\infty(M,T^\dual M)$
is given by the Lie algebra action as
$$
\other{D}_{X}e-D_{X}e = \Lieb{\christo,X}.e = - e.\Lieb{\christo,X}.
$$
\end{corollary}

\begin{discussion}\label{Fundamentaltheoremofcalculus}
\summary{Fundamental theorem of calculus} Let $\Nabla^D$ be a linear
differential operator defined using a Weyl derivative $D$ and the
conformal structure $\com$. If $D(s)$ is a smooth curve
of Weyl derivatives parameterized by $s\in [0,1]$ then $\Nabla^{D(s)}$
defines a curve of operators. The difference
$\Nabla^{D(1)}-\Nabla^{D(0)}$ can be calculated using the fundamental
theorem of calculus
$$
\Nabla^{D(1)}-\Nabla^{D(0)} 
= \int_{[0,1]} \delbydel{s} \Nabla^{D(s)}\, d\sigma.
$$
The velocity of $D(s)$ will be called $\christo:=\delbydel{s}D$. Hence
$\Nabla^D$ is independent of the choice of $D$ iff the derivative
$(\LieWeyl_\christo \Nabla)^D := \delbydel{s} \Nabla^{D(s)}$ vanishes for
all $\christo\in C^\infty(M,T^\dual M)$ and all Weyl derivatives
$D$. Hence the proof that a construction depending on a choice of Weyl
derivative is indeed conformally invariant can be simplified to a
linear situation.
\end{discussion}

\begin{examples}\label{Firstorderchange}
\summary{First order} The Weyl derivative itself, the exterior
derivative and the exterior divergence are first order operators
depending on a Weyl derivative: the induced derivative $D$ on an
associated bundle $EM$ itself depends affinely on the Weyl derivative
$D$ by the Koszul formula, and we simply have
$$
\LieWeyl_\christo D = \Lieb{\christo,\ }.
$$
Since the exterior derivative, see applications
\ref{Twistedexteriorderivative}, on multilinear forms of arbitrary
central weight $w$,
$$
\dR^D\from C^\infty(M,L^{w+k}\tens \Lambda^k T^\dual M) \to
   C^\infty(M,L^{w+k}\tens \Lambda^{k+1} T^\dual M),
$$
is simply the invariant $\dR $ twisted by the Weyl derivative on
$L^{w+k}$, we get
$$
\LieWeyl_\christo \dR^D = (w+k)\, \christo \wedge. 
$$
Similarly for the exterior divergence, see applications
\ref{Twistedexteriordivergence}, on multivector densities of arbitrary
central weight $w$,
$$
\divR^D\from C^\infty(M,L^{w-k}\tens \Lambda^k TM) \to
   C^\infty(M,L^{w-k}\tens \Lambda^{k-1} TM),
$$
we have
$$
\LieWeyl_\christo \divR^D = (w-k+n)\, \christo \inner. 
$$
\end{examples}

\begin{examples}\label{Secondorderchange}
\summary{Second order} For the induced second derivative
$D^2_{X,Y}e:=D_XD_Ye-D_{D_XY}e$ on an associated $EM$ we find
\begin{eqnarray*}
\LieWeyl_\christo( D^2)_{X,Y}e 
&=&  (\LieWeyl_\christo D)_X D_Ye + D_X ( (\LieWeyl_\christo D)_Ye) 
      -(\LieWeyl_\christo D)_{D_XY}e - D_{(\LieWeyl_\christo D)_XY}e\\
&=&  (\LieWeyl_\christo D)_X D_Ye + (\LieWeyl_{D_X\christo} D)_Ye) 
      + (\LieWeyl_\christo D)_Y D_Xe - D_{(\LieWeyl_\christo D)_XY}e\\
&=&  \Lieb{\christo,X}.D_Ye + \Lieb{D_X\christo,Y}.e 
      + \Lieb{\christo,Y}.D_Xe - D_{\Lieb{\christo,X}.Y}e.
\end{eqnarray*}
In particular the curvature tensor $R^{E,D}$ is defined to be the skew
part of the second derivative which simply gives
$$
(\LieWeyl_\christo R^{E,D})_{X,Y}e 
=  \Lieb{D_X\christo,Y}.e - \Lieb{D_Y\christo,X}.e.
$$
\end{examples}

\begin{application}
Notice that the curvature tensor only changes in its Ricci part, which
already shows that the Weyl curvature is conformally invariant,
i.e.~$\Weylcurv$ only depends upon the conformal structure.
\end{application}

\begin{example}\label{LinearchangeofRiccicurvature}
\summary{Linear change of Ricci curvature} More explicitly we can
calculate the linear change of the Faraday and Ricci curvature terms:
\begin{eqnarray*}
(\LieWeyl_\christo F^D)_{X,Y} 
&=& \trace (\Lieb{D_X\christo,Y} - \Lieb{D_Y\christo,X})\\
&=& \dR \christo(X,Y),\\
(\LieWeyl_\christo \ricci^D)_{Y,Z} 
&=& \trace(\LieWeyl_\christo R^D_{\ ,Y}Z-\LieWeyl_\christo F^D(\ ,Y)Z)\\ 
&=& \trace(\Lieb{D\christo,Y}.Z - \Lieb{D_Y\christo,\ }.Z) 
    - \dR \christo(Z,Y)\\
&=& (2-n) D_Y\christo(Z) - \cip{Y,Z}\divR^D\christo,\\ 
(\LieWeyl_\christo \nric^D)_{Y,Z}
&=& - D_Y\christo(Z).
\end{eqnarray*}
\end{example}

I like to thank D.~Calderbank for pointing out to me the following
observation:

\begin{discussion}\label{Taylorstheorem}
\summary{Taylor's theorem of calculus} Let $\Nabla^D$ be again a linear
differential operator defined using a Weyl derivative $D$ and the
conformal structure $\com$. For a smooth curve $D(s)$ of Weyl
derivatives parameterized by $s\in [0,1]$ we defined the first
derivative at $D=D(0)$ by $(\LieWeyl_\christo \Nabla)^D:= \delbydel{s}
\Nabla^{D(s)}$ with $\christo=D'(0)$. Similarly we can define higher
derivatives as
$$
(\LieWeyl^2_{\christo,\christo} \Nabla)^D
= \delbydel{s} (\LieWeyl_\christo \Nabla)^{D(s)}, \hbox{\ etc.}
$$
Since $\Nabla^D$ is assumed to be of finite order, all higher
derivatives will vanish eventually. Then Taylor's theorem applies to
give
$$
\Nabla^{D+\christo} = \Nabla^D
+ (\LieWeyl_\christo \Nabla)^D 
+ \oneover{2} (\LieWeyl^2_{\christo,\christo} \Nabla)^D 
+ \oneover{3!} (\LieWeyl^3_{\christo,\christo,\christo} \Nabla)^D + \ldots\ .
$$ 
\end{discussion}

\begin{example}
\summary{Ricci curvature} We can calculate the change of the Ricci
curvature as quadratic polynomial in $\christo$: we only need to
differentiate $(\LieWeyl_\christo \nric^D)_{Y,Z} = - D_Y\christo(Z)$
once more to obtain $(\LieWeyl^2_{\christo,\christo} \nric^D)_{Y,Z}
=-\Lieb{\christo,Y}.\christo(Z)$ and finally
\begin{eqnarray*}
\nric^{D+\christo}(Y,Z)
&=& \nric^D(Y,Z) 
- D_Y\christo(Z) + \oneover{2} \christo(\Lieb{\christo,Y}Z).
\end{eqnarray*}
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Differential Bianchi identities}
\label{DifferentialBianchiidentities}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Riemann curvature tensor of a covariant derivative $D$ of a vector
bundle $EM$ satisfies a first order integrability condition,
proposition \ref{Bianchiidentity}. Since there is no prefered
derivative on a conformal manifold we will construct conformally
invariant operators annihilating the Weyl curvature tensor. In $n=3$
dimensions the Weyl curvature vanishes and the Cotton York curvature
defined below is a first invariant obstruction against local flatness
in three dimensions. In this section we will follow Gauduchon's notes
\cite{GauduchonNotes}, personal communications with Calderbank and his
paper \cite{CalderbankFaraday}.

\begin{definition}\label{CottonYorktensor}
\summary{Cotton York tensor} For any Weyl derivative $D$ on a
conformal manifold $M$ define the \emph{Cotton York tensor}
$\Cotton^D$ to be the skew derivative of the normalized Ricci
curvature: $\Cotton^D := \dR^D\nric^D$, where $\nric^D$ is viewed as a
covector valued $1$-form and $\Cotton^D$ is a covector valued
$2$-form. On vector fields $X$, $Y$, $Z$ we have
$$
\Cotton^D_{X,Y}(Z) := D_X\nric^D(Y,Z)-D_Y\nric^D(X,Z).
$$
\end{definition}

\begin{proposition}
The linear change of the Cotton York tensor under a change of Weyl
derivative is given by the Weyl curvature tensor as:
$$
(\LieWeyl_\christo\Cotton)^D = \christo(\Weylcurv).
$$
Since the linear change is independent of the Weyl derivative $D$ we
can even conclude $\Cotton^{D+\christo}-\Cotton^D
=\christo(\Weylcurv)$.
\end{proposition}

\proof The calculation is a straight forward application of the method
developed in the previous chapter:
\begin{eqnarray*}
(\LieWeyl_\christo\Cotton)^D_{X,Y} 
&=& \Lieb{\christo,X}.\nric^D(Y) - \Lieb{\christo,Y}.\nric^D(X)
- D^2_{X,Y}\christo + D^2_{Y,X}\christo\\
&=& \Lieb{\nric^D(Y),X}.\christo - \Lieb{\nric^D(X),Y}.\christo
- R^D_{X,Y}.\christo\\
&=& -\Weylcurv_{X,Y}.\christo.
\end{eqnarray*}
The final remark is elementary calculus
e.g.~paragraph \ref{Fundamentaltheoremofcalculus} or \ref{Taylorstheorem}.\qed

\begin{corollary}
The Cotton York tensor is conformally invariant in $n=3$ dimensions
since then the Weyl curvature vanishes. Hence in three dimensions the
Cotton York tensor is an invariant obstruction against conformal
flatness.
\end{corollary}

\begin{discussion}\label{SecondBianchiidentity}
\summary{Second Bianchi identity} The differential Bianchi identity of
the Riemann curvature tensor $R^D$ of the tangent bundle of a Weyl
manifold reads:
$$
0 = \dR^D R^D_{X,Y,Z}:= D_XR^D_{Y,Z} + D_YR^D_{Z,X} + D_ZR^D_{X,Y}.
$$
With the splitting $R^D_{X,Y} =\Weylcurv_{X,Y} -\Lieb{\nric^D(X),Y}
+\Lieb{\nric^D(Y),X}$ in proposition \ref{Splittingthecurvature} of
$R^D$ into Weyl and Ricci part we can rewrite the Bianchi identity
using the Cotton York tensor:
$$
\dR^DR^D_{X,Y,Z} = \dR^D \Weylcurv_{X,Y,Z} 
- \Lieb{\Cotton^D_{X,Y},Z}
- \Lieb{\Cotton^D_{Y,Z},X}
- \Lieb{\Cotton^D_{Z,X},Y}.
$$
A first contraction of this expression gives a covector valued
$2$-form (note $\trace\Weylcurv =0$ and $\dR F^D=0$):
\begin{eqnarray*}
0 &=& \trace \dR^DR^D_{Y,Z} \\
%&=& \trace D\Weylcurv_{Y,Z} 
%- (n-2)\Cotton_{Y,Z} 
%+ D_Y\nric^D(\ ,Z) - D_Z\nric^D(\ ,Y)
%- D\nric^D(Y,Z) + D\nric^D(Z,Y)
%+ \,\flat Z\tens \trace_\com D\nric^D(Y) - \,\flat Y\tens \trace_\com D\nric^D(Z)
%- \flat Z \tens D_Y\nscal^D + \,\flat Z\tens D_Z\nscal^D\\
&=& \trace D\Weylcurv_{Y,Z} 
- (n-3)\Cotton_{Y,Z}\\ 
&&
+ \,\flat Z\tens \trace_\com D\nric^D(Y) - \flat Y\tens \trace_\com D\nric^D(Z)
- \flat Z \tens D_Y\nscal^D + \,\flat Y\tens D_Z\nscal^D.
\end{eqnarray*}
One further contraction using $\com$ shows (note $\trace_\com
\Weylcurv = 0$):
\begin{eqnarray*}
0 &=& - (n-3)\trace_\com D\nric^D(Z) + (n-3) D_Z\nscal^D\\ 
&&
+ \trace_\com D\nric^D(Z) - n \trace_\com D\nric^D(Z)
- D_Z\nscal^D + n D_Z\nscal^D\\
&=& - (2n-4)\trace_\com D\nric^D(Z) + (2n-4) D_Z\nscal^D.
\end{eqnarray*}
We summarize these calculations in the following:
\end{discussion}

\begin{proposition}\label{ContractedBianchiidentity}
\summary{Contracted Bianchi identity}
The derivative of the normalized scalar curvature is a divergence of
the normalized Ricci curvature:
$$
\trace_\com D\nric^D(Z) = D_Z\nscal^D,
$$
and a multiple of the Cotton York tensor is a divergence of the Weyl
curvature:
$$
\divR^D\Weylcurv_{Y,Z} := \trace D\Weylcurv_{Y,Z} = (n-3)\Cotton_{Y,Z}.
$$  
\end{proposition}

\begin{remark}
The first equation means that the Cotton York tensor is trace-free and
by $\dR F^D=0$ it is also alternating-free, hence we have identified
$\Cotton^D$ as a section of $\Cotton^D\in
C^\infty(M,\Lambda^2T^\dual\cartan T^\dual M)$.
\end{remark}

\begin{remark}\label{Einsteinequation}
\summary{Einstein equation} In closed Weyl geometry, i.e.~$F^D=0$, the
Ricci curvature is symmetric and $\nric^D-\nscal^D\com$ is a symmetric
bilinear form, which is divergence-free by the above contracted Bianchi
identity. General relativity models spacetime by a $n=4$ dimensional
conformal manifold of Lorentzian signature together with a closed
(usually exact) Weyl derivative $D$. The geometry is coupled to the
matter in question by the \emph{Einstein equation}
$\nric^D-\nscal^D\com = T$ where $T$ is the central weight $-2$ symmetric
and divergence-free energy momentum stress density tensor of the
matter. The dimensional analysis uses Galilei's constant as was
explained in paragraph \ref{Einsteinconstants}.
\end{remark}

By the second part of the above proposition
\ref{ContractedBianchiidentity} together with paragraph
\ref{SecondBianchiidentity}, it is clear that the Weyl curvature tensor
satisfies its own first order Bianchi identity in $n\geq 5$
dimensions: $0 =\Bianchi(\Weylcurv)$ with the operator $\Bianchi$
given by the following:

\begin{proposition}\label{FirstorderBianchiidentity}
\summary{First order Bianchi identity} The following linear first
order differential operator
$$
\Bianchi\from 
C^\infty(M,\Lambda^2T^\dual \cartan\so(TM))
\to C^\infty(\Lambda^3T^\dual\cartan \so(TM)),
$$
defined on $W\in C^\infty(M,\Lambda^2T^\dual \cartan\so(TM))$ by 
$$\Bianchi(W)_{X,Y,Z}
:= \dR^D W_{X,Y,Z} 
- \oneover{n-3}
(\Lieb{\divR^D W_{X,Y},Z} 
+ \Lieb{\divR^D W_{Y,Z},X}
+ \Lieb{\divR^D W_{Z,X},Y}),
$$
is nonzero and conformally invariant in $n\geq 5$ dimensions. It
provides an integrability condition for the Weyl curvature tensor in
$n\geq 5$ dimensions: $0=\Bianchi(\Weylcurv)$.
\end{proposition}

\proof That this definition is independent of the choice of the Weyl
derivative $D$ is a lengthy but elementary calculation. It is also a
special case of Fegan's result, theorem
\ref{Fegansfirstorderconformalinvariants}. In $n=4$ dimensions this
operator is zero. \qed

The next aim will be to find a conformally invariant second order
Bianchi identity in $n=4$ dimensions.

\begin{proposition}\label{ThirdBianchiidentity}
\summary{Third Bianchi identity}
The Cotton York tensor viewed as a covector valued $2$-form satisfies
on vector fields $X$, $Y$, $Z$ and $U$ the following equation:
$$
\dR^D \Cotton^D_{X,Y,Z}(U) 
+ \nric^D(X,\Weylcurv_{Y,Z}U)
+ \nric^D(Y,\Weylcurv_{Z,X}U)
+ \nric^D(Z,\Weylcurv_{X,Y}U) = 0.
$$
\end{proposition}
\proof Applying $\dR^D$ to $\dR^D\nric^D$ gives $R^D.\wedge\nric^D$,
i.e.~cyclic permutations like:
\begin{eqnarray*}
\dR^D\Cotton^D_{X,Y,Z}
&=& (R^D_{X,Y}.\nric^D) (Z) + \cyclic(X,Y,Z)\\
&=&  R^D_{X,Y}.(\nric^D(Z)) + \cyclic(X,Y,Z),
\end{eqnarray*}
by the first Bianchi identity for $R^D$. The splitting of $R^D$ gives
$$
R^D_{X,Y}.(\nric^D(Z))
= \Weylcurv_{X,Y}.(\nric^D(Z))
- \Lieb{\nric^D(X),Y}.(\nric^D(Z)) + \Lieb{\nric^D(Y),X}.(\nric^D(Z)).
$$
Summing the cyclic permutations of the last two terms gives zero
because $\Lieb{\alpha,v}.\beta$ is symmetric in $\alpha$ and $\beta$.\qed

%\begin{corollary}
%\summary{Contracted third Bianchi identity}.
%Taking a trace over $\com$ gives
%$$
%\trace_\com D \Cotton^D_{X,Y} 
%- {1\over 2} \Sum_i F^D(\sharp\theta^i,\Weylcurv_{X,Y}t_i)
%=0,
%$$
%with a dual basis $t_i$, $\theta^i$ of $TM$.
%\end{corollary}
%
\begin{remark}
In three dimensions the Weyl curvature vanishes and the Cotton York
tensor is conformally invariant. Hence the above leads to a first
order integrability condition $\dR\Cotton=0$ for the Cotton York
tensor, where the operator $\dR\from
C^\infty(M,\Lambda^2T^\dual\cartan T^\dual)\to C^\infty(M,\Lambda^3
T^\dual\cartan T^\dual)$ is also conformally invariant (either
by an elementary calculation or by theorem
\ref{Fegansfirstorderconformalinvariants}).
\end{remark}

The contracted Bianchi identity $(n-3)\Cotton^D_{X,Y}U =
\divR^D\Weylcurv_{X,Y}U$ together with the third Bianchi identity
gives:

\begin{corollary}\label{SecondorderBianchiidentity}
The Weyl curvature tensor is in the kernel of a second order
differential operator:
$$
(\dR^D \divR^D \Weylcurv)_{X,Y,Z}U 
+ (n-3)( \nric^D(X,\Weylcurv_{Y,Z}U)
+ \nric^D(Z,\Weylcurv_{X,Y}U) 
+ \nric^D(Y,\Weylcurv_{Z,X}U) )
= 0.
$$
\end{corollary}

In $n=4$ this is a conformally invariant Bianchi identity since the
above operator is conformally invariant which we will prove in
proposition \ref{Simplesecondorderdifferentialoperators}, see also
\cite{BransonSecond}:

\begin{proposition}\label{FourdimensionalBianchioperator}
\summary{Second Order Bianchi identity} For $n=4$ dimensions there is
a second order conformally invariant differential operators:
$\Bianchi\from C^\infty(M,\Lambda^2 T^\dual\cartan\so(TM)) \to
C^\infty(M,\Lambda^3 T^\dual\cartan T^\dual M)$ given on an
endomorphism valued $2$-form $W$ by
$$
\Bianchi(W)_{X,Y,Z}
:= (\dR^D \divR^D W)_{X,Y,Z} 
+ \nric^D(X,W_{Y,Z}) + \nric^D(Y,W_{Z,X}) + \nric^D(Z,W_{X,Y}).
$$
It provides an integrability condition for the Weyl curvature tensor
in $n=4$ dimensions $\Bianchi(\Weylcurv)=0$.
\end{proposition}

\begin{remark}
We have presented a number of integrability conditions on Weyl
manifolds and conformal manifolds. As was remarked before the Bianchi
identity provides a motivation for the Einstein equation in the
Riemannian case. It is not clear to us how to couple matter to Weyl's
geometric theory of gravity and electromagnetism (briefly described in
remark \ref{Weylsunifiedtheory}), i.e.~it is not clear how to relate
the curvature invariants in Weyl geometry to the matter energy
density.
\end{remark}

We will finish this section by introducing the \emph{Bach tensor}
$\Bachtensor$ in $n=4$ dimension which is a curvature invariant of
quartic order. In fact $\Bachtensor$ is obtained by applying a
conformally invariant second order operator to the Weyl curvature
tensor. The invariance of the following operator will be proved in
proposition \ref{Simplesecondorderdifferentialoperators}, see also
\cite{BransonSecond}:

\begin{proposition}\label{Bachoperator}
In arbitrary dimensions $n\geq 3$ there is a second order conformally
invariant differential operator: $\Bach\from C^\infty(M,L^{4-n}\tens
\Lambda^2 T^\dual\cartan\so(TM)) \to C^\infty(M,L^{2-n}\tens
T^\dual\cartan T^\dual M)$ given on an endomorphism valued $2$-form
$W$ by
$$
\Bach(W)_{Y,U} := \Sum_{i,j} (D^2_{t_i,t_j} + \nric^D(t_i,t_j))
\cip{W(\theta^i,Y)U,\theta^j} + \swap(Y,U),
$$
where $t_i$, $\theta^i$ is a dual basis of $TM$. 
\end{proposition}

\begin{definition}\label{Bachtensor}
\summary{Bach tensor} On a four dimensional conformal manifold we
define the \emph{Bach tensor} to be $\Bachtensor:=\Bach(\Weylcurv)$,
which is a trace-free symmetric tensor of central weight $-4$.
\end{definition}

\begin{proposition}
The Bach tensor on a conformal four manifold is divergence-free:
$0=\divR\Bachtensor$, where $\divR\from C^\infty(M,L^{2-n}\tens
T^\dual\cartan T^\dual M) \to C^\infty(M,L^{-n}\tens T^\dual M)$ is
invariant, see theorem \ref{Fegansfirstorderconformalinvariants}.
\end{proposition}

\proof This is an application of the Bianchi identities. For
simplicity choose a flat Weyl derivative $F^D=0$, then $\nric^D$ is
symmetric. Weyl curvature and Cotton York tensor are trace-free and
alternating-free. For $n=4$ we have from proposition
\ref{ContractedBianchiidentity} that $\divR^D\Weylcurv = \Cotton^D$
and from proposition \ref{ThirdBianchiidentity} we know
$\Sum_iD_{t_i}\Cotton^D_{Y,Z}\sharp\theta^i = 0$.  A lengthy but
elementary calculation shows the result.\qed \hidden{
\begin{eqnarray*}
\divR\Bachtensor(U)
&=& \Sum_k D_{t_k}\Bachtensor(\sharp\theta^k,U)\\
&=& \Sum_{i,j,k} D^3_{t_k,t_i,t_j}
\ip{\Weylcurv_{\sharp\theta^i,\sharp\theta^k}U,\theta^j}
+ D^3_{t_k,t_i,t_j}
\ip{\Weylcurv_{\sharp\theta^i,U}\sharp\theta^k,\theta^j}\\
&&
+ \Sum_{i,j,k} 
D_{t_k}\nric^D(t_i,t_j)\ip{\Weylcurv_{\sharp\theta^i,\sharp\theta^k}U,\theta^j}
+ \nric^D(t_i,t_j)\ip{D_{t_k}\Weylcurv_{\sharp\theta^i,\sharp\theta^k}U,\theta^j}\\
&&
+ \Sum_{i,j,k} 
D_{t_k}\nric^D(t_i,t_j)\ip{\Weylcurv_{\sharp\theta^i,U}\sharp\theta^k,\theta^j}
+ \nric^D(t_i,t_j)\ip{D_{t_k}\Weylcurv_{\sharp\theta^i,U}\sharp\theta^k,\theta^j}\\
&=& \Sum_{i,k} D^2_{t_k,t_i}
\divR^D\Weylcurv_{\sharp\theta^i,\sharp\theta^k}U
+ D^2_{t_k,t_i}
\divR^D\Weylcurv_{\sharp\theta^i,U}\sharp\theta^k\\
&&
+ \Sum_{i,j,k} 
\Cotton^D_{t_k,t_i}t_j\ip{\Weylcurv_{\sharp\theta^i,\sharp\theta^k}U,\theta^j}
+ \Sum_{i,j}\nric^D(t_i,t_j)\divR^D\Weylcurv_{U,\theta^j}\sharp\theta^i\\
&&
- \Sum_{i,j}\nric^D(t_i,t_j)\divR^D\Weylcurv_{\sharp\theta^i,U}\sharp\theta^j\\
&=& \Sum_{i,k} D^2_{t_k,t_i}
\Cotton^D_{\sharp\theta^i,\sharp\theta^k}U
- D^2_{t_k,t_i}
\Cotton^D_{\sharp\theta^k,\sharp\theta^i}U
- D^2_{t_k,t_i}
\Cotton^D_{U,\sharp\theta^k}\sharp\theta^i\\
&&
+ \Sum_{i,j,k} 
\Cotton^D_{t_k,t_i}t_j\ip{\Weylcurv_{\sharp\theta^i,\sharp\theta^k}U,\theta^j}
+ \Sum_{i,j}\nric^D(t_i,t_j)\Cotton^D_{U,\theta^j}\sharp\theta^i\\
&&
- \Sum_{i,j}\nric^D(t_i,t_j)\Cotton^D_{\sharp\theta^i,U}\sharp\theta^j\\
&=& \Sum_{i,k} R^D_{t_k,t_i}.\Cotton^D_{\sharp\theta^i,\sharp\theta^k}U\\
&&
+ \Sum_{i,j,k} 
\Cotton^D_{t_k,t_i}t_j\ip{\Weylcurv_{\sharp\theta^i,\sharp\theta^k}U,\theta^j}
+ \Sum_{i,j}\nric^D(t_i,t_j)\Cotton^D_{U,\theta^j}\sharp\theta^i\\
&&
- \Sum_{i,j}\nric^D(t_i,t_j)\Cotton^D_{\sharp\theta^i,U}\sharp\theta^j\\
&=& -\Sum_{i,k}\Lieb{\nric^D(t_k),t_i}.\Cotton^D_{\sharp\theta^i,\sharp\theta^k}U
+\Sum_{i,k}\Lieb{\nric^D(t_i),t_k}.\Cotton^D_{\sharp\theta^i,\sharp\theta^k}U\\
&&
+ \Sum_{i,j}\nric^D(t_i,t_j)\Cotton^D_{U,\theta^j}\sharp\theta^i\\
&&
- \Sum_{i,j}\nric^D(t_i,t_j)\Cotton^D_{\sharp\theta^i,U}\sharp\theta^j\\
&=& -\Sum_{i,k}\Lieb{\nric^D(t_k),t_i}.\Cotton^D_{\sharp\theta^i,\sharp\theta^k}U
+\Sum_{i,k}\Lieb{\nric^D(t_i),t_k}.\Cotton^D_{\sharp\theta^i,\sharp\theta^k}U\\
&&
+ \Sum_{i,j}\nric^D(t_i,t_j)\Cotton^D_{U,\theta^j}\sharp\theta^i\\
&&
- \Sum_{i,j}\nric^D(t_i,t_j)\Cotton^D_{\sharp\theta^i,U}\sharp\theta^j\\
\end{eqnarray*}}

\begin{remark}\label{Bachstheoryofgravity}
\summary{Bach's theory of gravity} A couple of years after Weyl
published his unified theory of electromagnetism and gravity, Bach
obtained in \cite{Bach} his conformal curvature invariant
$\Bachtensor$ together with its conservation law $\divR \Bachtensor
=0$ on a $n=4$ dimensional conformal manifold. Using fundamental
constants as in paragraph \ref{Gravitationaldimensions} the Bach
tensor $\Bachtensor$ has the dimension $\Dim{energy}/\Dim{volume}$.
The field equation in Bach's conformal theory of gravity is $\Bachtensor
=T$ where $T$ is the conserved energy momentum stress tensor of the
matter in question. This theory is nonlinear in the conformal structure
$\com$ since $\com\mapsto \Weylcurv$ is nonlinear. Bach's field
equation is fourth order in the conformal structure. This was one
reason for most physicists to reject this theory. The induced
conformal structure of the Schwarzschild geometry solves the source
free Bach equation. On the other hand, the cosmological models due to
Friedmann Robertson Walker in Einstein's theory are conformally flat
and hence only provide trivial or flat solutions of Bach's equation.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Algebra of conformal geometry}
\label{Algebraofconformalgeometry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This chapter uses the Taylor expansion of smooth functions to
determine in section \ref{Liederivativesonaffinespace} and
\ref{SectionConformalKillingfields} all vector fields on an affine
conformal space $\A$ over $V$ which leave the (flat) conformal metric
$\com$ invariant. The space of these fields forms the conformal Lie
algebra $\g$, which is also called the \emph{M\"obius Lie
algebra}. The algebra structure provides a link between $V^\dual$, the
infinitesimal translations at infinity of $\A$, and the linear change
of a Weyl derivative on a curved conformal manifolds, see remark
\ref{RemarkChangeofWeylderivative}. The M\"obius algebra acts on
smooth polynomial functions through the Lie derivative, see paragraph
\ref{grepfromlrep}. The above mentioned link to linear changes of Weyl
derivatives guides the search for higher jet operators on conformal
manifolds, see proposition \ref{Linearchangeofthesecondjet}.

In section \ref{Tensorialtwistorsinconformalgeometry} we identify the
M\"obius Lie algebra as a linear Lie algebra $\g=\so(\asso V)$, where
$\asso V$ is a natural vector space associated to a conformal affine
space $\A$. This allows to use tensor products of
$\asso V$ as representations $W$ for $\g$. An element of such a
representation space is called a \emph{twistor} and it induces a
\emph{twistor field} on $\A$, i.e.~a polynomial with values in the
vector space $W_{V^\dual}$ of \emph{coinvariants} of $W$. Twistors and
twistor fields have a vivid history and play the key role in this
dissertation. They occur here in their most elementary context. The
inclusion $W\to C^\infty(\A,W_{V^\dual})$ is the beginning of the
conformal Bernstein Gelfand Gelfand complex, see theorem
\ref{ConformalBGGcomplex}. In the last section
\ref{Projectivelightcone} we will identify the conformal sphere as a
homogeneous space.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lie derivatives on affine space}
\label{Liederivativesonaffinespace}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\A$ be an affine space modelled on a vector space $V$. Let $F$ be
a finite dimensional right representation space of $\gl(V)$
(convention: for $A,B\in\gl(V)$ we have $\Lieb{A,B}=A\circ B- B\circ
A$). Denote by $f\in C^\infty(\A,F)$ a function on $\A$ with values in
$F$ (i.e.~a section of $F$). If $F\tens\gl(V)\to F$ comes from a group
representation $F\cross\GL(V)\to F$ and if $\phi\from A \to A$ is a
(local) diffeomorphism, then the \emph{pull back} of $f$ with respect
to $\phi$ is defined to be $\phi^\dual f(x) :=
f(\phi(x)).(\del\phi\restr{x})$. For any vector field $X\in
C^\infty(\A,V)$ let $\phi_t$ be its local flow, i.e.~${\del\over\del
t} \phi_t(x) = X\restr{\phi_t(x)}$. We define the \emph{Lie derivative
of $f$ in the direction of $X$} by (convention):
$$
\Lie_Xf := {\del\over\del t}\restr{t=0} {\phi_t}^\dual f
        = \del_X f + f.\del X.
$$
Here $\del$ denotes the affine derivative on $\A$ in $V$ and $F$. In
particular, if $Y\in C^\infty(\A,V)$ is another vector field, then
$\Lie_XY=\del_XY - \del_YX$. The space of smooth vector fields forms a
Lie algebra under $\Lie$. This Lie algebra structure turns the space
of sections with the above Lie derivative $\Lie$ into a left
representation of $C^\infty(\A,V)$, since
$$
\Lie_X\Lie_Y f - \Lie_Y\Lie_X f = \Lie_{\Lie_XY} f.
$$
In what follows we like $\Lie$ to be a right representation of
$C^\infty(\A,V)$ hence we will use in this section the opposite Lie
bracket between smooth vector fields: $\Lieb{X,Y} :=-\Lie_XY
=\del_YX-\del_XY$. With these convention every right $\gl(V)$
representation $F$ induces a right $C^\infty(\A,V)$-representation:
$$
\Lie\from C^\infty(\A,F)\tens C^\infty(\A,V) \to C^\infty(\A,F).
$$

\begin{discussion}
\summary{Taylor expansion} We fix a point $x\in\A$. Any smooth section
$f\in C^\infty(\A,F)$ has a Taylor expansion at $x$: if $v\in V$
denotes a direction then $(f + \del_vf + \ldots +
\oneover{k!}\del^k_{v^k}f)\restr{x}$ is a polynomial in $v$ of degree
$k$. We assume, that $F$ has a certain central weight $w\in\R$. We
will label tensors in $\Sym^k(V^\dual)\tens F$ by their central weight
$w-k$. An element
$$
f=(f_w,f_{w-1},\ldots,f_{w-k}) \in \Sym(V^\dual) \tens F := (\R \dsum 
V^\dual \dsum \Sym^2V^\dual \dsum \ldots) \tens F
$$
can be viewed as an $F$-valued polynomial on $V$ (of finite degree
$k$) given by ($e^v=1+v+\oneover{2}v^2+\ldots$):
$$
v\mapsto f(e^v)
:= f_{w} + f_{w-1}(v) + \ldots + \oneover{k!} f_{w-k}(v^k).
$$
The Lie derivative of smooth sections in the direction of smooth vector 
fields $\Lie_Xf=\del_Xf + f.\del X$ induces a natural right action of 
polynomial vector fields $X=(X_1,X_0,\ldots)\in\Sym(V^\dual)\tens V$ on 
polynomial sections:
$$
(\Sym(V^\dual)\tens F)\ \tens\ (\Sym(V^\dual)\tens V) 
\to \Sym(V^\dual)\tens F,
$$
given by $f\tens X \mapsto \Lie_Xf$. We use the symmetric algebra 
$\Sym(V)$ where the product of $a$ and $b$ is denoted by $ab$ and $e^v$ 
is viewed as a formal element $e^v=1+v+\oneover{2}v^2+\ldots$. Then we 
find:
$$
\Lie_X f (e^v) = f(X(e^v)e^v) + f(e^v).(X(e^v)).
$$
\end{discussion}

The Lie derivative on smooth vector fields $C^\infty(V,V)$ induces a
Lie algebra structure on $\Sym(V^\dual) \tens V$. Note, that the Lie
algebra $\Sym(V^\dual) \tens V$ comes with a natural grading, such
that elements in $\Sym^k(V^\dual)\tens V$ are labled by $1-k$. We will
specialize the above formula to the Lie bracket $\Lieb{X,Y}=-\Lie_XY$
(note that the right action of $\gl(V)$ on $V$ is given by
$v.A=-A(v)$):
\begin{eqnarray*}
\Lieb{X_1,Y_1} &=& 0,\\
\Lieb{X_0,Y_1} &=& X_0(Y_1),\\
\Lieb{X_{-1},Y_1}(v) &=& X_{-1}(Y_1,v),\\
\Lieb{X_0,Y_0}(v) &=& X_0(Y_0(v))-Y_0(X_0(v)),\\
\Lieb{X_{-1},Y_0}(v,v) &=& 2 X_{-1}(Y_0(v),v) - Y_0(X_{-1}(v,v)),\\
\Lieb{X_{-1},Y_{-1}}(v,v,v)
&=& 3 X_{-1}(Y_{-1}(v,v),v) - 3 Y_{-1}(X_{-1}(v,v),v).
\end{eqnarray*}
In general we have
$$
\Lieb{X,Y}(e^v) = X(Y(e^v)e^v) - Y(X(e^v)e^v).
$$
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conformal Killing fields}
\label{SectionConformalKillingfields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\A$ be an affine space modelled over the vector space $V$ which
is equipped with a conformal inner product $\com$ (see definition
\ref{Conformalvectorspace}). In this chapter we characterize
polynomial vector fields $K$ on $\A$ which have local diffeomorphisms
leaving the conformal structure of the affine space invariant:
$\Lie_K\com=0$. Such vector fields are called \emph{conformal Killing
fields} and represent the infinitesimal symmetries of conformal
geometry. In $n\geq 3$ dimensions the space of these fields forms a
finite dimensional Lie algebra.

\begin{proposition}
The following sub space $\g$ of quadratic vector fields on a
conformal vector space $V$, $\com$
$$
\g:= V \dsum \co(V) \dsum V^\dual 
\quad\subset\quad V \,\dsum\, V^\dual\tens V \,\dsum\, \Sym^2 V^\dual \tens V,
$$ 
with inclusion $V^\dual \inclusion \Sym^2 V^\dual \tens V$ defined by
$$
\alpha\mapsto
((v,w)\mapsto \alpha(v)w + \alpha(w)v - \cip{v,w}\sharp \alpha
               = \Lieb{\Lieb{\alpha,v},w}),
$$
defines a Lie algebra of polynomial vector fields $K$, which leave the
conformal structure invariant: $\Lie_K\com=0$. 
\end{proposition}

\proof If $(a,A,\alpha)$ and $(b,B,\beta)$ denote elements in $\g=V
\dsum \co(V) \dsum V^\dual$ then the Lie bracket in
$\Sym(V^\dual)\tens V$ under the above inclusion is given by:
\begin{eqnarray*}
\Lieb{a,b} &=& 0,\\
\Lieb{A,b} &=& Ab,\\
\Lieb{\alpha,b} &=&
   \alpha(b)\id + \alpha\tens b - \flat b\tens\sharp\alpha,\\
\Lieb{A,B} &=& A\circ B - B\circ A,\\
\Lieb{\alpha,B} &=& \alpha\circ B,\\
\Lieb{\alpha,\beta} &=& 0.
\end{eqnarray*}
This shows that $\g$ is indeed a subalgebra. Since the conformal
structure on affine space is parallel we know $\Lie_a\com=0$. Also
$\Lie_A\com=\com.A=0$ by definition of $\co(V)$. Finally
$\Lie_\alpha\com(v) =\com.\Lieb{\alpha,v} =0$.\qed

Note, that the Lie algebra $\g$ inherits the grading from
$\Sym(V^\dual)\tens V$ namely $\g_1 := V$, $\g_0 := \co(V)$ and
$\g_{-1} := V^\dual$. The subalgebra of vector fields which
vanish at the origin will be called $\p:=\co(V)\dsum V^\dual$.

\begin{remark}\label{Translationsatinfinity}
\summary{Translations at infinity} The Abelian subalgebra $V^\dual$
generates local diffeomorphisms which can be called \emph{translations
at infinity}: if $\mu\in L^1$ is a length scale and $g=\mu^{-2}\com$ a
Euclidean metric then the \emph{inversion at the unit sphere} $I$ is
defined to be a map $V\setminus{\setof{x\property\cip{x,x}\neq 0}} \to
V$ with $I(x) := {x\over \letterip{g}{x,x}}$. It leaves the conformal
structure invariant $I^\dual\com =\com$ and is idempotent
$I^2=\id$. Finally, for $\alpha\in V^\dual$ define $x\mapsto
I(I(x)+\alpha)$, which is independent of $\mu$ and defines a group
homomorphism between $V^\dual,+$ and conformal diffeomorphisms of $V$.
\end{remark}

\begin{remark}\label{RemarkChangeofWeylderivative}
\summary{Change of Weyl derivative} The Lie bracket
$\Lieb{\christo,v}$ between a linear form $\christo\in V^\dual$ and a
vector $v\in V$ is an element in $\co(V)$ and hence acts on any
representation of $\CO(V)$. The corresponding bundle map on a
conformal manifold coincides with the linearized Koszul formula
\ref{LinearizedKoszulformula}, where $\christo$ represents a (linear)
change of Weyl derivative and $v$ a direction in which to
differentiate. Hence the translations at infinity on the affine space
can be interpreted as linear changes of Weyl derivatives on a curved
conformal manifold. 
\end{remark}

\begin{proposition}
For $n\ge 3$ the Lie subalgebra of all polynomial vector fields $K$
with $\Lie_K\com=0$ is finite dimensional and given by the above $\g:=
V \dsum \co(V) \dsum V^\dual$.
\end{proposition}

\proof If $K\in\Sym^k V^\dual\tens V$ with $k\geq 1$ represents a
polynomial vector field, then $0=\Lie_K\com$ forces $K$ to be in the
kernel of the map
$$
\Sym^kV^\dual\tens V
\to \Sym^{k-1}V^\dual\tens (V^\dual\cartan V),
$$
where $V^\dual\cartan V$ denote the endomorphisms of $V$ which are
symmetric and trace-free, i.e.~$V^\dual\tens V =(V^\dual\cartan
V)\dsum\co(V)$. For $n\geq 3$ and $k\geq 3$ the above map is
injective. For $k=2$ we have a short exact sequence
$$
0\to V^\dual
\to \Sym^2V^\dual\tens V
\to V^\dual\tens (V^\dual\cartan V)
\to 0.\qed
$$

\begin{remark}
Conformal geometry in two dimensions $n=2$ is different reflected
by the richness of holomorphic functions (which are conformal
diffeomorphisms at points where the first derivative is
nonzero). Indeed for $n=2$ and all $k\geq 2$ we have a short exact
sequence
$$
0\to \Sym_0^{k-1} V^\dual
\to \Sym^{k}V^\dual\tens V
\to \Sym^{k-1} V^\dual\tens (V^\dual\cartan V)
\to 0,
$$
where $\Sym_0^{k-1} V^\dual$ is the two dimensional space of trace-free
symmetric forms. Consequently the Lie algebra of polynomial vector
fields is infinite dimensional and given by
$$
V\dsum
\co(V) \dsum
V^\dual \dsum
\Sym^2_0 V^\dual \dsum 
\Sym^3_0 V^\dual \dsum \ldots\quad.
$$
This dissertation only covers the $n\geq 3$ dimensional case. Conformal
vector fields on a two dimensional conformal space which also leave the
additional choice of a \emph{M\"obius structure} invariant (see
\cite{KulkarniPinkall} for the flat case, and \cite{CalderbankMoebius} for
the curved case) are then again only given by the above quadratic
polynomials.
\end{remark}

\begin{discussion}\label{grepfromlrep}
\summary{$\g$-action on functions} Let $F\tens\co(V)\to F$ be a right
representation. As before we assume, that $F$ has a certain central
weight $w\in\R$. Polynomials with values in $F$ are elements
$f=(f_w,f_{w-1},\ldots,f_{w-k}) \in \Sym(V^\dual) \tens F$ (of
finite degree $k$) given by $v\mapsto f(e^v):= f_{w} + f_{w-1}(v) +
\ldots + \oneover{k!} f_{w-k}(v^k)$. The Lie derivative $\Lie$ induces
a right action of $(a,A,\alpha)\in \g=V\dsum\co(V)\dsum V^\dual$ on
$f\in \Sym(V^\dual)\tens F$:
$$
\Lie\from\Sym(V^\dual)\tens F\ \tens\ (V\dsum\co(V)\dsum V^\dual) 
\to \Sym(V^\dual)\tens F,
$$
given by
%\begin{eqnarray*}
%\Lie_a f_w &=& 0,\\
%\Lie_A f_w &=& f_w.A,\\
%\Lie_\alpha f_w(v) &=& f_w.\Lieb{\alpha,v},\\
%\Lie_a f_{w-1} &=& f_{w-1}(a),\\
%\Lie_A f_{w-1}(v) &=& f_{w-1}(Av) + f_{w-1}(v).A,\\
%\Lie_\alpha f_{w-1}(v,v) &=& 
%   f_{w-1}(\Lieb{\alpha,v}v) + 2f_{w-1}(v).\Lieb{\alpha,v},\\
%\Lie_a f_{w-2}(v) &=& f_{w-2}(a,v),\\
%\Lie_A f_{w-2}(v,v) &=& 2f_{w-2}(Av,v) + f_{w-2}(v,v).A,\\
%\Lie_\alpha f_{w-2}(v,v,v) &=& 
%   3f_{w-2}(\Lieb{\alpha,v}v),v) + 3f_{w-2}(v,v).\Lieb{\alpha,v}.\\
%\end{eqnarray*}
%In general we have
%\begin{eqnarray*}
%\Lie_a f_w &=& 0,\\
%\Lie_a f_{w-k} (v^{k-1}) &=& f_{w-k}(a,v^{k-1}),\\
%\Lie_A f_{w-k} (v^{k})
%&=& k f_{w-k}(Av,v^{k-1}) + f_{w-k}(v^k).A,\\
%\Lie_\alpha f_{w-k} (v^{k+1})
%&=& \smallfrac{k(k+1)}{2} f_{w-k}(\Lieb{\alpha,v}v, v^{k-1})
%    + (k+1) f_{w-k}(v^k).\Lieb{\alpha,v}.\\
%\end{eqnarray*}
\begin{eqnarray*}
\Lie_a f(e^v) &=& f(ae^v),\\
\Lie_A f(e^v) &=& f((Av)e^v) + f(e^v).A,\\
\Lie_\alpha f(e^v) 
&=& f(\oneover{2}(\Lieb{\alpha,v}v)e^v) + f(e^v).\Lieb{\alpha,v}.
\end{eqnarray*}
\end{discussion}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Tensorial twistors in conformal geometry}
\label{Tensorialtwistorsinconformalgeometry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Penrose discovered the physical significance of elements in a
representation space of the Lie algebra of conformal Killing fields
$\g$. He called these elements \emph{twistors} (see
\cite{PenroseRindler}) and in the present context of affine conformal
geometry any such element induces a special section of a certain
bundle, a \emph{twistor field}. Constant functions and conformal
Killing fields are particular examples. Twistors play a central role
in this dissertation and occur here as polynomials. Later on they will
be realized as solutions of overdetermined partial differential
equations and will play the role of generalized charges (such as
gravitational masses) in linear field theories.

\begin{definition}
\summary{Conformal twistors} Let $W$ be a right representation space
of the conformal Lie algebra $\g=V\dsum\co(V)\dsum V^\dual$. An
element of $W$ is called a \emph{conformal twistor}.
\end{definition}

Our first aim is to construct such representations $W$ by identifying
$\g$ as a linear Lie algebra: if $V$ is a vector space then we denote
by $V^0:=L^{-1}\tens V$ the weightless vector space.

\begin{definition}\label{InnerproductonassoV}
If $V$ carries a conformal structure $\com$ with signature
$\sig(\com)=(p,n-p)$ then the $(n+2)$-dimensional space
$$
{\asso V}:=L^1 \dsum V^0 \dsum L^{-1}
$$
carries a natural (real valued) inner product of signature
$\sig(\com)+(1,1)=(p+1,n-p+1)$. It is given on two vectors ${\asso
v}=(v_1,v_0,v_{-1})$ and ${\asso w}=(w_1,w_0,w_{-1})$ in $\asso V$ by
$$
\ip{{\asso v},{\asso w}}
   := v_1 w_{-1} + \cip{v_0,w_0} + v_{-1} w_1.
$$
\end{definition}

The space $\Skew(\asso V)=\so(\asso V)$ of skew symmetric
endomorphisms of $\asso V$ forms a Lie algebra under $\Lieb{\asso
A,\asso B}={\asso A}\circ{\asso B} - {\asso B}\circ{\asso A}$.

\begin{proposition}
The following identification:
$$
V\dsum\co(V)\dsum V^\dual\stackrel{\simeq}{\longrightarrow}\so({\asso V})\ ;\ 
(a,A,\alpha)\mapsto {\asso A}
$$
defined on ${\asso v}=(v_1,v_0,v_{-1})\in {\asso V}$ by
$$
{\asso A}{\asso v}
  := \left( \oneover{n}(\trace A) v_1 + \cip{a,v_0}\, , \,
       A_0 v_0 - a \tens v_{-1} - \alpha \tens v_1\, , \,
       \alpha(v_0) - \oneover{n}(\trace A) v_{-1} \right),
$$
is a Lie algebra isomorphism, where $A=A_0+\oneover{n}\trace
A\id$.
\end{proposition}

Tensor products of $\asso V=L^1\dsum V^0\dsum L^{-1}$ are therefore
examples of representations for $\g=V\dsum\co(V)\dsum V^\dual$. Let
$\A$ be an affine space over $V$ and $x\in\A$ a point. Each element of
$\so({\asso V})$ (the right adjoint representation) induces a
polynomial vector field on $\A$, a so called \emph{conformal Killing
field} via $\so({\asso V})=V\dsum\co(V)\dsum
V^\dual\to\Sym(V^\dual)\tens V\to C^\infty(\A,V)$. In general if $W$
is a right representation of $\g=V\dsum\co(V)\dsum V^\dual$ then
$W\tens \g\to W$ induces a map $W\tens V^\dual\to W$ with image
denoted by $W.V^\dual \subset W$.

\begin{definition}\label{Coinvariants}
\summary{Coinvariants} Elements of the quotient space $W_{V^\dual}
:=W/W.V^\dual$ are called \emph{coinvariants} of $W$ with respect to
$V^\dual$.
\end{definition}

Since the map $W\tens V^\dual\to W$ is $\p=\co(V)\dsum V^\dual$
equivariant the space of coinvariants $W_{V^\dual}$ is indeed a
$\p$-representation. The nilpotent part $V^\dual$ of $\p$ acts
trivially on $W_{V^\dual}$.

\begin{examples}
If $W$ is an irreducible subspace of a tensor product of $\asso
V=L^1\dsum V^0 \dsum L^{-1}$ with itself, then $W$ splits under the
action of the centre of $\co(V)$ into a direct sum of subspaces with
elements of the same central weight $W=W_w\dsum W_{w-1}\dsum\ldots
\dsum W_{-w}$. The linear action of elements in $V^\dual$ lowers the
central weight by $-1$. From this it is clear that the space of
coinvariants $W_{V^\dual}$ is isomorphic to $W_w$ as
$\co(V)$-representation (we will use the following notation: if $E$
is a $\co(V)$-representation of central weight $w_E$, then we denote
by $E^w:=L^{w-w_E}\tens E$ the same $\so(V)$-representation but of
central weight $w$):
\begin{eqnarray*}
W &=& \asso V = L^1\dsum V^0\dsum L^{-1},
\hbox{\ hence\ } W_{V^\dual} = L^1,\\
W &=& \so(\asso V) = V\dsum\co(V)\dsum V^\dual,
\hbox{\ hence\ } W_{V^\dual} = V,\\
W &=& \Lambda^{k+1}(\asso V) 
   = (\Lambda^k V)^1
      \dsum \Lambda^{k-1}V^0 
      \dsum \Lambda^{k+1}V^0
      \dsum (\Lambda^k V)^{-1},
\hbox{\ hence\ } W_{V^\dual} = L^{k+1}\tens\Lambda^k V^\dual,\\
W &=& \Sym_0^2(\asso V) = 
  L^2 \dsum V 
  \dsum \Sym^2 V^0 
  \dsum V^\dual
  \dsum L^{-2},
\hbox{\ hence\ } W_{V^\dual} = L^2.
\end{eqnarray*}
\end{examples}

\begin{remark}
Suppose that a finite dimensional irreducible representation $W$ of
$\so({\asso V})$ is characterized by a Young diagram (see
\cite{FultonHarris}) for $\asso V$. In that case, the coinvariants
$W_{V^\dual}$ as $\so(V)$-representation can be characterized by the
Young diagram for $V$ which arises after deleting the first row. The
central weight of $W_{V^\dual}$ is given by the number of entries of
the first row, which determines $W_{V^\dual}$ as
$\co(V)$-representation.  The nilpotent part of $\p=\co(V)\dsum
V^\dual$ will act trivially on $W_{V^\dual}$.
\end{remark}

Each element of $W$ induces (in a linear way) a section with values in
$W_{V^\dual}$. More precisely:
 
\begin{proposition}\label{Conformaltwistorfields}
\summary{Conformal twistor fields} The following linear inclusion is
equivariant under the right $\g$-actions:
$$
\iota\from W\to \Sym(V^\dual)\tens W_{V^\dual};\quad 
\iota(w)(v^k)=[w(.v)^k]_{V^\dual}.
$$ 
The resulting sections of $C^\infty(\A,W_{V^\dual})$ are called
\emph{conformal twistor fields} induced by $W$.
\end{proposition}

\proof Let $(a,A,\alpha)\in\g$ be a conformal Killing field then
$$
\Lie_a(\iota(w))(e^v) 
= \iota(w)(ae^v)
= [w.a.e^v]_{V^\dual}
=\iota(w.a)(e^v).
$$
As elements acting from the right on $W$ (alternatively in the
universal enveloping algebra $\U(\g)$) we have $.A.e^v =
\Lieb{A,v}.e^v + .e^v.A$ from which we get
\begin{eqnarray*}
\Lie_A(\iota(w))(e^v)
&=&\iota(w)((Av)e^v) + \iota(w)(e^v).A\\
&=& [w.\Lieb{A,v}.e^v + w.e^v.A]_{V^\dual}\\
&=& [w.A.e^v]_{V^\dual} =\iota(w.A)(e^v).
\end{eqnarray*}
Finally for the action of $\alpha$ note $.\alpha.e^v =
.(\oneover{2}\Lieb{\alpha,v}v).e^v + .e^v.\Lieb{\alpha,v} +
.e^v.\alpha$ hence
\begin{eqnarray*}
\Lie_\alpha(\iota(w))(e^v)
&=& \oneover{2}\iota(w)((\Lieb{\alpha,v}v)e^v) + \iota(w)(e^v).\Lieb{\alpha,v}\\
&=& [\oneover{2}w.(\Lieb{\alpha,v}v).e^v + w.e^v.\Lieb{\alpha,v}]_{V^\dual}\\
&=& [w.\alpha.e^v - w.e^v.\alpha]_{V^\dual}\\
&=& [w.\alpha.e^v]_{V^\dual} = \iota(w.\alpha)(e^v).\qed
\end{eqnarray*}

\begin{example}\label{Conformaltwistorfieldsofweightone}
The conformal twistor fields induced by $W=\Lambda^{k+1}\asso V$ with
$k=0\ldots n$ are quadratic polynomials with values in $L^{k+1}\tens
\Lambda^kV^\dual$. In the case $k=0$ the scalar valued polynomial coming
from $\asso a=(a_1,a_0,a_{-1})\in L^1\dsum V^0\dsum L^{-1}$ is a
called a \emph{Fierz twistor (for gravity)} and is given by
\begin{eqnarray*}
\iota(\asso a)(e^v)
&=& [\asso a + \asso a.v + \oneover{2}\asso a.v.v + \ldots]_{V^\dual}\\
&=& a_1 - \cip{a_0,v} - \oneover{2}a_{-1}\cip{v,v}.
\end{eqnarray*}
In the case $k=1$ we reproduce the \emph{conformal Killing fields}:
the element $\asso A=(a,A,\alpha)\in V\dsum\co(V)\dsum V^\dual$
induces
\begin{eqnarray*}
\iota(\asso A)(e^v)
&=& [\asso A + \asso A.v + \oneover{2}\asso A.v.v + \ldots]_{V^\dual}\\
&=& [\asso A + \Lieb{\asso A,v} + \oneover{2}\Lieb{\Lieb{\asso A,v},v}]_{V^\dual}\\
&=& a + Av + \oneover{2}\Lieb{\alpha,v}v.
\end{eqnarray*}
In the case $k\ge 2$ the twistor fields induced by
$\asso A=(A_1,A_0^-,A_0^+,A_{-1})\in (\Lambda^k V)^1\dsum (\Lambda^{k-1} V)^0
\dsum (\Lambda^{k+1} V)^0 \dsum (\Lambda^k)^{-1}$ are given by
\begin{eqnarray*}
\iota(\asso A)(e^v)
&=& [\asso A + \asso A.v + \oneover{2}\asso A.v.v + \ldots]_{V^\dual}\\
&=& A_1 + v \wedge A_0^- + v\inner A_0^+ + \oneover{2}\cip{v,v}A_{-1}. 
\end{eqnarray*}
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Projective light cone}\label{Projectivelightcone}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A projective model of conformal geometry can be constructed as
follows: let ${\asso V}$ be a $(n+2)$-dimensional vector space with
inner product of signature $(p+1,n-p+1)$. A nonzero vector ${\asso l}
\in {\asso V}$ is called \emph{lightlike}, \emph{null} or
\emph{isotropic}, if $\ip{{\asso l},{\asso l}}=0$. The set of light
like vectors forms the light cone $\Cone := \setof{{\asso l} \in {\asso
V}\minuszero \property \ip{{\asso l},{\asso l}}=0}$.

\begin{definition}\label{Conformalsphere}
\summary{Conformal sphere} The set of null lines
$S:=\Cone/{\R^\times} =\setof{\R{\asso l} \property {\asso l} \in
\Cone}$ defines a compact real smooth manifold and will be called the
\emph{real conformal sphere of signature $(p,n-p)$}.
\end{definition}

\begin{remark} 
The real conformal sphere is diffeomorphic to a product of spheres
$S^p\cross S^{n-p}/\sim$ where points like $(x,y)\sim(-x,-y)$ are
identified. It plays the role of the \emph{flat model} of conformal
geometry. The case $p=n-1$ and $n=4$ refers to the \emph{real
conformal Minkowski space} of special relativity.
\end{remark}

\par From this definition there are various tautological vector bundles
over $S$: the trivial bundle $\asso V \cross S$ has two natural sub
bundles: the canonical real line bundle denoted by $\Cano \to S$ with
fibre $\R {\asso l}$ over $\R{\asso l} \in S$ and the bundle of
tangent spaces to the light cone with fibre at $\R{\asso l} \in S$
given by $T_{\asso l} \Cone =\setof{{\asso j} \in {\asso V}\property
\ip{{\asso j},{\asso l}}=0}$. Since the canonical line sits inside the
tangent cone $\Cano \subset T\Cone$ the quotient bundle $T\Cone/\Cano$ is
also natural. Indeed all intrinsic tensor bundles of $S$ can naturally
be rediscovered: for the tangent, cotangent and density bundles we
have $TS=\Cano^\dual \tens T\Cone/\Cano$, $T^\dual S=\Cano \tens
(T\Cone/\Cano)^\dual$ and $L^w=L^w(TS)=\Cano^{-w}$.

Notice that the weightless tangent bundle $L^{-1} \tens TS=T\Cone/\Cano$
inherits a Riemannian metric of signature $(p,n-p)$ from $\asso V$, and
this defines a conformal inner product on each tangent space of $S$,
i.e.~a \emph{conformal structure} on $S$.

\begin{theorem}
\summary{M{\"o}bius group} The orthogonal group $\O({\asso V})$ acts
linearly upon $\asso V$ and leaves the light cone invariant, which
induces a smooth transitive action on the conformal sphere $\O({\asso
V}) \cross S \to S$ by conformal diffeomorphisms, the so called
\emph{M\"obius transformations}. For $n\ge 3$ all conformal
diffeomorphisms of $S$ come from $\O({\asso V})$.
\end{theorem}

For a proof of the last part of this theorem we refer to
\cite{KulkarniPinkall}.
 
\begin{remark}
Let $G:=\O({\asso V})$ be a short hand for the M\"obius
transformations and $\R\asso l\in S$ be a point, then $P:=\setof{g\in
G\property \R {\asso {g l}} = \R\asso l}$ denotes the stabilizer sub
group. By use of the orbit stabilizer theorem $S$ can be identified as
homogeneous space $S=G/P$. In the case ${\asso V}=L^1\dsum V^0\dsum
L^{-1}$ the sub line $L^{-1}$ is null and the stabilizer group of
$\R\asso l = L^{-1}$ can be identified with $P=\CO(V)\semidirect
V^\dual$.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Differential conformal invariants}
\label{Differentialconformalinvariants}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The aim of this chapter is to study local differential invariants on
conformal manifolds. The conformal metric itself and the Weyl
curvature are simple examples. We are concerned here with first and
second order invariants in $n\geq 3$ dimensions. On conformal
manifolds of indefinite signature lightlike geodesics are well known
to be invariant and because of their physical significance we
investigate them in the first section. Conformal geodesics generalize
circles to curved conformal manifolds. We derive new conserved
properties along conformal geodesics using twistors. In the last two
sections we discuss differential invariants for sections of associated
bundles. We recall Fegan's classification of linear conformally
invariant differential operators of first order and apply it to give
new bilinear invariants. These bilinear invariants are crucial when
dealing with linear field theories as we will explain in the next
chapter. The key tool to study second order invariants is the
definition \ref{Secondjetoperator} of a $2$-jet operator in terms of a
Weyl derivative, which behaves well under a change of Weyl
derivative. With this operator in hand all second order invariants of
the affine space have analogues on curved conformal manifolds.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conformal invariants along lightlike curves}
\label{Conformalinvariantsalonglightlikecurves}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A curve $\alpha\from\R\to M$ in a conformal manifold of indefinite
signature is called \emph{lightlike} if $\ip{\dot\alpha,\dot\alpha}
=0$. Lightlike curves play the role of light rays in relativistic
geometric optics. If the acceleration $D_{\dot\alpha}\dot\alpha$ is
proportional to the tangent vector $D_{\dot\alpha}\dot\alpha \prop
\dot\alpha$ the curve is called a \emph{lightlike geodesic}. The above
condition is independent of the parameterization and of the choice of
the Weyl derivative $D$. Hence lightlike geodesics are conformally
invariant. Together with the conformal invariance of electromagnetism
we suggest a simple law how an electromagnetic field can influence a
light ray. Unfortunately it is very difficult to verify such a law in
a laboratory, since the interaction is expected to be very
weak. Motivated by astronomical distance measurements we will define a
new invariant distance on a conformal manifold along the backward
lightcone of an observer.

One way to see the conformal invariance of lightlike acceleration is
to attach a never vanishing vector field $L_\alpha\in L^{w-1}\tens
T_\alpha M$ of central weight $w$ along $\alpha$ which is proportional
to the tangent vector $L_\alpha\prop\dot\alpha$. For the acceleration
in terms of $L$ we find for a $1$-form $\christo$ representing a
linear change of Weyl derivative, see example \ref{Firstorderchange}:
$$
\LieWeyl_\christo D_LL
= w\christo(L)L + \christo(L)L - \ip{L,L}\sharp\christo,
$$
hence for a lightlike $L$ of central weight $w=-1$ the acceleration
$D_LL$ is conformally invariant.

\begin{definition}
\summary{Lightlike geodesic} A lightlike curve $\alpha\from\R\to M$
with an attached cotangent vector $0\neq L_\alpha\in T^\dual_\alpha M$
with $L_\alpha\prop\dot\alpha$ is called a \emph{lightlike geodesic}
if $D_LL=0$.
\end{definition}

\begin{remark}
\summary{Frequency} Lightlike acceleration being conformally invariant
for vectors $L$ of central weight $-1$ corresponds to the fact that in
physics one usually attaches the dimension $\Dim{frequency}$ to the
tangent vector $L$: with the velocity of light as fundamental constant
this has geometric dimension $\Dim{length}^{-1}$, see
paragraph \ref{Kinematicdimensions}. Indeed the \emph{frequency of $L$ measured
by an observer $N$} (with $\ip{N,N}=-1$) is defined to be
$-\ip{L,N}$. This is a way to normalize lightlike vectors in a
conformally invariant and physically meaningful way.
\end{remark}

\begin{proposition}\label{Lightlikeconservation}
\summary{Conservation law along lightlike geodesics} If $K$ denotes a
conformal Killing field on $M$ and $L$ is the attached covector of a
lightlike geodesic, then $\ip{L,K}$ is a weightless conformally
invariant constant of motion.
\end{proposition}

\proof This is a simple application of the product rule:
\begin{eqnarray*}
\del_L(\ip{L,K})
&=&\ip{D_LL,K}+\ip{L,D_LK}\\
&=&\ip{D_LL,K}+\ip{L,\Killing(K)(L)} + \oneover{n}\ip{L,L}\divR^DK,
\end{eqnarray*}
and all three summands are zero by assumption.\qed

The conformal invariance of the lightlike acceleration $D_LL$ can be
used to suggest an interaction between $L$ and a Faraday $2$-form $F$
as follows:

\begin{remark}
\summary{Interaction with $2$-forms} Let $F\in
C^\infty(M,\Lambda^2T^\dual M)$ denote a closed $2$-form. A
conformally invariant interaction between a lightlike curve $L$ and
$F$ is given by $D_LL = \lambda \sharp (F(L))$, where $\lambda\in\R$
is a coupling constant. If $F$ is a kinematic electromagnetic field,
then the above is a simple description of a light-light interaction in
terms of classical geometric optics. The number $\lambda$ needs to be
determined by experiment: according to the above law a monochromatic
laser light ray of (low) frequency $\nu$ should bend in a (small)
circle of radius $R$ when moving through a (strong) homogeneous
magnetic field $B$ such that:
$$
R =\oneover{\lambda}
\sqrt{{\hbar\over\oneovercoulomb\light^3}}\, \smallfrac{\nu}{B}\quad .
$$
\end{remark}

\begin{remark}
If $K\in C^\infty(M,TM)$ denotes a vector field and $A\in
C^\infty(M,T^\dual M)$ a (local) potential of $F=\dR A$ then
$q:=\ip{L,K} - \lambda A(K)$ is a weightless function along the light
ray of $L$, which satisfies
\begin{eqnarray*}
\del_L q
&=& \ip{D_LL,K} + \ip{L,D_LK} - \lambda \del_L( K \inner A ) \\
&=& -\lambda (K\inner \dR A)(L) 
+ \ip{L,\Killing(K)(L)} - \lambda \dR( K \inner A )(L)\\
&=& -\lambda \Lie_KA(L) + \ip{L,\Killing(K)(L)}.
\end{eqnarray*}
Hence if $K$ is a conformal Killing field representing a symmetry of
$A$, i.e.~$\Lie_K(A)=0$, then $q$ is a conformally invariant conserved
quantity along $L$.  
\end{remark}

\begin{remark}\label{RedshiftandBendingoflight}
\summary{Redshift and bending of light} We will apply the study of
motion in a Coulomb field, see paragraph \ref{Coulombfield}, also to
the case of a lightlike solution curve $L=\dot c$ and $\ip{\dot c,\dot
c}=0$ subject to the equation $D_LL=\lambda F(L)$. The variables $a$,
$b$, $E$, $l$ are defined in paragraph \ref{Motioninacentralfield} and
we will use their properties here. The law of motion fixes the
parameter $a$ to be $a=-\lambda Q$. The frequency of $L$ measured by
the static observer $N$ is $- \ip{\dot c,N}=b + a/r$ hence $b>0$ has
the interpretation of the frequency at infinity. In the case of an
attractive potential $a>0$ the light has a higher frequency (it gains
energy) when it comes nearer to the centre (when it looses potential
energy). This is a redshift phenomenon.  From $0=\ip{\dot c,\dot c}$
we know $2E=b^2$ and the extremal distances $r_0$ with $\dot r=0$
satisfy $E =\oneover{2}l^2/r_0^2 + U(r_0)$, i.e.~$r_0 =(-a\pm l)/b$
which has exactly one positive solution if $l>|a|>0$ given by
$r_0=(l-a)/b$. In the case $l>|a|$ the light ray passes the centre and
comes nearest at $t=0$ with distance $r_0$. The total polar angle is
given by
\begin{eqnarray*}
\phi(\infty)-\phi(-\infty)
&=& 2(\phi(\infty)-\phi(0))\\
&=& 2\int_{r_0}^\infty {\dot\phi(t)\over {\dot r}} \,dr
=2\int_{r_0}^\infty {l\over r^2 \dot r} \,dr
= 2\int_0^\infty {1\over \cosh(x)-a/l }\,dx\\
&=& {4\over \sqrt{1 -(a/l)^2}}
\arctan\Big({e^x-a/l\over \sqrt{1 -(a/l)^2}}\Big)\Bigr\vert_0^\infty\\
&\approx& \pi + 2(a/l) + {\pi\over 2} (a/l)^2 + \ldots.
\end{eqnarray*}
The qualitative behavior of this bending depends on the sign of $a=-
\lambda Q$. Moreover from $l = a+br_0$ we find $\phi(\infty)-\phi(-
\infty)\approx \pi + 2a/(br_0)$ which shows that the bending also 
depends upon the frequency $b$ of the light. The bending of light in the 
Schwarzschild model (see e.g.~\cite{ONeill} p.~384) does not depend upon 
the frequency (in consistency with the equivalence of energy and 
gravitational mass) and is given by $\phi(\infty)-\phi(-\infty)\approx 
\pi + 4(a/r_0)$.
\end{remark}

\begin{remark}\label{Luminositydistance}
\summary{Luminosity distance} In special relativity the intensity of
electromagnetic radiation (also called luminosity) falls off like
$1/r^2$ where $r$ is the (retarded) spacelike distance to the source
(the intensity is an energy density quadratic in the field). This fall
off behavior allows one to measure astronomical distances of shining
stars by comparing their radiation intensity to that of comparable
stars with known distance. In special relativity this so called
\emph{luminosity distance} satisfies a differential equation (see
paragraph \ref{ElectromagneticLienardWiechertfields}) which
generalizes to the following equation on a (curved) conformal
manifold: along a lightlike geodesic $\alpha\from\R\to M$ with
$D_LL=0$ we can define a section $r_\alpha\in L^1_\alpha$ subject to
$D_LD_Lr + \nric^D(L,L)r=0$, where the initial conditions $r(0)=0$ and
$D_Lr(0)=-\ip{L,N}$ depend upon the choice of an observer
$\ip{N,N}=-1$ at $\alpha(0)$ and $\nric^D$ is the normalized Ricci
curvature of $D$, see proposition \ref{Splittingthecurvature}. The
resulting length scale along $\alpha$ is called the \emph{luminosity
distance} with respect to the observer $N$, the latter can be thought
of as the radiation source or as the measuring astronomer.
\end{remark}

\begin{definition}
A vector field $L$ of central weight $-1$ on a conformal manifold of
indefinite signature is called \emph{lightlike} if $\ip{L,L}=0$. It
is called \emph{geodetic} if $D_LL=0$ (which is also a conformally
invariant condition). The \emph{frequency} measured by an observer $N$
with $\ip{N,N}=-1$ is a density of weight $-1$ defined to be
$-\ip{L,N}$.
\end{definition}

\begin{example}\label{Lightfromapointsource}
\summary{Light from a point source} Let $c(t)$ be a parameterized
timelike curve representing an emitter. Let $M$ be a neighborhood of
the worldline $c$ such that the spacetime double lightcone of each
point $x\in M$ intersects the worldline $c$ twice: once with the forward
lightcone at $c(\phi^+(x))$ and once with the backward lightcone at
$c(\phi^-(x))$. This defines two function $\phi^\pm\from M\to \R$
(\emph{advanced} and \emph{retarded} parameter). The two gradients
$L^\pm:=-\grad \phi^\pm$ are lightlike geodetic vector fields. Along
the worldline $c$ we have $t=\phi^\pm(c(t))$ which gives
$1=\del\phi^\pm(\dot c)=-\ip{L^\pm,\dot c}$, hence $L^\pm$ are fields
pointing in the same time direction as $\dot c$ and the emitted
frequency (relative to $c$) is incorporated into the parameterization of
$c$.
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conformal invariants along nonlightlike curves}
\label{Conformalinvariantsalongnonlightlikecurves}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we focus on invariants along timelike or spacelike
curves. The Riemannian geodesic equation is not conformally invariant,
but there is a conformally invariant third derivative of a
curve. Curves which are uniformly accelerated are called
\emph{conformal geodesics}. The Fermi transport along arbitrary curves
is a simpler example of an invariant, which we will need in the next
section to construct conserved properties along conformal
geodesics. None of the results of this section are new, although we
simplified the conformal geodesic equation to a single third order
equation, which will turn out to be useful in the next section.

\begin{discussion}\label{Geometryofcurves}
\summary{Geometry of curves} Let $c\from\R\to M$ with $t\mapsto c(t)$
be a parameterized nonlightlike curve $\ip{\dot c,\dot c}\neq 0$ on a
conformal manifold $M$. The parameterization induces a positive section
$\nu$ of $L^{-1}$ along $c$ defined by $\pm 1=\nu^2\ip{\dot c,\dot
c}$. The normalized weightless tangent vector $N:=\nu\dot c$ is
independent of the parameterization. Given a Weyl derivative $D$ we
call $D_NN$ the \emph{acceleration} of $c$. The skew symmetric
endomorphism $K^D \in L^{-1}_c \tens \so(T_cM)$ defined by $\ip{N,N}
K^D := N \skew D_NN=\ip{\ ,N}D_NN-\ip{\ ,D_NN}N$ is called the
\emph{curvature} of $c$ with respect to $D$.
\end{discussion}

\begin{definition}
\summary{Fermi derivative along curves} Let $X$ be a vector field
along a nonlightlike curve $c$, then $N$ splits $X$ into tangential
and normal parts $X=X^\tang + X^\normal$, where $\ip{N,N}
X^\tang:=\ip{X,N}N$. The Fermi derivative with respect to a Weyl
derivative $D$ is defined by
\begin{eqnarray*}
\Fermi^D_NX
&:=& (D_N{X^\tang})^\tang + (D_N{X^\normal})^\normal\\
%&=& (D_N({\ip{X,N}\over\ip{N,N}}N))^\tang
%    + (D_N(X-{\ip{X,N}\over\ip{N,N}}N))^\normal\\
%&=& (D_NX)^\tang + {\ip{X,D_NN}\over\ip{N,N}}N
%    + (D_NX)^\normal - {\ip{X,N}\over\ip{N,N}}D_NN\\
&=& D_NX - K^D(X).
\end{eqnarray*}
\end{definition}

This definition on vector fields extends to arbitrary sections
associated to a $\co(V)$-representation by means of the left $K^D$
action. The Fermi derivative leaves the conformal structure and the
projections on tangential respectively normal parts parallel. A vector
field $X$ with $0=\Fermi^D_NX$ is called \emph{minimally rotating} or
\emph{Fermi parallel}.

\begin{proposition}
The Fermi derivative is conformally invariant for weightless tensors
and therefore weightless tensors can be transported in a conformally
invariant way along any curve.
\end{proposition}
\proof Let $Z$ a vector field of central weight $w$ along $c$. Under a
linear change of Weyl derivative $\christo$, the Fermi derivative of
$Z$ in the direction $N$ changes as follows:
\begin{eqnarray*}
\LieWeyl_\christo \Fermi^D_NZ
&=& \LieWeyl_\christo D_NZ - \LieWeyl_\christo K^D(Z)\\
&=& w\christo(N)Z.\qed
\end{eqnarray*}

On the conformal sphere $S$ (see definition \ref{Conformalsphere})
circles are preferred curves, since they are mapped onto circles under
all M\"obius transformations. In Euclidean geometry, circles have
constant geodesic curvature, i.e.~their third derivative is
proportional to their velocity. On an arbitrary conformal manifold
$M$, $\com$ with $n\ge 3$ dimensions, there is a conformally invariant
third derivative for non lightlike curves $c\from\R\to M$: denote by
$N_c$ the normalized tangent vector and choose a Weyl derivative
$D$. From $\ip{N,N}=\pm 1$ we have $\ip{D_NN,N}=0$ and for the
tangential component of $D_ND_NN$ we find
$\ip{D_ND_NN,N}=-\ip{D_NN,D_NN}$.

\begin{proposition}\label{Conformalgeodesics}
\summary{Conformal geodesics} In $n\geq 3$ dimensions the following
parameter independent third derivative of $c$
$$
\CGeod{N}:= ( D_ND_NN - \ip{N,N}\sharp\nric^D(N) )^\normal
$$
is conformally invariant. A nonlightlike curve $c$ with $\CGeod{N}=0$
is called a \emph{conformal geodesic}.
\end{proposition}
\proof Under a linear change $\christo$ of the Weyl derivative $D$ the
normal component of the third derivative changes like the normalized
Ricci curvature, see example \ref{LinearchangeofRiccicurvature}:
\begin{eqnarray*}
\LieWeyl_\christo (D_ND_NN)^\normal
&=& (\Lieb{\christo,N}.D_NN)^\normal + (D_N(\Lieb{\christo,N}.N))^\normal\\
&=& -\christo(N)D_NN + (D_N(\christo(N)N - \ip{N,N}\sharp\christo))^\normal\\
&=& \ip{N,N}(-D_N\sharp\christo)^\normal\\
&=& \ip{N,N}(\sharp \LieWeyl_\christo\nric^D(N))^\normal.\qed
\end{eqnarray*}
%For a parameterization $c\from\R\to M$ we find the
%following ordinary differential equation of third order:
%$$
%( {D\over\del t}{D\over\del t}{\dot c} 
%  - 3{ {\ip{\dot c,{D\over\del t}\dot c}}\over
%       \ip{\dot c,\dot c} }{D\over\del t}\dot c 
%  - \ip{\dot c,\dot c}\nric^D(\dot c) )^\normal
%= 0.
%$$
%This equation is still conformally invariant and independent of the
%parameterization. After the choice of a Weyl derivative $D$ one can
%find a preferred parameterization by solving:
%$$
%  {D\over\del t}{D\over\del t}{\dot c} 
%  + {\ip{{D\over\del t}\dot c,{D\over\del t}\dot c} 
%   \over \ip{\dot c,\dot c}}\dot c 
%  - \ip{\dot c,\dot c}\nric^D(\dot c)^\normal
%= 0,
%$$ 
%together with the initial condition $\ip{{D\over\del t}\dot c,\dot
%c}(0)=0$. The solution satisfies $\ip{{D\over\del t}\dot c,\dot
%c}=0$. The parameterization is now determined up to a real factor which
%corresponds to the $D$-parallel length of $\dot c$.

\begin{remark}
The above third derivative occurs as \emph{Abraham vector} in the
Lorentz Dirac equation for the classical motion of a charged particle
influenced by its own electromagnetic field in four dimensional
Minkowski space (see Rohrlich \cite{Rohrlich}).
\end{remark}

\begin{remark}
In a conformal affine space $\A$ with base point $x_0$ we could use
the translations at infinity $\phi\from\A\to\A$ with $\phi(x):=x_0 +
I(I(x-x_0)+\alpha)$ with $\alpha\in V^\dual$ and the inversion $I$
(see paragraph \ref{Translationsatinfinity}) to produce general
conformal geodesics $t\mapsto \phi(x_0+tN)$ i.e.~(hyperbolic) circles
out of straight lines $t\mapsto x_0+tN$. The conformal invariance of
electromagnetism allows to pull back the Coulomb field
$F_{\mathrm{Coul}}$, see paragraph \ref{Coulombfield},
associated to the straight worldline $x_0+tN$ to produce a solution of
Maxwell's equations $\phi^\dual F_{\mathrm{Coul}}$ with source
supported along the conformal geodesic $\phi(x_0+tN)$. These solutions
are called \emph{radiation fields from uniformly accelerated charges}
see Rohrlich \cite{Rohrlich}. They are special cases of the Lienard
Wiechert fields discussed in paragraph
\ref{ElectromagneticLienardWiechertfields}.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weight one twistors and conservation laws}
\label{Weightonetwistorsandconservationlaws}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This section contains new conserved conformal invariants along
conformal geodesics. In Riemannian geometry it is well known that a
Killing field $\Lie_Kg=0$ (leaving the Riemannian metric invariant)
can be contracted with the velocity of a curve $g(\dot c,K_c)$ to
produce a number along $c$ which is constant if $\ddot c=D_{\dot
c}\dot c =0$, i.e.~if $c$ is a Riemannian geodesic (see also
paragraph \ref{Lightlikeconservation}). We will present such pairings in
conformal geometry where we need to consider bilinear differential
pairings rather then just tensorial contractions as in $g(\dot
c,K)$. Moreover we will use a class of twistor fields rather than only
Killing fields to construct these conserved invariants.

In the case of an affine conformal space $\A$ over $V$ we already
constructed conformal twistor fields in proposition
\ref{Conformaltwistorfields} which were induced by elements in a
representation space $W$ for the Lie algebra of conformal Killing
fields $\g=\so(\asso V)$, with $\asso V=L^1\dsum V^0\dsum
L^{-1}$. Differential operators annihilating such twistor fields are
also called \emph{twistor operators}. In this section we define
twistor operators covering the cases $W=\Lambda^{k+1}\asso V$ for
$k=0\ldots n$. The twistor fields take values in the coinvariants
$W_{V^\dual}=L^{k+1}\tens\Lambda^k V^\dual$, see the examples
following definition \ref{Coinvariants}. These twistor operators
annihilating the twistor fields on affine space make sense on any
(curved) conformal manifold. Hence we will define twistor fields on
manifolds to be sections of the bundle $L^{k+1}\tens\Lambda^k T^\dual
M$ which lie in the kernel of the twistor operator. The existence of a
nontrivial twistor places constraints on the curvature.

The Fermi derivative is defined on weightless tensors and the above
mentioned twistor fields $f$ (including Killing fields) have central
weight $+1$. Along conformal geodesics we will construct weightless
Fermi parallel tensors out of twistor fields. From the dimensional
analysis we expect first order pairings between $f$ and $N$ the
normalized weightless tangent vector. 

We begin with the defining $\g$-representation $W=\asso V=L^1\dsum
V^0\dsum L^{-1}$. In the conformal affine space the resulting twistor
fields are scalar valued quadratic polynomials with values in $L^1$,
see paragraph \ref{Conformaltwistorfieldsofweightone}. There is no
restriction on the linear part of these polynomials, but the quadratic
part has to be tracelike. Therefore the annihilating operator must be
second order:
 
\begin{discussion}\label{Secondderivativeonscalars}
\summary{Second derivative on scalars} Let $D$ be a Weyl derivative on
a conformal manifold $M$ and $\christo$ a $1$-form. The linear change
$\LieWeyl_\christo D^2$ of the second covariant derivative $D^2$ applied
to scalar densities $f$ of weight $w$ is given in terms of vector
fields $X$, $Y$ by example \ref{Secondorderchange}:
\begin{eqnarray*}
{\LieWeyl_\christo D^2}_{X,Y}f
&=& \Lieb{\christo,X}.D_Yf + \Lieb{D_X\christo,Y}.f 
    + \Lieb{\christo,Y}.D_Xf - D_{\Lieb{\christo,X}.Y}f\\
&=& w\christo(X)D_Yf + wD_X\christo(Y)f 
    + w\christo(Y)D_Xf - D_{\christo(X)Y+\christo(Y)X-\ip{X,Y}\sharp\christo}f\\
&=& (w-1)\christo(X)D_Yf + wD_X\christo(Y)f 
    + (w-1)\christo(Y)D_Xf + \ip{X,Y}D_{\sharp\christo}f.
\end{eqnarray*}
To find conformally invariant parts of the second derivative we like
to add on the right amount of Ricci curvature to kill the $D\christo$
term in the above expression. This is possible in $n\ge 3$ dimensions:
\begin{eqnarray*}
\LieWeyl_\christo (D^2 + w\nric^D)_{X,Y}f
&=& (w-1)\christo(X)D_Yf + (w-1)\christo(Y)D_Xf + \ip{X,Y}D_{\sharp\christo}f.
\end{eqnarray*}
This proves the following:
\end{discussion}

\begin{proposition}\label{ConformalHessian}
\summary{Conformal Hessian} For scalar densities of weight
$w=+1$ in $n\ge 3$ dimensions the trace-free part of the Ricci
corrected second covariant derivative is a conformally invariant
differential operator called \emph{conformal Hessian}:
\begin{eqnarray*}
\Hesse &\from& C^\infty(M,L^1)
\to C^\infty(M,L^1\tens\Sym_0^2\,T^\dual M),\\
\Hesse f\, (X,Y)
&:=& \Sym_0^2 (D^2 + \nric^D)f\, (X,Y)\\
&=& \oneover{2}( D^2_{X,Y}f + D^2_{Y,X}f) -
\oneover{n}\trace_\com D^2f\cip{X,Y} + \nric_0^D(X,Y)f.
\end{eqnarray*}
In the case of a conformal affine space this operator annihilates the
twistor fields coming from $W=\asso V$, hence $\Hesse$ is a twistor
operator.
\end{proposition}

\begin{remark}\label{Einsteinmetricsfromtheconformalclass}
\summary{Einstein metrics from the conformal class} Sections $f$ of $L^1$ 
which are in the kernel of $\Hesse$ are also called \emph{Fierz twistors} 
(for gravity): $\Hesse(f)=0$. In a region where a Fierz twistors is 
positive $f>0$, it defines a length scale and induces a metric $g=f^{-
2}\com$ from the conformal structure which is \emph{Einstein}: for the 
induced exact Weyl derivative $D$ we have $Df=0$, hence
$0=\Hesse(f)=\Sym_0(D^2f + \nric^Df)=\oneover{n-2}f\ricci_0^D$. Hence the 
Ricci curvature of $g$ is a constant multiple of $g$.
\end{remark}

\begin{proposition}\label{Fierzconservation}
\summary{Conservation law along conformal geodesics} Let $f\in
C^\infty(M,L^1)$ be a density of weight $+1$ and $c$ a nonlightlike
curve with normalized weightless tangent vector $N$, then
$$
f.N :=\ip{N,N}(\grad^Df)^\perp + f D_NN
$$ 
is independent of the Weyl
derivative $D$ and hence defines a conformally invariant weightless
vector field along $c$ perpendicular to $N$. For the Fermi derivative
we have
$$
\Fermi_N f.N = f \CGeod(N) + \ip{N,N}(\sharp\Hesse f(N))^\perp.
$$
Hence if $f$ is a twistor field $\Hesse f=0$ and $c$ is a conformal
geodesic, then the induced $f.N$ is Fermi parallel.
\end{proposition}

\proof The term $f.N$ changes under a change of Weyl derivative
by the $1$-form $\christo$ as 
$$
\LieWeyl_\christo (f.N)^D
= \ip{N,N}\sharp\christo\tens f
  -\christo(N)fN + f\christo(N)N - f\ip{N,N}\sharp\christo = 0.
$$
It is straight forward to calculate the Fermi derivative of $f.N$:
\begin{eqnarray*}
\Fermi_N(f.N)
&=& (D_N(f.N))^\perp\\
&=& \ip{N,N}(D_N\grad^Df)^\perp - (D_Nf)D_NN + D_Nf D_NN + f (D_ND_NN)^\perp\\
&=& \ip{N,N}\sharp(D^2_N f + \nric^D(N))^\perp 
    + f (D_ND_NN - \ip{N,N}\nric^D(N))^\perp\\
&=& \ip{N,N}\sharp(\Hesse f(N))^\perp + f \CGeod(N).\qed
\end{eqnarray*}

\begin{application}
In the case of an affine conformal space $\A$ we have an injection
$\iota\from\asso V\to C^\infty(\A,L^1)$ form the $(n+2)$ dimensional
vector space $\asso V=L^1\dsum V^0\dsum L^{-1}$ into the kernel of the
conformal Hessian. If $c$ is a conformal geodesic and $J$ a Fermi
parallel vector field along $c$, then $\ip{J,\iota(\asso v).N}$ is a
constant real number linear in $\asso v\in\asso V$. Hence a given
conformal geodesic with attached Fermi parallel vector field induces a
linear from on $\asso V$. Since $\asso V$ is equipped with an inner
product the conformal geodesic with attached Fermi parallel vector
field provides us with a unique twistor.
\end{application}

\begin{definition}\label{Killingoperator}
\emph{Conformal Killing fields} $K\in C^\infty(M,TM)$ are in the
kernel of the \emph{conformal Killing operator} $\Killing\from
C^\infty(M,TM)\to C^\infty(M,L^2\tens\Sym^2_0(T^\dual M))$ which in
terms of a Weyl derivative $D$ is given by
\begin{eqnarray*}
\Killing(K)(X,Y)
&:=& \oneover{2}(\Lie_K\com)(X,Y) \\ 
&=& \oneover{2}\ip{D_XK,Y} + \oneover{2}\ip{X,D_YK} 
    - \oneover{n}(\divR^DK) \ip{X,Y}\\
&=& \ip{D_XK,Y} - \oneover{2} \dR^D (\flat K)(X,Y)
    - \oneover{n}(\divR^DK) \ip{X,Y},
\end{eqnarray*}
where $X$,$Y$ are vector fields.
\end{definition}

\begin{proposition}\label{Killingconservation}
Let $K\in C^\infty(M,TM)$ be a vector field (of central weight $+1$)
and $c$ a nonlightlike curve with normalized weightless tangent vector
$N$. Let ${t_i}$, ${\theta^i}$ be a dual basis of $T_cM$, then
$$
K.N
:= K^\perp\wedge D_NN 
+\oneover{2}\ip{N,N}\Sum_i (D_{t_i}K)^\perp\wedge (\sharp\theta^i)^\perp
$$ 
is independent of the Weyl derivative $D$ and hence defines a
conformally invariant weightless bivector field along $c$
perpendicular to $N$. For the Fermi derivative of $K.N$ we have:
$$
\Fermi_N(K.N) = K^\perp \wedge \CGeod(N) 
 + \Killing(K).N
 + \ip{N,N}\Sum_i (\Weylcurv_{N,t_i}K)^\perp\wedge(\sharp\theta^i)^\perp,
$$
where $\Killing(K).N$ is another conformally invariant bilinear
combination along $c$: if $h\in C^\infty(M,L^2\tens
\Sym_0^2(T^\dual))$ is a weightless bilinear form then define:
$$
h.N := (\sharp h(N))^\perp\wedge D_NN 
       + \ip{N,N}\Sum_i (\sharp D_{t_i}h(N))^\perp \wedge (\sharp 
       \theta^i)^\perp.
$$
Hence if $K$ is a conformal Killing field $\Killing K=0$ in $n=3$
dimensions (where $\Weylcurv=0$) and $c$ is a conformal geodesic, then
the induced $K.N$ is Fermi parallel.
\end{proposition}

\proof The term $K.N$ changes under a change of Weyl derivative
by the $1$-form $\christo$ as 
\begin{eqnarray*}
\LieWeyl_\christo (K.N)^D
&=& -K^\perp\wedge \ip{N,N}(\christo)^\perp 
+\oneover{2}\ip{N,N}\Sum_i 
(\Lieb{\christo,t_i}K)^\perp\wedge (\theta^i)^\perp\\
&=& -\ip{N,N} K^\perp\wedge (\christo)^\perp 
+\ip{N,N} K^\perp\wedge (\christo)^\perp 
+ 0 = 0.
\end{eqnarray*}
Let $U$ be a weightless vector and $t$ a vector of central weight $+1$, then
\begin{eqnarray*}
\Lieb{\christo,t}.h (N,U)
&=& - h(\Lieb{\christo,t}.N,U) - h(N,\Lieb{\christo,t}.U)\\
&=& -\christo(N)h(t,U) + \ip{t,N}h(\christo,U) 
- \christo(U)h(N,t) + \ip{t,U}h(N,\christo).
\end{eqnarray*}
Hence we find
\begin{eqnarray*}
\LieWeyl_\christo (h.N)^D
&=& -\ip{N,N}(h(N))^\perp\wedge \christo^\perp 
+\ip{N,N}\Sum_i (\Lieb{\christo,t_i}.h (N))^\perp
\wedge (\sharp\theta^i)^\perp\\
&=& -\ip{N,N}(h(N))^\perp\wedge \christo^\perp 
-\ip{N,N} \christo^\perp \wedge (h(N))^\perp =0.
\end{eqnarray*}
Let $t_i$ and $\theta^i$ be Fermi parallel with respect to $D$, hence
$0=D_Nt_i-K^D(t_i)$. The claimed identification of the Fermi
derivative of $K.N$ is an elementary application of the product rule
using the above Fermi parallel basis.\qed
%\begin{eqnarray*}
%\lefteqn{\Fermi_NK.N}&&\\
%&=& (D_N(K-\smallfrac{\ip{K,N}}{\ip{N,N}}N))^\perp\wedge D_NN
%+ K^\perp\wedge (D_ND_NN)^\perp\\
%& &\quad
%+\oneover{2}\ip{N,N}\Sum_i 
%(D_N(D_{t_i}K-\smallfrac{\ip{D_{t_i}K,N}}{\ip{N,N}}N))^\perp
%\wedge {\theta^i}^\perp\\
%&=& (D_NK)^\perp \wedge D_NN
%+ K^\perp\wedge (D_ND_NN)^\perp
%+\oneover{2}\ip{N,N}\Sum_i (D^2_{N,t_i}K)^\perp\wedge{\theta^i}^\perp\\
%&&
%+\oneover{2}\ip{N,N}\Sum_i (D_{D_Nt_i}K)^\perp\wedge{\theta^i}^\perp
%-\oneover{2}\Sum_i \ip{D_{t_i}K,N} D_NN\wedge {\theta^i}^\perp\\
%&=& \Killing(K)(N)^\perp \wedge D_NN
%+ K^\perp\wedge (D_ND_NN)^\perp
%+\oneover{2}\ip{N,N}\Sum_i (D^2_{N,t_i}K)^\perp\wedge{\theta^i}^\perp\\
%&=& \Killing(K).N 
%+ K^\perp\wedge(D_ND_NN)^\perp 
%- \ip{N,N}\Sum_i (D_{t_i}\Killing(K)(N))^\perp\wedge{\theta^i}^\perp\\
%&&
%+ \oneover{2}\ip{N,N}\Sum_i (D^2_{N,t_i}K)^\perp\wedge{\theta^i}^\perp\\
%&=& \Killing(K).N 
%+ K^\perp\wedge (D_ND_NN)^\perp
%- \ip{N,N}\Sum_i (D^2_{t_i,N}K)^\perp\wedge{\theta^i}^\perp\\
%&&
%+ \oneover{2}\ip{N,N}\Sum_i(D_{t_i}\dR^DK(N))^\perp \wedge{\theta^i}^\perp
%+ \oneover{2}\ip{N,N}\Sum_i (D^2_{N,t_i}K)^\perp\wedge{\theta^i}^\perp\\
%&=& \Killing(K).N 
%+ K^\perp\wedge(D_ND_NN)^\perp
%- \ip{N,N}\Sum_i (R^D_{t_i,N}K)^\perp\wedge{\theta^i}^\perp\\
%&&
%+ \oneover{2}\ip{N,N}\Sum_i(D_{t_i}\dR^DK(N))^\perp \wedge{\theta^i}^\perp
%- \oneover{2}\ip{N,N}\Sum_i (D^2_{N,t_i}K)^\perp\wedge{\theta^i}^\perp\\
%&=& \Killing(K).N 
%+ K^\perp\wedge \CGeod(N) 
%- \ip{N,N}\Sum_i (\Weylcurv_{t_i,N}K)^\perp\wedge{\theta^i}^\perp,
%\end{eqnarray*}
%where we have used the first Bianchi identity for $R^D$ in the last
%step.

On a conformal vector space $V$ we define the \emph{Cartan product} of
a $1$-form $\alpha\in V^\dual$ with a $k$-form
$\beta\in\Lambda^kV^\dual$ to be an element $\alpha\cartan\beta\in
V^\dual\tens \Lambda^kV^\dual$ which is alternating-free and trace-free:
evaluation on $v\in V$ gives
$$
(\alpha\cartan\beta)(v)
:= \alpha(v)\beta 
- \oneover{k+1}v\inner \alpha\wedge\beta 
- \oneover{n-(k-1)}\sum_i\cip{v,t_i}\theta^i\wedge (\sharp\alpha\inner\beta).
$$
The subspace spanned by these elements is denoted by $V^\dual\cartan
\Lambda^kV^\dual$.

\begin{definition}\label{Firstorderoperatoronforms}
\summary{First order operator on forms} In terms of a Weyl derivative
$D$ we define a linear first order operator acting on alternating
$k$-forms (with $k\ge 1$) of central weight $w$ as follows:
\begin{eqnarray*}
\Twistor^D&\from& C^\infty(M,L^{w+k} \tens \Lambda^k T^\dual)\to
C^\infty(M,L^{w+k} \tens T^\dual \cartan \Lambda^k T^\dual);\\
\Twistor^D(f)&:=& \Sum_i \theta^i\cartan D_{t_i}f.
\end{eqnarray*}
\end{definition}

\begin{proposition}\label{Conformaltwistoroperatorforweightone}
\summary{Conformal twistor operator for weight one} For $w=1$ the
above operator is conformally invariant. In case of conformal affine
space it annihilates the twistor fields coming from
$W=\Lambda^{k+1}\asso V$, hence $\Twistor$ is a twistor operator.
\end{proposition}
\proof
The linear change under a change of Weyl derivative is given
by:
\begin{eqnarray*}
\LieWeyl_\christo \Twistor^D (f)(X)
&=&  \Lieb{\christo,X}.f - \smallfrac{w+k}{k+1} X \inner (\christo\wedge f)
- \smallfrac{w-k+n}{n-(k-1)} \ip{X,\ }\wedge (\christo\inner f)\\
&=&  w\christo(X)f + \ip{X,\ }\wedge(\christo\inner f) - \christo\wedge (X\inner f)\\
&&
- \smallfrac{w+k}{k+1} X \inner (\christo\wedge f)
- \smallfrac{w-k+n}{n-(k-1)} \ip{X,\ }\wedge (\christo\inner f)\\
&=&  (w-1)(\christo(X) f - \oneover{k+1} X \inner \christo\wedge f
- \oneover{n-(k-1)} \ip{X,\ }\wedge (\christo\inner f))\\
&=&  (w-1)(\christo\cartan f)(X).\qed
\end{eqnarray*}

\begin{proposition}\label{Twistorconservation}
Let $f\in C^\infty(M,L^3\tens\Lambda^2T^\dual M)$ be a $2$-form of
central weight $+1$ and $c$ a nonlightlike curve with normalized weightless
tangent vector $N$, then
$$
f.N := f(D_NN,N) + \oneover{n-1} \ip{N,N} \divR^D f(N)
$$ 
is independent of the Weyl derivative $D$ and hence defines a
conformally invariant real number along $c$. For the
derivative of $f.N$ we have
$$
\del_N\ f.N = f(\CGeod(N),N) + (\Twistor f).N,
$$
where $(\Twistor f).N$ is another invariant pairing: for a section
$h\in C^\infty(M,L^{3} \tens T^\dual \cartan \Lambda^2 T^\dual)$ the
following scalar of weight $-1$ is invariant:
$$
h.N := h(N,D_NN,N) + \oneover{n-2}\ip{N,N}\trace D h(N,N),
$$
with $\trace D h(N,N):=\Sum_i D_{t_i} h(N,\sharp\theta^i,N)$. 
Hence if $f$ is a twistor field $\Twistor f=0$ and $c$ is a conformal
geodesic, then the induced $f.N$ is constant.
\end{proposition}

\proof
The term $f.N$ changes under a change of Weyl derivative
by the $1$-form $\christo$ as 
$$
\LieWeyl_\christo (f.N)^D
= -\ip{N,N}f(\christo,N) 
  + \smallfrac{1-(2-n)}{n-1} \ip{N,N} f(\christo,N)
= 0.
$$
To calculate the derivative of $f.N$ notice
$$
\Twistor f(N,D_NN,N) 
= D_Nf(D_NN,N) + \oneover{n-1}\ip{N,N}\divR^D f(D_NN),
$$
and 
\begin{eqnarray*}
\trace D\Twistor f(N,N)
%&:=& \Sum_i D_{t_i} \Twistor f(N,\theta^i,N)\\
%&=& \Sum_i D^2_{t_i,N} f(\theta^i,N)
% - \oneover{n-1} D_N\divR^D f(N)
% + \Sum_i \oneover{n-1} \ip{N,N}\wedge D_{t_i}\divR^D f(\theta^i)\\
%&=& \Sum_i D^2_{t_i,N} f(\theta^i,N) 
% - \oneover{n-1} D_N\divR^D f(N)\\
%&=& \Sum_i R^D_{t_i,N}.f(\theta^i,N) 
% + \Sum_i D^2_{N,t_i} f(\theta^i,N) - \oneover{n-1} D_N\divR^D f(N)\\
%&=& \Sum_i R^D_{t_i,N}.f(\theta^i,N) 
% + (1-\oneover{n-1}) D_N\divR^D f(N)\\
% &=& -\Sum_i f(R^D_{t_i,N}\theta^i,N)
% - \Sum_i f({}\theta^i,R^D_{t_i,N}N)
% + \smallfrac{n-2}{n-1} D_N\divR^D f(N)\\
%&=& f({} \ricci^D(N),N) + \smallfrac{n-2}{n-1} D_N\divR^D f(N)\\
&=& (n-2) \bigl( f(\nric^D(N),N) + \oneover{n-1} D_N\divR^D f(N) \bigr).
\end{eqnarray*}
Finally we obtain:
\begin{eqnarray*}
\lefteqn{\del_N\ f.N}&&\\
&=& D_Nf(D_NN,N) + f(D_ND_NN,N) \\
&&
    + \oneover{n-1}\ip{N,N}D_N\divR^D f(N) 
    + \oneover{n-1}\ip{N,N}\divR^D f(D_NN)\\
&=& \Twistor f(N,D_NN,N) + f(D_ND_NN,N) 
    + \oneover{n-1}\ip{N,N}D_N\divR^D f(N)\\
&=& \Twistor f(N,D_NN,N) + f(\CGeod(N),N) 
    + \oneover{n-2}\ip{N,N}\trace D\Twistor f(N,N).\qed
\end{eqnarray*} 

\begin{application}\label{Associateddistribution}
Any nonlightlike curve $c$ induces a distributional section $J$ of
$L^{-n-3}\tens TM\cartan\Lambda^2 TM$ as follows: the evaluation on a
test section $h\in C^\infty_0(M,L^{3}\tens T^\dual M\cartan\Lambda^2
T^\dual M)$ is defined by $\ipp{J,h} := \int_c h.N$. (Note that $h.N$
is a section in $L^{-1}(TM)$ which can be integrated over $c$ using
the conformal structure to identify $L^{-1}(Tc)\isomorph L^{-1}(TM)$,
see remark \ref{Pullbackofdensities}). The above theorem shows that for any
$f\in C^\infty_0(M,L^3\tens\Lambda^2T^\dual M)$ we have
$$
\ipp{J,\Twistor f} = \int_c (\Twistor f).N = - \int_c f(\CGeod(N),N).
$$
Hence the distributional $J$ is divergence-free $\Twistor^\dual J=0$
iff $c$ is a conformal geodesic.
\end{application}

\begin{remark}\label{PointlikesourcesinEinsteinstheory}
\summary{Pointlike sources in Einstein's theory} The above application
is a conformal analogue of a well known observation in general
relativity: a curve gives rise to a distributional symmetric tensor
$T$ (of central weight $1-n$) defined by $\ipp{T,\alpha}:=\int_c
\alpha(N,N)$ and $T$ is divergence-free iff $c$ is a geodesic
$D_NN=0$. If $m$ denotes a mass then $mT$ has the interpretation being
an energy momentum tensor of the particle. Viewed as a source for
gravity, the conservation law $0=\divR^D (mT)$ forces the particle to
move along a geodesic.
\end{remark}

\begin{remark}
For twistors induced by $W=\Lambda^{k+1}{\asso V}$ with $k\ge 2$
(i.e.~elements in the kernel of the above defined operator $\Twistor$
on $k$-forms) it is possible to construct an invariant weightless
$(k-2)$-form and a weightless $(k+1)$-form along a curve. In the flat
case they are Fermi parallel along conformal geodesics, but on a
curved conformal manifold the Weyl curvature prevents a true
conservation law. An example of this phenomenon was given in
proposition \ref{Killingconservation} with Killing fields in $n>3$
dimensions.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{First order conformal invariants}
\label{Firstorderconformalinvariants}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The previous three sections dealt with conformal invariants along
curves. We will now focus on conformal differential invariants for
sections of associated bundles. We begin with the first order linear
invariants of Fegan \cite{Fegan}, who classified all conformally invariant
differential operators of first order. For the proof we will follow
Gauduchon \cite{Gauduchon}. We will apply this result to give new bilinear
invariants: a favourite example of such a pairing is the Lie bracket
between vector fields:
$$
\Lieb{,}\from
C^\infty(M,TM)\cross C^\infty(M,TM)
\to C^\infty(M,TM).
$$
Another examples is a pairing between two scalars to give a $1$-form
of weight $+1$:
$$
\vee\from
C^\infty(M,L^1)\cross C^\infty(M,L^1) \to
C^\infty(M,L^{2}\tens T^\dual M),
$$
which is defined on scalar densities $f_1$ and $f_2$ by
$$
f_1\vee f_2 := Df_1\tens f_2 - f_1\tens Df_2.
$$
Here $D$ is a chosen Weyl derivative, but $\vee$ is indeed independent
of that choice: if $\other{D}=D+\christo$ is another Weyl derivative,
then $\other{D}f_1-Df_1 =\christo\tens f_1$ and $\other{D}f_2-Df_2
=\christo\tens f_2$. These two examples of pairings in fact only need
the smooth structure, whereas we are interested in invariants
involving the conformal structure as well: note that the Lie bracket
of two conformal Killing fields is a conformal Killing field, since
the following product rule holds:
$$
\Lie_{\Lieb{X,Y}}\com = \Lie_X\Lie_Y\com - \Lie_Y\Lie_X\com.
$$
Likewise the above pairing $\vee$ between scalar densities gives a
vector field which can be applied to the Killing operator $\Killing(X)
= \oneover{2} \Lie_X\com = \Sym_0^2 DX$ (see definition
\ref{Killingoperator}) to give:
\begin{eqnarray*}
\Killing (f_1\vee f_2)
&=& \Killing (Df_1\tens f_2 - f_1\tens Df_2)\\
&=& (\Sym_0^2 D^2f_1) \tens f_2 + Df_1\cartan Df_2 
- Df_1\cartan Df_2 - f_1\tens \Sym^2_0 D^2f_2\\
&=& (\Hesse f_1) \tens f_2 - f_1\tens (\Hesse f_2).
\end{eqnarray*}
(The two individual Ricci corrections for $\Hesse$, see proposition
\ref{ConformalHessian}, cancel in the difference). In particular two
Fierz twistors $f_1$, $f_2$ (in the kernel of the conformal Hessian)
induce a conformal Killing field. In the case of an affine conformal
space $\A$ over $V$ we already constructed conformal twistor fields in
proposition \ref{Conformaltwistorfields} which were induced by
elements in a representation space $W$ for the Lie algebra of
conformal Killing fields $\g=\so(\asso V)$. Fierz twistors are induced
by elements in the defining representation $\asso V$ (see remark
\ref{Einsteinmetricsfromtheconformalclass}) and Killing fields are
induced by elements in the adjoint representation $\so(\asso V)$. On
the algebraic level we have a skew symmetric $\g$-equivariant map
$$
{\asso V} \tens {\asso V} \to \so(\asso V),
$$
and the above pairing $\vee$ is an extension of that map to curved
conformal manifolds. Likewise the Lie bracket between smooth vector
fields is an extension of the algebraic Lie bracket
$$
\so({\asso V}) \tens \so({\asso V}) \to \so(\asso V).
$$ From these two examples where algebraic pairings correspond to
differential pairings we also expect a differential pairing
corresponding to $\so(\asso V)\tens {\asso V} \to {\asso V}$. This is
given by the Lie derivative of a scalar density in the direction of a
vector field:
$$
\Lie \from C^\infty(M,TM)\cross C^\infty(M,L^1)\to C^\infty(M,L^1).
$$
The induced product rule $\Hesse( \Lie_Xf) =\ldots $ becomes more
involved, since not only a second order pairing will be needed on the
right hand side (between $\Killing X$ and $f$) but there will be also
an additional curvature term. This and the next section cover first
and second order operators and pairings. More examples of product
rules can be found in chapter
\ref{Conformallinearfieldtheories} on linear field
theories. We obtain a general result on the conformal affine space,
see theorems \ref{ConformalBGGcomplex} and
\ref{Conformalexteriorpairings}.

We turn now to a systematic study of first order invariants: in terms
of a Weyl derivative $D$ we will define a first order differential
operator, encoding, in a trivial fashion, all the information of the
$1$-jet of a section: $\jet^1(e)\in C^\infty(M,\Jet^1(EM))$. It will
turn out that the operator behaves well if we would change the Weyl
derivative by a $1$-form $\christo\in C^\infty(M,T^\dual M)$ in the
sense that the first order information transforms as expected from the
algebraic action of $V^\dual\subset\g$ on affine functions
(polynomials of degree $1$) see paragraph \ref{grepfromlrep}. This allows to
classify algebraically all first order invariants discussed here.

\begin{definition}
A Weyl derivative $D$ induces a first order jet operator defined by
$$
\jet^{1,D}\from C^\infty(M,EM)
\to C^\infty(M,E \dsum T^\dual \tens EM);\quad
\jet^{1,D}e := e \dsum De.
$$
\end{definition}

Any linear first order differential operator $\Nabla\from
C^\infty(M,EM)\to C^\infty(M,FM)$ between associated bundles in Weyl
geometry is given by a $\CO(V)$-equivariant linear map $E\dsum
V^\dual\tens E\to F$, which is nontrivial on $V^\dual\tens E$. This in
terms gives a bundle map $\pi\from EM\dsum T^\dual M\tens EM\to FM$
and finally the differential operator $\Nabla e=\pi\circ \jet^{1,D}$.
If $E$ and $F$ are irreducible then $E$ and $F$ have central weights
$w_E$ and $w_F$ and $V^\dual\tens E\to F$ being nontrivial implies
$w_F=w_E-1$ and $E\to F$ is trivial. Hence in Weyl geometry the study
of first order operators between irreducible representations reduces
to the classification of irreducible sub representations of
$V^\dual\tens E$: an application of Weyl's character formula in
representation theory (see \cite{Fegan} or \cite{FultonHarris}) gives
the following

\begin{proposition}\label{Weylcharacterformula}
If $E$ is a finite dimensional irreducible representation of $\co(V)$,
then $V^\dual\tens E$ splits under the action of $\co(V)$ into a
direct sum of pairwise nonisomorphic irreducible representations.
\end{proposition}

\begin{example}
Let $E$ be a $\co(V)$-representation of central weight $w_E$, then we
denote by $E^w:=L^{w-w_E}\tens E$ the same $\so(V)$-representation but
of central weight $w$. With this notation we give the following
examples to the above proposition (to ensure irreducibility of all the
below representations we may assume $n>5$ and $n>2(k+1)$):
\begin{eqnarray*}
V^\dual\tens L^w
&=& V^{w-1},\\
V^\dual\tens V^w
&=& L^{w-1} \dsum (\Lambda^2 V^\dual)^{w-1} \dsum 
(\Sym^2_0(V^\dual))^{w-1},\\
V^\dual\tens (\Lambda^k V^\dual)^w
&=& (\Lambda^{k-1} V^\dual)^{w-1}
    \dsum (\Lambda^{k+1} V^\dual)^{w-1}
    \dsum (V^\dual \cartan \Lambda^k V^\dual)^{w-1}.
\end{eqnarray*}
\end{example}

\begin{definition}\label{SteinWeiss}
\summary{Stein Weiss operators} Let $E$ and $F$ be a finite
dimensional irreducible $\CO(V)$-representation and denote by
$\Hom_{\CO(V)}(V^\dual\tens E,F)$ the space of $\CO(V)$-e\-qui\-var\-iant
linear maps $\pi\from V^\dual\tens E\to F$. Associated to a nontrivial
$\pi$ is a linear first order differential operator on a conformal
manifold $M$ with Weyl derivative $D$:
$$
\Nabla^D\from C^\infty(M,EM) \to C^\infty(M,FM);\quad 
\Nabla^D(e) := \pi( De ).
$$
\end{definition}

\begin{remark}\label{AdjointSteinWeiss}
A Stein Weiss operator as above on a Weyl manifold always has an
adjoint which is also Stein Weiss: the zero order pairing $X\from
E\tens L^{-n}\tens F^\dual \to L^{-n}\tens V$ is given by the symbol
$X(e,\phi):=\Sum_i t_i \tens \ip{\pi(\theta^i\tens e),\phi}$ with a
dual basis $t_i$ and $\theta^i$. The divergence formula follows
\begin{eqnarray*}
\divR (X(e,\phi))
&=& \ip{\pi(De),\phi} + \Sum_i \ip{\pi(\theta^i\tens e),D_{t_i}\phi}\\
&=& \ip{\Nabla^De,\phi} - \ip{e,{\Nabla^\dual}^D\phi}.
\end{eqnarray*}
\end{remark}

The next aim is to study the behavior of a Stein Weiss operator
$\Nabla^D$ under a change of Weyl derivative. In example
\ref{Firstorderchange} we already observed that on a conformal
manifold $M$ the covariant derivative of an associated bundle $EM$
changes under a linear change of Weyl derivative $\christo\in
C^\infty(M,T^\dual M)$ as:
$$
(\LieWeyl_\christo D)_{X}e 
= \Lieb{\christo,X}.e,
$$
where $e\in C^\infty(M,EM)$ is a section and $X\in C^\infty(M,TM)$ is a
vector field. As was explained in paragraph \ref{grepfromlrep} for a conformal
vector space $V$ any $\co(V)$-representation $E$ induces a
$\g=V\dsum\co(V)\dsum V^\dual$ representation on $\Sym V^\dual\tens E$
via the Lie derivative. In what follows we like to view $E$ as a left
$\co(V)$-representation and we will use the induced left action of
$\g$ on $\Sym V^\dual\tens E$. The restriction of this action to the
Abelian subalgebra $V^\dual\subset \g$ is then given by
$$
\Lie\from V^\dual \tens \Sym^{k-1} V^\dual \tens E \to
\Sym^{k} V^\dual \tens E,
$$
with 
$$
\oneover{k!} \Lie_\christo f(v^k)
:= \oneover{(k-1)!} \Lieb{\christo,v}.(f(v^{k-1})) 
  - \oneover{2(k-2)!} f((\Lieb{\christo,v}v)v^{k-2}).
$$
Special cases are as follows:
\begin{eqnarray*}
k=0&:& \Lie_\christo f =\Lie(\christo\tens f)
= 0,\\
k=1&:& \Lie_\christo f (v) =\Lie(\christo\tens f)(v) 
= \Lieb{\christo,v}.f.
\end{eqnarray*}

The cases $k=0$ and $k=1$ show the following:

\begin{proposition}
Under a change of Weyl derivative the linear change of the above first
jet operator is given by the Lie derivative applied to the zero jet of
the section (with a dual basis $t_i$, $\theta^i$ of $TM$):
$$
(\LieWeyl_\christo \jet^{1,D})(e)
= \Lie_{\christo} e
= 0\dsum \Sum_i \theta^i\tens \Lieb{\christo,t_i}.e.
$$
\end{proposition}

For a Stein Weiss operator $\Nabla^D$ we get
$(\LieWeyl_\christo\Nabla^D)(e)=\pi\circ (\LieWeyl_\christo D)(e)
=\pi\circ \Lie(\christo\tens e)$. Those operators which do not change
$\LieWeyl_\christo\Nabla^D=0$ for all $\christo$ and $D$ are
conformally invariant. Algebraically this question reduces to the
study of the composition:
$$
V^\dual\tens E \stackrel{\Lie}{\longrightarrow}
V^\dual\tens E \stackrel{\pi}{\longrightarrow}
F:
$$

\begin{proposition}
The equivariant linear maps $\pi\in\Hom_{\CO(V)}(V^\dual\tens E,F)$
with $\pi\circ\Lie=0$ classify all conformally invariant linear first
order operators between bundles associated to irreducible
$\CO(V)$-representations $E$ and $F$.
\end{proposition}

Following Gauduchon \cite{Gauduchon} we will split
$\Lieb{\christo,X}\in\co(V)$ into its trace and skew part: if $w_E$
denotes the central weight of $E$ (under the action of the centre of $\co(V)$)
then define $B$ to be the induced $\so(V)$-action
$$
B\from V^\dual \tens E \to  V^\dual \tens E;\quad
\Lie(\christo\tens e) 
= \sum_i \theta^i \tens \Lieb{\christo,t_i}.e
= w_E\ \christo\tens e + B(\christo \tens e),
$$
where ${t_i}$, ${\theta^i}$ is a dual basis
of $V$. Explicitly $B(\christo\tens e)=\sum_i \theta^i
\tens (\christo\tens t_i - \flat t_i\tens\sharp\christo).e$.

According to proposition \ref{Weylcharacterformula} the tensor product
$V^\dual\tens E$ splits into a direct sum of pairwise nonisomorphic
$\co(V)$-sub representations $F^k\subset V^\dual\tens E$ with
$k=1\ldots N$. We will use the inclusions $\iota_k\from F^k\to
V^\dual\tens E$ and projections $\pi_k\from V^\dual\tens E\to F^k$
with normalization $\pi_k\circ\iota_k=\id_{F^k}$.  For any
$\co(V)$-equivariant map $B\in\gl(V^\dual\tens E)$ we know that
$\pi_l\circ B\circ \iota_k$ is trivial for $l\neq k$, since the $F^k$
are irreducible and pairwise nonisomorphic. In case $l=k$ we apply
Schur's lemma (see e.g.~\cite{FultonHarris}) to get multiples of the
identity: $\pi_k\circ B\circ \iota_k= b_k \id_{F^k}$. Therefore any
such $\co(V)$-equivariant map $B$ is determined by its eigenvalues
$b_k$:
$$
B = \Sum_{k=1}^N b_k \pi_k\circ \iota_k.
$$

\begin{theorem}\label{Fegansfirstorderconformalinvariants}
\summary{Fegan's first order conformal invariants} Let $E$ and $F$ be
irreducible representations of $\CO(V)$ and
$\pi\in\Hom_{\CO(V)}(V^\dual\tens E,F)$ a nontrivial map. The
associated Stein Weiss operator $\Nabla^D=\pi\circ D$ changes under a
change of Weyl derivative $D$ by $\christo$ as
$$
\Nabla^{D+\christo} e - \Nabla^D e
= \LieWeyl_\christo \Nabla^D (e)
= \pi\circ\Lie(\christo\tens e)
\stackrel{!}{=} (w_E+b)\pi(\christo\tens e),
$$
where $e$ is a section of $EM$, $w_E$ is the central weight of $E$ and
$b$ is a number directly associated to the two
$\so(V)$-representations $E$ and $F$. Conformally invariant linear
first order operators are characterized by $w_E=-b$.
\end{theorem}

\proof After writing $B$ in terms of its eigenvalues $B = \sum_k^N b_k
\pi_k\circ \iota_k$ on the invariant subspaces $F^k\subset
V^\dual\tens E$ we find for the difference (with $F=F^l$):
\begin{eqnarray*}
\pi\circ\Lie(\christo\tens e)
&=& \pi \circ \sum_i \theta^i \tens \Lieb{\christo,t_i}.e\\
&=& \pi \circ (w_E\ \christo\tens e + B(\christo\tens e))\\
&=& (w_E + b_l) \pi \circ (\christo\tens e).\qed
\end{eqnarray*}

\begin{remark}
The \emph{Killing form} on $\so(V)$ is proportional to the inner product
induced from the defining representation $V$ which is given by
$\ip{A,B}:=-\oneover{2}\trace(A\circ B)$ with $A,B\in\so(V)$. An
orthonormal basis in $\so(V)$ is given by $E_{ij}:=\ip{e_i,\ }e_j -
\ip{e_j,\ }e_i \in \so(V)$ with $1\leq i<j\leq n$ and $e_i\in V$ an
orthonormal basis of $V$. With this convention the \emph{Casimir
element} $\Casimir \in \Centre(\U(\so(V)))$ is $\Casimir = \Sum_{i<j}
E_{ij} E_{ij}$. Denote by $\Casimir(W)\in\End(W)$ the \emph{induced
endomorphism} on the $\so(V)$-representation $W$. Applied to the
defining representation we find $\Casimir(V)=(1-n)\id_V$ and
$\Casimir(\Lambda^kV)=k(k-n)\id$. In general, if $W$ is irreducible
then by Schur's lemma $\Casimir(W)$ is a multiple of the identity and
that multiple is called the \emph{Casimir value}
$\Casimir(W)=\casimir(W)\id_W$. Fegan observed, that the endomorphism
$B$ can be written in terms of various images of the Casimir element:
$$
B
=\oneover{2}
 ( \Casimir(V^\dual \tens E)
   -\id_{V^\dual} \tens \Casimir(E)
   -\Casimir(V^\dual) \tens \id_{E} ).
$$
This allows to calculate eigenvalues of $B$ once the Casimir values of 
the irreducible sub representations $F^k\subset V^\dual \tens E$ are 
known:
$$
b_k
=\oneover{2}( \casimir(F^k) -\casimir(E) -(1-n) ).
$$
\end{remark}

\begin{discussion}
\summary{First order pairings} The next aim is to study first order
bilinear pairings in general. In the introduction of this chapter we
mentioned two simple examples. In general suppose $E_1$, $E_2$ and $F$
are $\CO(V)$-representations with a $\CO(V)$-equivariant linear map
$$
Q\from 
(E_1\dsum V^\dual\tens E_1) \tens
(E_2\dsum V^\dual\tens E_2) \to F,
$$
then $\vee^D\from C^\infty(M,E_1M)\cross C^\infty(M,E_2M) \to
C^\infty(M,FM)$ defined by $e_1 \vee^D e_2
:=Q(\jet^{1,D}(e_1)\tens\jet^{1,D}(e_2))$ is a bilinear differential
pairing of first order in Weyl geometry. We will consider those
pairings which are truly first order in both $e_1$ and $e_2$, hence we
assume such a $Q$ to be nontrivial when restricted to $(V^\dual\tens
E_1) \tens (E_2\dsum V^\dual\tens E_2)$ and $(E_1\dsum V^\dual\tens
E_1) \tens (V^\dual\tens E_2)$. If $E_1$, $E_2$ and $F$ are
irreducible the action of the centre of $\CO(V)$ determines such a $Q$
to be nontrivial only on the following subspace
$$
Q\from 
(V^\dual\tens E_1 \tens E_2)\dsum (V^\dual\tens E_1\tens E_2) \to F.
$$
The corresponding pairing is then clearly given by 
$$
e_1 \vee^D e_2
:=Q(\Sum_i \theta^i\tens D_{t_i}e_1\tens e_2\dsum \Sum_i \theta^i\tens
e_1\tens D_{t_i}e_2).
$$
This dependence of $\vee^D$ on the Weyl derivative leads to study the
following composition of maps:
$$
V^\dual\tens E_1 \tens E_2 
\stackrel{\Lie\tens\id_{E_2}\dsum \id_{E_1}\tens\Lie}{\longrightarrow}
(V^\dual\tens E_1 \tens E_2)\dsum (V^\dual\tens E_1\tens E_2)
\stackrel{Q}{\longrightarrow}
F,
$$
where the first map is given by $\christo\tens e_1\tens e_2\mapsto
\sum_i \theta^i\tens\Lieb{\christo,t_i}.e_1\tens e_2 \dsum \sum_i
\theta^i\tens e_1\tens \Lieb{\christo,t_i}.e_2$.
\end{discussion}

\begin{proposition}\label{Algebraicfirstorderpairings}
Conformally invariant bilinear first order differential pairings
$$
\vee\from C^\infty(M,E_1M)\cross C^\infty(M,E_2M)\to C^\infty(M,FM),
$$
between associated bundles from irreducible $\CO(V)$-representations
$E_1$, $E_2$ and $F$ are classified by equivariant linear maps
$$
Q\in\Hom_{\CO(V)}(V^\dual\tens E_1\tens E_2
\dsum V^\dual\tens E_1\tens E_2,F),
$$ 
such that $Q\circ (\Lie\tens\id_{E_2}\dsum \id_{E_1}\tens\Lie)=0$. 
\end{proposition}

A simple subclass of examples is given in the following case:

\begin{definition}
\summary{Simple projections} Suppose that $E_1$, $E_2$ and $F\subset
V^\dual\tens E_1 \tens E_2$ are irreducible $\CO(V)$-representations
with irreducible $F_1\subset V^\dual\tens E_1$ and $F_2\subset V^\dual
\tens E_2$, such that the projection
$$
Q\from V^\dual\tens E_1\tens E_2 \to F
$$
factors through $F_1\tens E_2 \to F$ and $E_1\tens F_2\to F$. Such a
map $Q$ will be called \emph{simple}.
\end{definition}

\begin{remark}
If $F$ has multiplicity one inside $V^\dual\tens E_1\tens E_2$,
i.e.~$1=\dim(\Hom_{\CO(V)}(V^\dual \tens E_1 \tens E_2,F))$ then there
are unique spaces $F_1$, $F_2$ as required above and $Q\from
V^\dual\tens E_1\tens E_2 \to F$ is simple.
\end{remark}

\begin{proposition}\label{Simpleconformallyinvariantfirstorderpairings}
\summary{Simple conformally invariant first order pairings} Suppose
that $Q\from V^\dual \tens E_1 \tens E_2 \to F$ is simple. Then there
is a conformally invariant first order differential pairing
$$
\vee\from C^\infty(M,E_1M) \cross C^\infty(M,E_2M) \to C^\infty(M,FM),
$$
which in terms of a Weyl derivative $D$ applied to sections $e_1$ and 
$e_2$ is given by
$$
e_1 \vee e_2
:=(w_2+b_2) Q(\Sum_i \theta^i\tens D_{t_i}e_1\tens e_2)
- (w_1+b_1) Q(\Sum_i \theta^i\tens e_1\tens D_{t_i}e_2),
$$
where $w_1$, $w_2$ are the central weights of $E_1$, $E_2$ (the
central weight of $F$ is clearly $w_1+w_2-1$) and $-b_1$, $-b_2$ are
the weights for which the Stein Weiss operators associated to $E_1$,
$F_1$ and $E_2$, $F_2$ are conformally invariant.
\end{proposition}

\proof Applying Fegan's result, theorem
\ref{Fegansfirstorderconformalinvariants}, the following composition is
zero:
\begin{eqnarray*}
\lefteqn{
((w_2+b_{2}) Q\dsum (-w_1-b_{1}) Q)
\circ (\Lie\tens\id_{E_2}\dsum \id_{E_1}\tens\Lie)
(\christo\tens e_1\tens e_2)}&&\\
&=& (w_2+b_{2})(w_1+b_{1}) Q(\christo\tens e_1\tens e_2)
  - (w_1+b_{1})(w_2+b_{2}) Q(\christo\tens e_1\tens e_2)
= 0.\qed
\end{eqnarray*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Second order conformal invariants}
\label{Secondorderconformalinvariants}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In terms of a Weyl derivative $D$ we will define a second order
differential operator, encoding all the information of the $2$-jet of
a section: $\jet^2(e)\in C^\infty(M,\Jet^2(EM))$. We will use the
Ricci curvature to correct the second covariant derivative such that
this operator behaves well if we change the Weyl derivative
by a $1$-form $\christo\in C^\infty(M,T^\dual M)$. This allows to
classify algebraically all second order operators and pairings. We
will give examples which will become relevant in the next chapter on
linear field theories for gravity.

In example \ref{Secondorderchange} we already observed that on a
conformal manifold $M$ the second covariant derivative of an
associated bundle $EM$ changes under a change of Weyl derivative as:
$$
(\LieWeyl_\christo D^2)_{X,Y}e 
=  \Lieb{\christo,X}.D_Ye + \Lieb{D_X\christo,Y}.e + 
      \Lieb{\christo,Y}.D_Xe - D_{\Lieb{\christo,X}.Y}e,
$$
where $e\in C^\infty(M,EM)$ is a section, $X,Y\in C^\infty(M,TM)$ are 
vector fields and $\christo\in C^\infty(M,T^\dual M)$ is the change of 
the Weyl derivative $D$. The $D\christo$ term can be 
expressed as the linear change of the normalized Ricci curvature:
$$
(\LieWeyl_\christo \nric^D)_{X,Y}
= - D_X\christo(Y).
$$

\begin{definition}\label{RiccicorrectedsecondWeylderivative}.
For a Weyl derivative $D$ we call
$$
D^2_{X,Y}e + \Lieb{\nric^D(X),Y}.e
$$
the \emph{Ricci corrected second Weyl derivative} of the bundle $EM$. 
\end{definition}

The skew part of the above Ricci corrected second Weyl derivative is
given by the Weyl curvature tensor, hence
$$
D^2_{X,Y}e + \Lieb{\nric^D(X),Y}.e - \oneover{2}\Weylcurv_{X,Y}.e
$$
is symmetric in $X$ and $Y$. Note that this is not simply the
symmetric part of the second derivative. Instead it is characterized
by the fact that under a linear change of the Weyl derivative only
first order terms algebraic in $\christo$ and no $D\christo$ terms
occur. In terms of a Weyl derivative $D$ we will next define a linear
second order operator containing all the information of the $2$-jet of
a section $\jet^2(e)\in C^\infty(M,\Jet^2(EM))$ (we will use the
following notation: $\Sym^{\leq k} V^\dual := \R \dsum V^\dual \dsum
\ldots \dsum\Sym^k V^\dual$):

\begin{definition}\label{Secondjetoperator}
\summary{Second jet operator} In Weyl geometry we define the
\emph{$2$-jet operator} to be the following linear differential
operators of second order:
$$
\jet^{2,D}\from C^\infty(M,EM)
\to C^\infty(M,\Sym^{\leq 2}T^\dual\tens EM),
$$
with
$$
\jet^{2,D}e
:= e \dsum De \dsum D^2e 
+ \Lieb{\nric^D,\ }.e -\oneover{2}\Weylcurv.e\quad.
$$
\end{definition}

Any linear second order differential operator $\Nabla\from
C^\infty(M,EM)\to C^\infty(M,FM)$ between associated bundles in Weyl
geometry is given by a linear map $(\Sym^{\leq 2}V^\dual)\tens E\to
F$, which is $\CO(V)$-equivariant and nontrivial on
$\Sym^2V^\dual\tens E$. This in terms gives a bundle map and the
differential operator $\Nabla^D =\pi\circ \jet^{2,D}$.

If $E$ and $F$ are irreducible then $E$ and $F$ have central weights
$w_E$ and $w_F$ and $\Sym^2V^\dual\tens E\to F$ being nontrivial
implies $w_F=w_E-2$ and $E\to F$ as well as $V^\dual\tens E\to F$ are
trivial. Hence in Weyl geometry the study of second order operators
between irreducible representations reduces to the classification of
irreducible sub representations of $\Sym^2V^\dual\tens E$. Hence given
a nonzero $\CO(V)$-equivariant linear map $\pi\from \Sym^2V^\dual\tens
E\to F$ with $E$ and $F$ irreducible we get a differential operator
$\Nabla^D:=\pi\circ(D^2 + \Lieb{\nric^D,\ }. -\oneover{2}\Weylcurv.)$.

As for the $1$-jet in Weyl geometry, the linear change of the
curvature corrected second derivative under a change of Weyl
derivative is given by the algebraic Lie derivative:
$$
\Lie\from V^\dual \tens \Sym^{k-1} V^\dual \tens E \to
\Sym^{k} V^\dual \tens E,
$$
with 
$$
k=2\ :\ \Lie_\christo f (v^2) =\Lie(\christo\tens f)(v^2) 
= 2\Lieb{\christo,v}.(f(v)) - f(\Lieb{\christo,v}v).
$$

\begin{proposition}\label{Linearchangeofthesecondjet}
\summary{Linear change of the second jet} Under a change of Weyl
derivative the linear change of the above jet operator is given by the
Lie derivative applied to the lower jet:
\begin{eqnarray*}
(\LieWeyl_\christo \jet^{2,D})(e)
&=& \Lie_\christo (\jet^{1,D}e)
= 0\dsum \Lieb{\christo,\ }.e \dsum
(\Lieb{\christo,\ }.De + \Lieb{\christo,\ }.De - D_{\Lieb{\christo,\ }\ }e).
\end{eqnarray*}
\end{proposition}

The linear change of a second order operator $\Nabla^D$ between
irreducible $E$, $F$ is then $\LieWeyl_\christo(\Nabla^D)(e)
=\pi\circ\Lie_\christo(De)$. Hence $\Nabla^D$ is conformally invariant
if the following composition of maps is zero:
$$
V^\dual\tens (V^\dual\tens E)
\stackrel{\Lie}{\longrightarrow}
\Sym^2 V^\dual\tens E
\stackrel{\pi}{\longrightarrow}
F:
$$

\begin{proposition}
The linear equivariant maps $\pi\in\Hom_{\CO(V)}(\Sym^2V^\dual\tens
E,F)$ with $\pi\circ\Lie=0$ classify all conformally invariant linear
second order operators between bundles associated to irreducible
$\CO(V)$-representations $E$ and $F$.
\end{proposition}

Such a classification has been given by Branson in
\cite{BransonNonlinear} and \cite{BransonSecond}. Instead of
presenting the full classification, we will only construct examples
directly or with the help of Fegan's result.

\begin{example}\label{ConformalLaplacian}
\summary{Conformal Laplacian} For $E=L^w$ and $F=L^{w-2}$ the only map
$\Sym^2 V^\dual\tens E \to F$ is given by (multiples of) the conformal
structure: $\pi(\alpha^2\tens \mu)=\cip{\alpha,\alpha}\mu$. In terms
of a Weyl derivative $D$ this defines a Laplace operator
$$
\Laplace^D\from C^\infty(M,L^{w})\to C^\infty(M,L^{w-2}),
$$
given by
\begin{eqnarray*}
\Laplace^D(e)
&=&\trace_\com \jet^{2,D}(e) 
= \trace_\com D^2e + \trace_\com\Lieb{\nric^D,\ }.e
= \trace_\com D^2e + w\trace_\com \nric^D\tens e\\
&=& \trace_\com D^2e + w\nscal^D e,
\end{eqnarray*}
with the normalized scalar curvature $\nscal^D
:=\trace_\com\nric^D$. For the linear change we directly find:
\begin{eqnarray*}
\LieWeyl_\christo \Laplace^D (e)
&=& \Sum_{i,j} \cip{\theta^i,\theta^j}(
\Lieb{\christo,t_i}.D_{t_j}e 
+ \Lieb{\christo,t_j}.D_{t_i}e
- D_{\Lieb{\christo,t_i}t_j}e)\\
&=& 2(w-(2-n)/2) D_{\sharp\christo} e.
\end{eqnarray*}
Hence for $w=(2-n)/2$ this composite is zero and we get a conformally
invariant Laplace operator
$$
\Laplace\from C^\infty(M,L^{(2-n)/2})\to C^\infty(M,L^{(-2-n)/2}).
$$
\end{example}

\begin{example}\label{Bransonsoperator}
\summary{Branson's operator on forms} We will define Branson's
operator acting between $k$-forms for $k\geq 1$ which is part of his
classification \cite{BransonNonlinear}. There are three Stein Weiss
operators defined on $k$-forms according to the decomposition
$V^\dual\tens \Lambda^k V^\dual= \Lambda^{k+1}V^\dual \dsum
L^{-2}\tens\Lambda^{k-1}V^\dual \dsum V^\dual\cartan \Lambda^k
V^\dual$, which are the exterior derivative $\dR^D$, exterior
divergence $\divR^D$ and the twistor operator $\Twistor^D$ see
definition \ref{Firstorderoperatoronforms}. The second order operator
we are looking for is based upon a linear combination of
$\dR^D\circ\divR^D$ and $\divR^D\circ\dR^D$ acting on $k$-forms $f\in
C^\infty(M,L^{w+k}\tens\Lambda^kT^\dual M)$ of central weight
$w$. Hence the symbol is a linear combination of $\wedge
(\id_{V^\dual}\tens \inner)$ and $\inner (\id_{V^\dual}\tens \wedge)$
which we will apply to the Ricci corrected second derivative. For the
individual linear changes we find using $\co(V)$-equivariance and the
examples in \ref{Firstorderchange}:
\begin{eqnarray*}
\lefteqn{\LieWeyl_\christo
\wedge (\id_{V^\dual}\tens \inner) (D^2 + \Lieb{\nric^D,\ })f}&&\\
%&=& \Sum_{ij} \theta^i\wedge \theta^j\inner \Lieb{\christo,t_i}.D_{t_j}f 
%+  \theta^i\wedge \theta^j\inner \Lieb{\christo,t_j}.D_{t_i}f 
%-  \theta^i\wedge \theta^j\inner D_{\Lieb{\christo,t_j}t_i}f\\
%&=& \Sum_{ij} \theta^i\wedge \Lieb{\christo,t_i}.(\theta^j\inner D_{t_j}f)
%+  (w+(n-k)) \Sum_i\theta^i\wedge \christo\inner D_{t_i}f \\
&=& (w+k-2)\, \christo \wedge \divR^D f
- (w+n-k)\, \christo\inner \dR^D f
+ (w+n-k) D_{\sharp\christo} f,\\
%\end{eqnarray*}
%Similarly
%\begin{eqnarray*}
\lefteqn{\LieWeyl_\christo
\inner (\id_{V^\dual}\tens \wedge)(D^2 + \Lieb{\nric^D,\ })f}&&\\
&=& -(w+k)\, \christo\wedge\divR^D f
+ (w+n-k-2)\, \christo\inner \dR^D f 
+ (w+k) D_{\sharp\christo} f.
\end{eqnarray*}
The linear combination killing the $D_{\sharp\christo} f$ terms is called
\emph{Branson's operator}:
\begin{eqnarray*}
\Branson^D
&\from& C^\infty(M,L^{w+k}\tens\Lambda^k T^\dual M)
\to C^\infty(M,L^{w-2+k}\tens\Lambda^k T^\dual M);\\
\Branson^D
&:=& ( (w+k)\wedge (\id_{V^\dual}\tens \inner)
- (w+n-k) \inner (\id_{V^\dual}\tens \wedge))(D^2 + \Lieb{\nric^D,\ }).
\end{eqnarray*}
The linear change under changing the Weyl derivative is therefore:
\begin{eqnarray*}
\LieWeyl_\christo\Branson^D
&=& ((w+k)(w+k-2) + (w+n-k)(w+k)) \christo \wedge \divR^D f\\
&& - ((w+k)(w+n-k)+(w+n-k)(w+n-k-2))  \christo\inner \dR^D f\\
&=& (2w+n-2) \Big( (w+k)\christo \wedge \divR^D f
- (w+n-k) \christo\inner \dR^D f \Big).
\end{eqnarray*}
Hence for $w=(2-n)/2$ we obtain a conformally invariant operator
$$
\Branson
\from C^\infty(M,L^{(2-n)/2+k} \tens \Lambda^kT^\dual M)
\to C^\infty(M,L^{(-2-n)/2+k} \tens \Lambda^kT^\dual M).
$$
\end{example}

The last two examples where selfadjoint operators. The conformal
Hessian from proposition \ref{ConformalHessian}, the Bianchi and Bach
operator in $n=4$ dimensions, see propositions
\ref{FourdimensionalBianchioperator} and
\ref{Bachoperator}, are examples of the following
definition:

\begin{definition}
\summary{Simple projection} Suppose that $E$,
$F$ and $G$ are irreducible $\CO(V)$-representations with irreducible
$F\subset V^\dual\tens E$ and $G\subset \Sym^2_0 V^\dual \tens E$,
such that the projection
$$
\pi_G\from V^\dual\tens V^\dual\tens E \to G
$$
factors through the projection $\pi_F\from V^\dual \tens E \to
F$. Such a map $\pi_G$ will be called \emph{simple}.
\end{definition}

Associated to a simple $\pi_G\from V^\dual\tens V^\dual\tens E \to G$
is a second order operator on a Weyl manifold defined by the projected
Ricci corrected second derivative on $EM$:
$$
\Nabla^D := \pi_G(D^2 + \Lieb{\nric^D,\ })
\from C^\infty(M,EM)\to C^\infty(M,GM).
$$
Such operators will be called \emph{simple second order operators}.

\begin{remark}
Using Fegan's result, theorem \ref{Fegansfirstorderconformalinvariants}
the Ricci curvature correction simplifies in the simple second order
case to
\begin{eqnarray*}
\Nabla^D e
&=& \pi_G(X,Y\mapsto D_{X,Y}^2e + \Lieb{\nric^D(X),Y}.e)\\
&=& \pi_G(X,Y\mapsto D_{X,Y}^2e + (w_E+b)\nric^D(X,Y)\tens e )\\
&=& \pi_G(D^2e) + (w_E+b)\pi_G(\nric^D\tens\, e),
\end{eqnarray*}
where $w_E$ is the central weight of $E$ and $-b$ is the weight on
which the Stein Weiss operator associated to $E$ and $F$ is
conformally invariant.  The significance of the Ricci correction for a
composite of two Stein Weiss operators as above was pointed out to me
by D.~Calderbank. This has been generalized here to a Ricci correction
of the full second derivative, definition
\ref{RiccicorrectedsecondWeylderivative}.
\end{remark}

\begin{proposition}\label{Simplesecondorderdifferentialoperators}
\summary{Simple second order differential operators} Let $\Nabla^D$ be
a simple second order operator associated to $E$, $F$, $G$ and denote
by $w_E$ the central weight of $E$. Let $-b$ be the weight on which
the Stein Weiss operator associated to $E$ and $F$ is conformally
invariant (see theorem \ref{Fegansfirstorderconformalinvariants}). The
linear change of $\Nabla^D$ under changing the Weyl derivative is then
given by
$$
(\LieWeyl_\christo \Nabla^D)e
= \LieWeyl_\christo \pi_G (D^2e + \Lieb{\nric^D,\ }.e)
\stackrel{!}{=} 2(w_E+b-1) \pi_G (\christo\tens D e).
$$
Hence, conformally invariant simple second order operators are
characterized by $w_E=1-b$ and the Ricci correction simplifies to:
$$
\Nabla e
= \pi_G(D^2e + \nric^D\tens\, e).
$$
\end{proposition}

\proof This is a direct computation using proposition
\ref{Linearchangeofthesecondjet} and theorem
\ref{Fegansfirstorderconformalinvariants}:
\begin{eqnarray*}
\lefteqn{\LieWeyl_\christo \pi_G (X,Y\mapsto D^2_{X,Y}e +
\Lieb{\nric^D(X),Y}.e)}&&\\
&=& \pi_G(X,Y\mapsto
\Lieb{\christo,X}.D_Ye + \Lieb{\christo,Y}.D_Xe -
D_{\Lieb{\christo,X}Y}e)\\
&=& \pi_G(X,Y\mapsto
2\Lieb{\christo,X}.D_Ye - D_{2\christo(X)Y}e)\\
&=& \pi_G(X,Y\mapsto 2
(w_E+b-1)\christo(X)D_Ye).\qed
\end{eqnarray*}

We now turn to the study of second order bilinear differential
pairings. The general theory is slightly more involved than in the
first order case. For this reason we start with three examples of
pairings:

\begin{example}\label{Laplacepairing}
\summary{Laplace pairing} There is a conformally invariant bilinear
differential pairing between scalar densities
$$
\vee
\from C^\infty(M,L^{w_1}) \cross C^\infty(M,L^{w_2})
\to C^\infty(M,L^{w_1+w_2-2}).
$$
It is based upon the Laplace operator example \ref{ConformalLaplacian}
and it is defined in terms of a Weyl derivative by
$$
e_1\vee e_2 
:= w_2q_2\, (\Laplace^D e_1)\tens e_2
- 2q_1q_2\, \cip{De_1,De_2}
+ w_1q_1\, e_1 \tens (\Laplace^D e_2),
$$
where $q_1:=(w_1-(2-n)/2)$ and $q_2:=(w_2-(2-n)/2)$. The linear change
of the Weyl derivative is zero, since
\begin{eqnarray*}
\lefteqn{\LieWeyl_\christo (e_1\vee^D e_2)}&&\\
&=& w_2q_2(2q_1)\, D_\christo e_1 \tens e_2
- 2q_1q_2 w_1 e_1\tens D_\christo e_2
- 2q_1q_2 w_2 D_\christo e_1 \tens e_2
+ w_1q_1(2q_2)\, e_1 \tens D_\christo e_2\\
&=& (2w_2q_2q_1 - 2q_1q_2w_2)D_\christo e_1 \tens e_2  
- (2q_1q_2w_1 - 2w_1q_1q_2)e_1 \tens D_\christo e_2.
\end{eqnarray*}
\end{example}

\begin{example}\label{Bransonpairing}
\summary{Branson pairing} There is also a scalar valued pairing
between $k$-forms based upon Branson's operator, example
\ref{Bransonsoperator}:
$$
\vee\from
C^\infty(M,L^{w_1+k}\tens\Lambda^kT^\dual M)\cross
C^\infty(M,L^{w_2+k}\tens\Lambda^kT^\dual M)\to
C^\infty(M,L^{w_1+w_2-2}).
$$
In terms of a Weyl derivative it is given by
\begin{eqnarray*}
f_1 \vee f_2
&:=&
p_2q_2r_2\, \cip{\Branson^D f_1,f_2} +
p_1q_1r_1\, \cip{f_1,\Branson^D f_2}\\
&&-2p_1p_2r_1r_2\, \cip{\divR^D f_1,\divR^D f_2}
+2q_1q_2r_1r_2\, \cip{\dR^D f_1, \dR^D f_2},
\end{eqnarray*}
with $p_1:=w_1-(-k)$, $q_1:=w_1-(k-n)$, $r_1:=w_1-(2-n)/2$ and similar
for the index $2$. It is straight forward to check that the linear
change by changing the Weyl derivative is zero using the results in
examples \ref{Bransonsoperator} and \ref{Firstorderchange}.
\end{example}

\begin{example}\label{Hessepairing}
\summary{Hesse pairing} There is a conformally invariant bilinear
differential pairing between scalar densities to give trace-free
symmetric bilinear forms:
$$
\vee
\from C^\infty(M,L^{w_1}) \cross C^\infty(M,L^{w_2})
\to C^\infty(M,L^{w_1+w_2}\tens \Sym^2_0T^\dual M).
$$
It is based upon the Hessian operator, proposition
\ref{ConformalHessian}, and it is defined in terms of a Weyl
derivative by
$$
e_1\vee e_2 
:= w_2q_2\, (\Hesse^D e_1)\tens e_2
- 2q_1q_2\, De_1\cartan De_2
+ w_1q_1\, e_1 \tens (\Hesse^D e_2),
$$
where $q_1:=w_1-1$ and $q_2:=w_2-1$.
\end{example}

The last example is a special a case of a class of pairings associated
to simple second order operators:

\begin{definition}
\summary{Simple projection} Let $E_1$, $E_2$ be irreducible
$\CO(V)$-representation and $G\subset \Sym^2_0V^\dual \tens E_1\tens
E_2$ an irreducible subrepresentation. If there are irreducible
subrepresentations $F_1\subset V^\dual\tens E_1$ and $F_2\subset
V^\dual \tens E_2$ such that $Q\from V^\dual\tens V^\dual \tens E_1\tens E_2
\to G$ factors through $V^\dual\tens F_1\tens E_2$ and through
$V^\dual\tens E_1\tens F_2$ then $Q$ is called \emph{simple}.
\end{definition}

\begin{proposition}\label{Simpleconformallyinvariantsecondorderpairings}
\summary{Simple conformally invariant second order pairings} Let $Q$
be a simple projection as above and denote by $w_1$ and $w_2$ the
central weights of $E_1$ and $E_2$. Let $-b_1$ respectively $-b_2$ be
the weights for which the Stein Weiss operators associated to $E_1$,
$F_1$ respectively $E_2$, $F_2$ are conformally invariant. Then the
following differential pairing is conformally invariant:
$$
\vee\from C^\infty(M,E_1M) \cross C^\infty(M,E_2M) \to C^\infty(M,GM),
$$
which in terms of a Weyl derivative $D$ applied to sections $e_1$ and 
$e_2$ is given by
\begin{eqnarray*}
e_1 \vee e_2
&:=& (w_2+b_2)(w_2+b_2-1) Q((D^2+\Lieb{\nric^D,\ })e_1\tens e_2)\\
&&- 2(w_1+b_1-1)(w_2+b_2-1) Q(De_1\tens De_2)\\
&&+ (w_1+b_1)(w_1+b_1-1) Q(e_1\tens (D^2 +\Lieb{\nric^D,\ })e_2 ).
\end{eqnarray*}
\end{proposition}

\proof From the propositions
\ref{Simplesecondorderdifferentialoperators} and
\ref{Fegansfirstorderconformalinvariants} we immediately
have:
\begin{eqnarray*}
&Q&(\LieWeyl_\christo((D^2+\Lieb{\nric^D,\ })e_1\tens e_2) )
= 2(w_1+b_1-1)Q(\christo\tens De_1\tens e_2),\\
&Q&(\LieWeyl_\christo(e_1\tens (D^2+\Lieb{\nric^D,\ })e_2))
= 2(w_2+b_2-1)Q(e_1\tens\christo\tens De_2),\\
&Q&(\LieWeyl_\christo (De_1\tens De_2))
= (w_1+b_1)Q(\christo\tens e_1\tens De_2) 
+ (w_2+b_2)Q(De_1\tens \christo\tens e_2).\qed
\end{eqnarray*}

\begin{discussion}
\summary{Second order pairings} We now turn to the study of second
order bilinear differential pairings in general. Suppose $E_1$, $E_2$
and $F$ are $\CO(V)$-representations and
$$
Q\from 
(\Sym^{\leq 2}V^\dual\tens E_1) \tens
(\Sym^{\leq 2}V^\dual\tens E_2) \to G
$$
is $\CO(V)$-equivariant, then $\vee^D\from
C^\infty(M,E_1M)\cross C^\infty(M,E_2M) \to C^\infty(M,GM)$ defined by
$e_1 \vee^D e_2 :=Q(\jet^{2,D}(e_1)\tens\jet^{2,D}(e_2))$ is a
bilinear differential pairing of second order in Weyl geometry. As in
the first order case we will consider those pairings which are truly
second order in both $e_1$ and $e_2$. If $E_1$, $E_2$ and $G$ are
irreducible the action of the centre of $\CO(V)$ determines such a $Q$
to be nontrivial only on the following subspace
$$
Q\from 
(\Sym^2 V^\dual\tens E_1 \tens E_2)
\dsum (V^\dual\tens V^\dual \tens E_1\tens E_2)
\dsum (\Sym^2 V^\dual\tens E_1 \tens E_2)
\to G.
$$
The corresponding pairing is then clearly given by 
$$
e_1 \vee^D e_2
:=Q(
((D^2+\Lieb{\nric^D,\ })e_1\tens e_2)
\dsum (De_1\tens De_2)
\dsum (e_1\tens (D^2 +\Lieb{\nric^D,\ })e_2 )).
$$
The dependence of $\vee^D$ on the Weyl derivative leads to study the
following map:
\begin{eqnarray*}
\lefteqn{
\Lie^{\mathrm{bi}}\from
V^\dual\tens (V^\dual\tens E_1 \tens E_2\dsum V^\dual\tens E_1 \tens E_2)}&&\\
&\to& 
(\Sym^2 V^\dual\tens E_1 \tens E_2)
\dsum (V^\dual\tens V^\dual\tens E_1\tens E_2)
\dsum (\Sym^2 V^\dual\tens E_1 \tens E_2)
\end{eqnarray*}
which is given by
\begin{eqnarray*}
\lefteqn{\Lie^{\mathrm{bi}}(\christo\tens
((\alpha_1\tens e_1\tens e_2)\dsum(e_1\tens \alpha_2\tens e_2)))}&&\\
&:=& (\Lie(\christo\tens\alpha_1\tens e_1)\tens e_2)\\
&& \dsum (\Lie(\christo\tens e_1)\tens\alpha_2\tens e_2 +
\alpha_1\tens e_1 \tens \Lie(\christo\tens e_2))\\
&& \dsum (e_1\tens \Lie(\christo\tens\alpha_2\tens e_2)).
\end{eqnarray*}
\end{discussion}

We summarize the above observations in the following:
 
\begin{proposition}\label{Algebraicsecondorderpairings}
Conformally invariant bilinear second order differential pairings
$$
\vee\from C^\infty(M,E_1M)\cross C^\infty(M,E_2M)\to C^\infty(M,GM),
$$
between associated bundles from irreducible $\CO(V)$-representations
$E_1$, $E_2$ and $G$ are classified by $\CO(V)$-equivariant linear maps
$$
Q\from (\Sym^2 V^\dual\tens E_1 \tens E_2)
\dsum (V^\dual\tens V^\dual \tens E_1\tens E_2)
\dsum (\Sym^2 V^\dual\tens E_1 \tens E_2) \to G,
$$ 
such that $Q\circ \Lie^{\mathrm{bi}}=0$. 
\end{proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Conformal linear field theories}
\label{Conformallinearfieldtheories}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Soon after Weyl developed differential conformal geometry and his
unified theory of electromagnetism and gravity (see \cite{WeylRZM}, or
remark \ref{Weylsunifiedtheory}) Bach came up with a nonlinear
conformally invariant geometric theory of gravity (see \cite{Bach}, or
remark \ref{Bachstheoryofgravity}). We will present its linearized
version here. Fierz in \cite{Fierz} invented a class of theories which
shares analogous properties to electromagnetism, except these theories
fail to be conformally invariant. We will give a variation of Fierz's
theory for gravity which is conformally invariant. The so called
\emph{zero rest mass field equations} due to Penrose (see
\cite{PenroseRindler}) are conformally invariant and provide another
large class of field equations analogous to Maxwell's equations. We
will discuss Penrose's linear theory of gravity here. I like to thank
D.~Calderbank for drawing my attention to the fact that these
conformal field theories are Bernstein Gelfand Gelfand complexes---the
latter were studied in Minkowski space by Eastwood and Rice in
\cite{EastwoodRice}.

The aim of this chapter is to analyse the structure of Maxwell's
equations and the Lorentz force law to extract general principles a
linear field theory should follow. This happens in the first three
sections. The key ingredient which extends the basic field equations
are bilinear differential pairings imitating the exterior and interior
multiplication of forms and multivectors known from
electromagnetism. These pairings allow us to suggest the beginning of
a theory of motion: charged particles induce fields which travel with
the speed of light and these fields influence the motion of charged
particles. This is summarized in section \ref{Generalfieldtheory}. In
the last three sections we investigate the three field theories in
connection with linear gravity. We explicitly define the field
equation and bilinear pairings needed and hence prove special cases of
theorem \ref{Conformalexteriorpairings}, on an elementary level. These
explicit results and calculations arose out of joint efforts with
D.~Calderbank to understand conformal linear field theories.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ingredients of a general field theory}
\label{Ingredientsofageneralfieldtheory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We would like a general linear field theory to share the following 
ingredients and properties with electromagnetism: a tensor field 
represents the \emph{source} and satisfies a \emph{conservation law}. A 
charged particles along an arbitrary worldline naturally gives rise to a 
conserved source. We distinguish between two realizations of the field 
itself, the \emph{dynamic and the kinematic field}. The dynamic field is 
coupled to the source by the \emph{dynamic field equation}. The kinematic 
field satisfies an \emph{integrability condition}. Kinematic and dynamic 
fields are related by a \emph{constitutive relation}, which in vacuum 
only involves the conformal structure. It is the kinematic field which 
determines the \emph{motion of charged test particles} by a general force 
law. The integrability condition of the kinematic field means that it is 
(locally) in the image of a linear differential operator applied to a 
\emph{potential}. This potential is not uniquely determined, instead it 
is subject to (local) \emph{gauge transformations}. The operator of the 
dynamic field equation is adjoint to the potential operator, similarly 
the operator for the conservation law of the source is adjoint to the 
operator between gauges and potentials.

To summarize, a relativistic linear field theory in vacuum takes place
on a $n=4$ dimensional conformal manifold $M$ of Lorentzian
signature. The beginning of a complex of conformally invariant linear
differential operators defines the \emph{kinematic sequence} of the
theory:
\[
\begin{array}{*{7}{c}}
\mathrm{gauge} &&
\mathrm{potential} &&
\mathrm{kinematic\ field} &&\\
C^\infty(M,H_0) &\stackyrel{\dH}{\to}&
C^\infty(M,H_1) &\stackyrel{\dH}{\to}&
C^\infty(M,H_2) &\stackyrel{\dH}{\to}&
C^\infty(M,H_3)
\end{array}
\]
Here $H_k$ denote associated vector bundles in which the fields (gauges,
potentials, kinematic fields) take their values. Adjoint to this is
the end of a complex defining the \emph{dynamic sequence} of the
theory. Sources and dynamic fields take values in $L^{-n}\tens H^k$
with $H^k={H_k}^\dual$:
\[
\begin{array}{*{7}{c}}
C^\infty(M, L^{-n}\tens H^0) &\stackyrel{\divH}{\ot}&
C^\infty(M, L^{-n}\tens H^1) &\stackyrel{\divH}{\ot}&
C^\infty(M, L^{-n}\tens H^2) &\stackyrel{\divH}{\ot}&\\
&&
\mathrm{source} &&
\mathrm{dynamic\ field} &&
\end{array}
\]
To be consistent with electromagnetism we like $-\divH$ to be adjoint
to $\dH$, i.e.~according to paragraph \ref{Adjoints} we postulate the
existence of bilinear differential pairings
$$
\innerH\from C^\infty(M,H_k)\cross C^\infty(M,L^{-n}\tens H^{k+1})
\to C^\infty(M,L^{-n}\tens TM),
$$
such that the following divergence formula holds:
$$
\divH (\alpha\innerH a)=\ip{\dH\alpha,a} + \ip{\alpha,\divH a}.
$$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Charges in field theories}
\label{Chargesinfieldtheories}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We will follow Penrose who interpreted the kernel of the gauge
operator
$$
W:=\kernel \dH\from C^\infty(M,H^0) \to C^\infty(M,H^1)
$$
as the \emph{space of charges} which characterize the field theory in
question. In case $M=\A$ being a conformal affine space $W$ is a
representation space for the M\"obius Lie algebra $\so(\asso V)$ of
conformal Killing fields in $M$, since $\dH$ is conformally
invariant. Penrose calls $\dH$ the \emph{twistor operator} of the
theory and the solutions $W$ are called \emph{twistors}. In
electromagnetism electric charges are real numbers, and we have
$\R=\kernel \dR \from C^\infty(M,\R) \to C^\infty(M,T^\dual M)$. For
the linearized conformal geometry, remark \ref{Bachstheoryofgravity},
the charges are given by the space of conformal Killing fields in
affine background given by $\so(\asso V)=\kernel\Killing$. In case of
a linear theory for gravity $W$ has the interpretation as \emph{space
of gravitational masses}. This interpretation is not entirely obvious
and this section is only a first step towards this task: we will show
how a dynamic field represents an element in $W$ by integrating it
over a cooriented sphere, how a worldline with associated element in
$W$ naturally gives rise to a conserved distributional source and how
a kinematic field induces an acceleration on a test particle with
associated element in $W$. To make these three constructions work in a
general field theory we need bilinear pairings imitating the interior
and exterior multiplication known from the deRham complex and its
adjoint (electromagnetism).

\begin{remark}
As will be explained in section \ref{Generalfieldtheory}, twistors in
$W$ behaves more like magnetic charges whereas \emph{dual twistors},
i.e.~elements in $W^\dual$, behave more like electric charges, see
definitions \ref{Electriccharge} and \ref{Magneticcharge}. Since $W$
is a $\so(\asso V)$ representation it carries an inner product which
allows to identify $W$ with its dual $W^\dual$. Hence we will not
distinguish twistors from dual twistors until section
\ref{Generalfieldtheory}.
\end{remark}

\begin{discussion}\label{Chargerepresentedbydynamicfields}
\summary{Charge represented by dynamic fields} We recall that in
electromagnetism the electric charge generating a field $G$ can be
rediscovered from $G$ by integration: Maxwell's second equation $\divR
G=j$ allows to define the charge contained inside a cooriented
$(n-2)$-dimensional sphere $S^{n-2}$ (away from the source
$\support(j)$) to be $q=\int_{S^{n-2}}\ip{G,\coorie}$. The divergence
theorem provides a conservation law for this integral. To make a
similar construction work for a general field theory we need the
following \emph{bilinear differential pairings}:
\begin{eqnarray*}
\innerH\from C^\infty(M,H_0)\cross C^\infty(M,L^{-n}\tens H^2)
&\to& C^\infty(M,L^{-n}\tens\Lambda^2TM),\\
\innerH\from C^\infty(M,H_1)\cross C^\infty(M,L^{-n}\tens H^2)
&\to& C^\infty(M,L^{-n}\tens TM),\\
\innerH\from C^\infty(M,H_0)\cross C^\infty(M,L^{-n}\tens H^1)
&\to& C^\infty(M,L^{-n}\tens TM),
\end{eqnarray*}
such that for sections $f\in C^\infty(M,H_0)$ and $g\in
C^\infty(M,H^2)$ the following \emph{codimension one divergence
formula} holds:
$$
\divR( f\innerH g ) = (\dH f) \innerH g + f \innerH \divH g.
$$
In that case we say that the gauge operator $\dH\from
C^\infty(M,H_0)\to C^\infty(M,H_1)$ and the operator of the dynamic
field equation $\divH\from C^\infty(M,L^{-n}\tens H^2)\to
C^\infty(M,L^{-n}\tens H^1)$ are \emph{codimension one adjoints} (see
the general definition below). From this its clear that for a charge
$q\in W=\kernel \dH$ and in a regions where the dynamic field $G$
satisfies $\divH G=0$ we can produce a divergence-free bivector field
$q\innerH G$. Hence if $S^{n-2}$ is a cooriented sphere (away from
$\support(j)$), then $\int_{S^{n-2}} \ip{ {q \innerH G},\coorie}$ is a
real number linear in $q$. This determines an element in the dual
space $\int_{S^{n-2}} \ip{ {. \innerH G},\coorie}\in W^\dual$. As was
already mentioned in the affine case $M=\A$ the space $W$ is a
representation of the M\"obius Lie algebra, hence it can carry an
invariant inner product to identify $W$ with its dual $W^\dual$.
\end{discussion}

\begin{definition}\label{Codimensiononeadjoints}
\summary{Codimension one adjoints}
Let $EM$, $FM$ and $GM$ be vector bundles over $M$ and $\Nabla_1\from 
C^\infty(M,EM)\to C^\infty(M,FM)$ and $\Nabla_2\from C^\infty(M,FM) \to 
C^\infty(M,GM)$ be two differential operators of order $d_1$ and $d_2$ 
respectively. Suppose that both operators have adjoints 
$\Nabla_1^\dual\from C^\infty(M,L^{-n}\tens F^\dual M) \to C^\infty(M,L^{-
n}\tens E^\dual M)$ and $\Nabla_2^\dual\from C^\infty(M,L^{-n}\tens 
G^\dual M) \to C^\infty(M,L^{-n}\tens F^\dual M)$ coming from $L^{-n} 
\tens TM$ valued bilinear differential pairings $X_1$ and $X_2$, such that
\begin{eqnarray*}
\divR (X_1(e,\phi))
&=& \ip{\Nabla_1 e,\phi} - \ip{e,\Nabla_1^\dual\phi},
\hbox{\quad and}\\
\divR (X_2(f,\psi))
&=& \ip{\Nabla_2 f,\psi} - \ip{f,\Nabla_2^\dual\psi}.
\end{eqnarray*}
Then $\Nabla_1$ and $\Nabla_2^\dual$ are called \emph{codimension one
adjoints} if there is a $L^{-n}\tens\Lambda^2TM$ valued bilinear
differential pairing $Y$ of order $d_1+d_2-2$ between sections of $EM$
and sections of $L^{-n}\tens G^\dual M$, such that
$$
\divR(Y(e,\psi))
= X_2(\Nabla_1 e,\psi) + X_1(e,\Nabla_2^\dual\psi).
$$
\end{definition}

\begin{remark}
Applying the operator $\divR$ once more to the codimension one
divergence formula we obtain a necessary condition to hold for
codimension one adjoints. This is satisfied if $\Nabla^1$, $\Nabla^2$
from a complex, i.e.~$\Nabla_2\circ\Nabla_1=0$ and hence also
$\Nabla_1^\dual\circ\Nabla_2^\dual=0$.
\end{remark}

\begin{remark} A complex of first order Stein Weiss 
operators always has codimension one adjoints: We assume that the
operators are given in terms of a Weyl derivative $D$ and symbols
$\pi_1\from T^\dual\tens E\to F$ and $\pi_2\from T^\dual\tens F\to G$
as $\Nabla_1=\pi_1\circ D$ and $\Nabla_2=\pi_2\circ D$. With a dual
basis $t_i$, $\theta^i$ of $TM$ the zero order pairings are
$X_1(e,\phi)=\Sum_i t_i\tens \ip{\pi_1(\theta^i\tens e),\phi}$ and
similar $X_2$. For $\alpha,\beta\in T^\dual M$ we note that the symbol
of $\Nabla_2\circ\Nabla_1$ is $\pi_{21}(\alpha\tens\alpha\tens e) =
\pi_2(\alpha\tens\pi_1(\alpha\tens e))$. Since
$0=\Nabla_2\circ\Nabla_1$ we know that $\pi_{21}(\alpha\tens\beta\tens
e) = \pi_2(\alpha\tens\pi_1(\beta\tens e))$ needs to be skew in
$\alpha$ and $\beta$, which defines
$Y(e,\psi):=\oneover{2}\Sum_{i,j}t_i\wedge t_j
\ip{\pi_{21}(\theta^j\tens\theta^i\tens e),\psi}$.
\end{remark}

\begin{discussion}\label{Chargedparticlesassources}
\summary{Charged particles as sources} An oriented worldline
$c\from\R\to M$, $t\mapsto c(t)$ of an electrically charged particle
$q\in\R$ gives rise to a distributional source $j\in {\cal
D}(M,L^{-n}\tens TM)$ in electromagnetism: $\ipp{j,\alpha}:=\int_\R
q\alpha(\dot c)$ with $\alpha\in C^\infty_o(M,T^\dual M)$. This source
is divergence-free, since for a real valued test function $f\in
C^\infty_o(M,\R)$ we have $\ipp{j,\dR f}:=\int_\R q\dR f(\dot
c)=\int_\R \delbydel{t}(qf(c(t)))=0$. A similar construction works for
a general linear field theory if it is equipped with two further
bilinear differential pairings
\begin{eqnarray*}
\wedgeH\from C^\infty(M,H_0)\cross C^\infty(M,H_0) &\to& 
C^\infty(M,\R),\\
\wedgeH\from C^\infty(M,H_1)\cross C^\infty(M,H_0) &\to& 
C^\infty(M,T^\dual M),
\end{eqnarray*}
such that the following product rule holds for any $f,g\in
C^\infty(M,H_0)$:
$$
\dR (f\wedgeH g) = (\dH f) \wedgeH g + f\wedgeH (\dH g).
$$
If $c\from\R\to M$ is an oriented worldline and $q\in W$ an associated
charge then this gives rise to a conserved distributional source
$j\in{\cal D}(M,L^{-n}\tens H^1)$ of the general linear field theory:
$\ipp{j,\alpha} := \int_\R (q\wedgeH \alpha)(\dot c)$, where
$\alpha\in C^\infty_o(M,H_1)$ is a test section. This source is
conserved, since for a test gauge section $f\in C^\infty_o(M,H_0)$ we
find
\begin{eqnarray*}
\ipp{j,\dH f}
&=& \int_\R (q\wedgeH \dH f)(\dot c)\\
&=& \int_\R \dR (q\wedgeH f)(\dot c) - ((\dH q)\wedgeH f)(\dot c)\\
&=& 0.
\end{eqnarray*}
\end{discussion}

\begin{discussion}\label{Lorentzforcelawingeneralfieldtheories}
\summary{Lorentz force law in general field theories} A worldline $c$
with normalized timelike tangent vector $N$, associated electric
charge $q\in\R$ and inertial mass $m\in L^{-1}$ can represent a test
particle moving in an electromagnetic field $F$. The Lorentz force law
determines the acceleration on $c$ to be $mD_NN=q\sharp(F(N))$ (in
terms of a Weyl derivative $D$ with $Dm=0$). To establish a simple
force law in a general linear field theory we need three further
bilinear differential pairings:
\begin{eqnarray*}
\wedgeH\from C^\infty(M,H_0)\cross C^\infty(M,H_2) &\to& 
C^\infty(M,\Lambda^2 T^\dual M),\\
\wedgeH\from C^\infty(M,H_1)\cross C^\infty(M,H_2) &\to& 
C^\infty(M,\Lambda^3 T^\dual M),\\
\wedgeH\from C^\infty(M,H_0)\cross C^\infty(M,H_3) &\to& 
C^\infty(M,\Lambda^3 T^\dual M),
\end{eqnarray*}
such that for sections $f\in C^\infty(M,H_0)$ and $F\in
C^\infty(M,H_2)$ the following product rule holds:
$$
\dR ( f\wedgeH F ) = (\dH f) \wedgeH F + f \wedgeH (\dH F).
$$ From this its clear that for a charge $q\in W=\kernel \dH$ and a
kinematic field $F$ subject to the integrability condition $\dH F=0$
we can produce a closed $2$-form $q\wedgeH F$. Hence a test particle
with inertial mass $m$ and charge $q\in W$ is accelerated due to the
kinematic field $F$ as
$$
mD_NN = \sharp( (q\wedgeH F)(N) ),
$$
where $D$ denotes a Weyl derivative of $M$ with $Dm=0$.
\end{discussion}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lienard Wiechert fields}
\label{LienardWiechertfields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The long history of electromagnetism provides us with a long list of
solutions of the field equations which have elaborated physical
interpretations. In this chapter we will describe how bilinear
differential pairings imitating the wedge product of the deRham
complex allow to generate solutions in general field theories from
solutions of Maxwell's equations. In particular a single electrically
charged particle along a worldline produces an electromagnetic
field. The corresponding solution of Maxwell's equation is due to
Lienard and Wiechert. We will describe how analogous solutions can be
obtained in any general linear field theory.

\begin{discussion}\label{Kinematichelicityraising}
\summary{Kinematic helicity raising} We begin with a general field
theory based upon the following kinematic sequence:
$$
W\to C^\infty(M,H_0)\to C^\infty(M,H_1)\to C^\infty(M,H_2)\to\ldots
$$
Suppose there are bilinear differential pairings like
\begin{eqnarray*}
\wedgeH &\from& 
C^\infty(M,H_0)\cross C^\infty(M,T^\dual M)
\to C^\infty(M,H_1),\\
\wedgeH &\from& 
C^\infty(M,H_0)\cross C^\infty(M,\Lambda^2T^\dual M)
\to C^\infty(M,H_2),\\
\wedgeH &\from& 
C^\infty(M,H_1)\cross C^\infty(M,T^\dual M)
\to C^\infty(M,H_2), \\
\end{eqnarray*}
so that for sections $f\in C^\infty(M,H_0)$ and $A\in
C^\infty(M,T^\dual M)$ the following product rule holds:
$$
\dH ( f\wedgeH A ) = (\dH f) \wedgeH A + f \wedgeH (\dR A).
$$
The first two pairings allow to produce a general potential $f\wedgeH
A$ respectively a general kinematic field $f\wedgeH F$ from an
electromagnetic potential $A$ respectively from an electromagnetic
field $F$ with the help of a general gauge $f$. If the gauge is indeed
a general charge $\dH f=0$ then the product rule ensures that the
general kinematic field comes from the general potential: $f\wedgeH
F=\dH (f\wedgeH A)$ if $F=\dR A$. Such a translation of solutions from
electromagnetism to a general field theory is called \emph{helicity
raising} (where the \emph{helicity} of a field theory is a number
reflecting a particular \emph{polarization property} of plane wave
solutions of that theory - for example electromagnetic waves have
helicity one and gravitational waves are expected to have helicity
two).
\end{discussion}

\begin{discussion}\label{Dynamichelicityraising}
\summary{Dynamic helicity raising} For the dynamic sequence of a
general field theory
$$
C^\infty(M,L^{-n}\tens H^0)\ot C^\infty(M,L^{-n}\tens H^1)
\ot C^\infty(M,L^{-n}\tens H^2)\ot\ldots
$$
suppose there are bilinear differential pairings like
\begin{eqnarray*}
\innerH &\from& 
C^\infty(M,H_0)\cross C^\infty(M,L^{-n}\tens \Lambda^2 TM)
\to C^\infty(M,L^{-n}\tens H^2),\\
\innerH &\from& 
C^\infty(M,H_0)\cross C^\infty(M,L^{-n}\tens TM)
\to C^\infty(M,L^{-n}\tens H^1),\\
\innerH &\from& 
C^\infty(M,H_1)\cross C^\infty(M,L^{-n}\tens \Lambda^2 TM)
\to C^\infty(M,L^{-n}\tens H^1), \\
\end{eqnarray*}
so that for sections $f\in C^\infty(M,H_0)$ and $G\in
C^\infty(M,L^{-n}\tens \Lambda^2 TM)$ the following product rule
holds:
$$
\divH ( f\innerH G ) = (\dH f) \innerH G + f \innerH (\divR G).
$$
The first two pairings allow to produce a general dynamic field
$f\wedgeH G$ respectively a general source $f\wedgeH j$ from an
electromagnetic field $G$ respectively from an electromagnetic source
$j$ with the help of a general gauge $f$. If the gauge is indeed a
general charge $\dH f=0$ then the product rule ensures that the
general source comes from the general dynamic field: $f\wedgeH j=\divH
(f\wedgeH G)$ if $j=\divR G$. 
\end{discussion}

\begin{discussion}\label{ElectromagneticLienardWiechertfields}
\summary{Electromagnetic Lienard Wiechert fields} In affine conformal
Minkowski space $\A$ modelled over the vector space $V$ let
$c\from\R\to\A$; $t\mapsto c(t)$ be a timelike curve, which represents
an emitter of light rays. The backward lightcone of each point
$x\in\A$ intersects the worldline at $c(\phi(x))$, which defines a
function $\phi\from\A\to\R$ the retarded parameter. Its gradient
$L:=-\grad\phi$ is a lightlike geodetic vector field in $\A$ (compare
example \ref{Lightfromapointsource}). This function $\phi$ satisfies
$0=\ip{x-c(\phi(x)),x-c(\phi(x))}$ and differentiating along a curve
$x\from\R\to\A$; $s\mapsto x(s)$ gives $0 =\ip{x'-\dot
c(\phi(x))\del_{x'}\phi\restr{x}, x-c(\phi(x))}$, hence the lightlike
vector field is explicit:
$$
L_x
={x-c(\phi(x))\over - \ip{\dot c(\phi(x)),x-c(\phi(x))}}.
$$
Let $N=\nu\dot c\in V^0$ with $\nu\in L^{-1}$ along $c$ denote the
normalized tangent vector $\ip{N,N}=-1$, whereas $\nu$ depends upon
the parameterization of $c$. It can be viewed as being the emitted
\emph{frequency} of $L$ with respect to $N$, since
$\nu=-\ip{L,N}$. The \emph{luminosity distance} to the worldline
$r\from\A\to L^1$ (compare remark \ref{Luminositydistance}) is defined
by
$$
r(x) := -\ip{N_{c(\phi(x))}, x-c(\phi(x))}.
$$
Along the light rays $r$ has initial conditions $r(c(t))=0$,
$D_Lr(c(t))=-\ip{L,N}=\nu$ and satisfies $D_LD_L r=0$ with the affine
derivative $D$.

With this in hand we can define the electromagnetic \emph{Lienard
Wiechert potential} $A\from\A\to V^\dual$ in $n=4$ dimensions to be
$$
A(x) := q {N\over r},
$$
where $q\in \R$ is a constant number. From the potential we have the
kinematic Lienard Wiechert field $F=\dR A$ and its dynamic field $G=F$
(identified using the conformal structure). The source of the dynamic
field is a distribution with support on the worldline given by (see
\cite{Thirring}):
$$
\ipp{\divR G,\alpha} = 4\pi \int_\R q\alpha(\dot c).
$$
Hence the Lienard Wiechert field solves Maxwell's equation with a
single charged particle along a worldline as source. The \emph{Coulomb
field} (compare paragraph \ref{Coulombfield}) is the case
when the worldline follows a straight line.
\end{discussion}

\begin{discussion}
\label{DistributionalsourcesandgeneralLienardWiechertfields}
\summary{Distributional sources and general Lienard Wiechert fields}
In paragraph \ref{Dynamichelicityraising} we explained what pairings
were needed to raise a smooth electromagnetic field $G$ with sources
$j=\divR G$ to a general field $q\innerH G$ with source $q\innerH
j=\divH(q\innerH G)$. We used the following pattern of pairings:
$$
\innerH\from
C^\infty(M,H_k)\cross C^\infty(M,L^{-n}\tens \Lambda^{k+l}TM)
\to C^\infty(M,L^{-n}\tens H^l),
$$
with $k=0,1$ and $k+l=1,2$. In paragraph
\ref{Chargedparticlesassources} we considered pairings like
\begin{eqnarray*}
\wedgeH\from C^\infty(M,H_k)\cross C^\infty(M,H_l) &\to& 
C^\infty(M,\Lambda^{k+l}T^\dual M),
\end{eqnarray*}
(with $k=0,1$ and $l=0$) to construct conserved distributional sources
from worldlines with an attached charge $q\in W$. More generally if
$j\in{\cal D}(M,L^{-n}\tens TM)$ is a distributional source in
electromagnetism, then $J\in{\cal D}(M,L^{-n}\tens H^1)$ defined by
$\ipp{J,\alpha} =\ipp{j,q\wedgeH\alpha}$ with $\alpha\in
C^\infty_o(M,H_1)$ is conserved if $j$ is conserved. To make the
smooth and distributional construction of sources consistent we need
the operators $q\wedgeH$ and $q\innerH$ to be adjoint in the sense
that there is a trilinear differential pairing:
$$
X\from C^\infty(M,H_0)\cross C^\infty(M,L^{-n}\tens\Lambda^l TM)\cross
C^\infty(M,H_l) \to C^\infty(M,L^{-n}\tens TM),
$$
such that 
$$
\divR(X(q,a,\alpha)) = \ip{q\innerH a,\alpha} - \ip{a,q\wedgeH\alpha},
$$
for $l=1$. Moreover for $l=2$ such a pairing guarantees $\ipp{ \divH
(q\innerH G),\alpha} = \ipp{\divR G, q\wedgeH \alpha}$ since then the
following calculation holds
\begin{eqnarray*}
\ipp{\divH (q\innerH G),\alpha}
&=&-\ipp{q\innerH G,\dH\alpha}\\
&=&-\ipp{G,q\wedgeH \dH\alpha}\\
&=&-\ipp{G,\dR (q\wedgeH \alpha) - (\dH q)\wedgeH \alpha}\\
&=&\ipp{\divR G,(q\wedgeH \alpha)}.
\end{eqnarray*}
With such pairings in hands the electromagnetic Lienard Wiechert
fields of the last paragraph
\ref{ElectromagneticLienardWiechertfields} can be raised into any
general field theory, and the sources of these general Lienard
Wiechert fields are the expected ones.
\end{discussion}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{General field theory}
\label{Generalfieldtheory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This section begins with quoting a special case of a general result
which we will prove at the end of this dissertation. The first of
these theorems is due to Eastwood and Rice \cite{EastwoodRice} whereas
the second statement concerning differential pairings is new: let $\A$
be an affine space modelled on a $n=4$ dimensional vector space $V$
with conformal metric $\com$. We already identified the M\"obius Lie
algebra (of vector fields on $\A$ leaving the affine conformal
structure invariant) with the graded Lie algebra $\g=V\dsum\co(V)\dsum
V^\dual$ and also with the orthogonal Lie algebra $\g=\so(\asso V)$
where $\asso V=L^1\dsum V^0\dsum L^{-1}$.

\begin{theorem}\label{ConformalBGGcomplex}
\summary{Conformal Bernstein Gelfand Gelfand complex} Associated to
any finite dimensional representation $W$ of the Lie algebra $\g$ is a
sequence of finite dimensional $\co(V)$-representations denoted by
$H_k(W)$ with $k=0\ldots n$, beginning with the coinvariants
$H_0(W)=W_{V^\dual}$ of $W$ with respect to $V^\dual$. The natural
inclusion $\iota_H\from W\to C^\infty(\A,W_{V^\dual})$ is the
beginning of a locally exact complex of $\g$-equivariant linear
differential operators:
$$
W \stackrel{\iota_H}{\rightarrow}
C^\infty(\A,H_0(W)) \stackrel{\dH}{\rightarrow}
C^\infty(\A,H_1(W)) \stackrel{\dH}{\rightarrow}
C^\infty(\A,H_2(W)) \stackrel{\dH}{\rightarrow} \ldots\quad.
$$
\end{theorem}

This theorem provides us with a kinematic sequence of a general linear
field theory characterized by $W$. For the trivial representation
$W=\R$ we recover the deRham complex $H_k(\R)=\Lambda^kV^\dual$. The
next theorem guarantees the existence of bilinear pairings imitating
the wedge product on forms in electromagnetism:

\begin{theorem}\label{Conformalexteriorpairings}
\summary{Conformal exterior pairings} If $W_1$, $W_2$ and $W_3$ are
three finite dimensional $\g$-representations and $F\from W_1\tens
W_2 \to W_3$ is a linear $\g$-equivariant map, then there is a
$\g$-equivariant bilinear differential \emph{exterior pairing}
$$
\wedgeH \from C^\infty(\A,H_k(W_1))
\cross C^\infty(\A,H_l(W_2))
\to C^\infty(\A,H_{k+l}(W_3)). 
$$
For $k=l=0$ this pairing is an extension of $F$ since it satisfies
$\wedgeH \circ (\iota_H\tens \iota_H) = \iota_H\circ F$. More
generally for $s\in C^\infty(\A,H_k(W_1))$ and $t\in
C^\infty(\A,H_l(W_2))$ the following Leibniz rule holds:
$$
\dH (s \wedgeH t) 
= (\dH s) \wedgeH t + (-1)^k s \wedgeH (\dH t).
$$
\end{theorem}

To construct an adjoint dynamic sequence we quote the relation between
$H_k(W)$ and its dual space $H^k(W^\dual) :=(H_k(W))^\dual$ which is
given by the following observation which will also be proved later:

\begin{proposition}
\summary{Duality} For the sequence of $\co(V)$-representation
$H_k(W^\dual)$ we have natural isomorphisms
$$
\inner\from
H^k(W^\dual)\tens \Lambda^nV^\dual
\stackrel{\isomorph}{\to}
H_{n-k}(W^\dual).
$$
\end{proposition} 

\begin{remark}
For the natural pairing $F\from W\tens W^\dual\to\R$ the bilinear
differential pairing $\wedgeH$ in degrees with $k+l=n$ is indeed zero
order and induced by the above duality rewritten as: $H_k(W)\tens
H_{n-k}(W^\dual)=\Lambda^nV^\dual$.
\end{remark}

The duality allows to twist the kinematic sequence $\dH$ induced by
$W^\dual$ with the bundle of pseudoscalar $L^{-n}\tens \Lambda^n V$ as
in paragraph \ref{Exteriordivergence}:

\begin{definition}
In analogy to the exterior divergence we define the \emph{divergence
operators associated to $W$}: $\divH\from C^\infty(M,L^{-n}\tens
H^k(W^\dual)) \to C^\infty(M,L^{-n}\tens H^{k-1}(W^\dual))$, by
$$
(\divH a)\inner\orie^\dual := (-1)^{k+1} \dH(a\inner\orie^\dual),
$$
where $\orie^\dual\in L^n\tens\Lambda^nV^\dual$ is a parallel element
of the (dual) space of pseudoscalars and the operator $\dH\from
C^\infty(M,H_{n-k}(W^\dual)) \to C^\infty(M,H_{n-k+1}(W^\dual))$ is
induced by $W^\dual$, see theorem \ref{ConformalBGGcomplex}. Moreover,
if $W_1$, $W_2$ and $W_3$ are three finite dimensional
$\g$-representations and $F\from W_1\tens W_2 \to W_3$ is a linear
$\g$-equivariant map, then we define bilinear differential
\emph{interior pairings}
$$
\innerH \from C^\infty(\A,H_k(W_1))
\cross C^\infty(\A,L^{-n}\tens H^{k+l}(W_2))
\to C^\infty(\A,L^{-n}\tens H^l(W_3)),
$$
by
$$
(\alpha \innerH a) \inner \orie^\dual
:= (-1)^{k(k+l+1)} \alpha \wedgeH (a\inner\orie^\dual).
$$
\end{definition}

\begin{corollary}\label{ConformalBGGdivergence}
The resulting sequence of differential operators $\divH$
$$
C^\infty(\A,L^{-n}\tens H^0(W^\dual))
\stackrel{\divH}{\ot}
C^\infty(\A,L^{-n}\tens H^1(W^\dual))
\stackrel{\divH}{\ot}
C^\infty(\A,L^{-n}\tens H^2(W^\dual))
\stackrel{\divH}{\ot} \ldots
$$
defines a locally exact complex which for $W=\R$ reduces to the
exterior divergence. For the interior pairing $\innerH$ induced by
$F\from W_1\tens W_2\to W_3$ we obtain from the Leibniz rule the
following formula:
$$
(-1)^k \divH (\alpha\innerH a)
= (\dH\alpha)\innerH a + \alpha\innerH(\divH a),
$$
with $\alpha\in C^\infty(M,H_k(W_1))$ and $a\in C^\infty(M,L^{-n}\tens
H^{k+l}(W_2))$. In particular applied to the natural pairing $W\tens
W^\dual\to\R$ with $l=1$ the above formula shows that $\dH$ of $W$ and
the above $-\divH$ are adjoints.
\end{corollary}

\proof Unraveling the definitions and using the above two theorems
gives
\begin{eqnarray*}
\divH(\divH (a)) \innerH \orie^\dual
&=& - (-1)^{k-1} \dH (\divR(a) \innerH \orie^\dual)\\
&=& -\dH(\dH(a\innerH\orie^\dual)) =0.
\end{eqnarray*}
Similarly for the divergence formula note
\begin{eqnarray*}
\lefteqn{(-1)^k (\divH (\alpha\innerH a)) \innerH \orie^\dual}&&\\
&=& -(-1)^{k+l} \dH ((\alpha\innerH a)\innerH \orie^\dual )\\
&=& (-1)^{(k+1)(k+l+1)} \dH (\alpha\wedgeH (a\innerH \orie^\dual) )\\
&=& (-1)^{(k+1)(k+l+1)} (\dH \alpha) \wedgeH (a\innerH \orie^\dual) 
+(-1)^{(k+1)(k+l+1)+k} \alpha \wedgeH \dH (a\innerH \orie^\dual) \\
&=& ((\dH \alpha) \innerH a) \innerH \orie^\dual 
+(-1)^{k(k+l)} \alpha \wedgeH ((\divH a)\innerH \orie^\dual) \\
&=& ((\dH \alpha) \innerH a) \innerH \orie^\dual 
+ (\alpha \innerH (\divH a)) \innerH \orie^\dual.
\end{eqnarray*}
For $F\from W\tens W^\dual\to\R$ the interior pairing $\innerH$
between $\beta\in C^\infty(\A,H_k(W))$ and $b\in
C^\infty(\A,L^{-n}\tens H^k(W^\dual))$ takes values in
$C^\infty(\A,L^{-n})$ and is zero order and induced by the natural
pairing $H_k(W)\tens H^k(W^\dual)\to \R$. Hence the pairings $\innerH$
on the right hand side of the divergence formula for $l=1$ are simple
contractions.\qed

We will now apply the above results to put the ideas of the previous two
sections \ref{Chargesinfieldtheories} and \ref{LienardWiechertfields}
into proper definitions: \emph{Dynamic helicity lowering} is based
upon the contraction $W\tens W^\dual\to \R$ inducing the following
interior pairing:
$$
\innerH \from C^\infty(\A,H_k(W))
\cross C^\infty(\A,L^{-n}\tens H^{k+l}(W^\dual))
\to C^\infty(\A,L^{-n}\tens \Lambda^lV ).
$$

\begin{definition}\label{Generalelectriccharge}
\summary{General electric charge} (Compare paragraphs
\ref{Electriccharge} and \ref{Chargerepresentedbydynamicfields}) For
a dynamic field $G\in C(\A,L^{-n}\tens H^{2}(W^\dual))$ of the theory
associated to $W$ we define the \emph{general electric charge}
(represented by $G$) contained in an cooriented $(n-2)$-dimensional
sphere $S^{n-2}\subset \A$ to be the following element in $W^\dual$:
$$
W\to \R;\quad q\mapsto \int_{S^{n-2}} \ip{ q \innerH G,\coorie}.
$$
\end{definition}

In regions where $\divH G=0$ this charge satisfies the conservation
law from the divergence theorem, see proposition
\ref{Vanishingelectriccharge}. \emph{Kinematic helicity lowering} is
based upon the contraction $W^\dual\tens W\to \R$ inducing the
exterior pairing:
$$
\wedgeH \from C^\infty(\A,H_k(W^\dual))
\cross C^\infty(\A,H_l(W))
\to C^\infty(\A,\Lambda^{k+l}V^\dual).
$$

\begin{definition}\label{Generalmagneticcharge}
\summary{General magnetic charge} (Compare paragraph
\ref{Magneticcharge}) For a kinematic field $F\in C(\A,H_{2}(W))$ of
the theory associated to $W$ we define the \emph{general magnetic
charge} (represented by $F$) contained in an oriented $2$-dimensional
sphere $S^{2}\subset \A$ to be the following element in $W$:
$$
W^\dual\to \R;\quad q\mapsto \int_{S^{2}} \ip{ q \wedgeH F,\orie}.
$$
\end{definition}

In regions where $\dH F=0$ this charge satisfies the conservation law
from Stoke's theorem, see proposition \ref{Vanishingmagneticcharge}.

\begin{definition}
\summary{General Lorentz force law} (Compare paragraphs
\ref{Lorentzforcelaw} and \ref{Lorentzforcelawingeneralfieldtheories})
The force of a kinematic field $F$ felt by a test particle $N$ with an
attached dual twistor $q\in W^\dual$ (interpreted as general electric
test charge) is given by $(q\wedgeH F)(N)$.
\end{definition}

\begin{definition}
\summary{General particles as sources} (Compare paragraph
\ref{Chargedparticlesassources}) A dual twistor $q\in W^\dual$ (a
general electric charge) attached to an oriented worldline $t\mapsto
c(t)$ induces a distributional source $j \in {\cal D}(\A, L^{-n}\tens
H^1(W^\dual)) = (C^\infty_o(\A,H_1(W)))^\dual$ by $\ipp{j,\alpha} :=
\int_\R (q\wedgeH \alpha)(\dot c)$.
\end{definition}

The induced exterior pairing from scalar multiplication $W\tens\R\to
W$ allows \emph{kinematic helicity raising}, as in paragraph
\ref{Kinematichelicityraising}, from electromagnetism to the general
field theory associated to $W$:
$$
\wedgeH \from C^\infty(\A,H_k(W))
\tens C^\infty(\A,\Lambda^lV^\dual)
\to C^\infty(\A,H_{k+l}(W)).
$$
This establishes Lienard Wiechert fields in general field
theories. The interior pairing from scalar multiplication $W\tens\R\to
W$ allows \emph{dynamic helicity raising}, as in
paragraph \ref{Dynamichelicityraising}, from electromagnetism to the
general field theory associated to $W$:
$$
\innerH \from C^\infty(\A,H_k(W))
\tens C^\infty(\A,L^{-n}\tens\Lambda^{k+l}V)
\to C^\infty(\A,L^{-n}\tens H^l(W^\dual)).
$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fierz's theory of gravity}
\label{Fierzstheoryofgravity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The next three sections study in some detail the three linear field
theories which are candidates for a conformally invariant linear field
theory for gravity. To relate physical dimensions and central weights
of tensors we will use fundamental constants as in example
\ref{Gravitationaldimensions}. The three theories are based upon the
representations $W=\Lambda^k{\asso V}$ with $k=1,2,3$. Hence we will
prove special cases of the claims about sequences of differential
operators and pairings quoted in the last section. The conformal
differential geometry developed in chapters
\ref{Elementaryconformalgeometry} and
\ref{Differentialconformalinvariants} is sufficient to develop these
theories on an arbitrary conformal manifold rather than on the
conformal affine space. The presence of curvature introduces some
phenomena that do not arise in the flat case.

Twistor fields induced by elements in the defining representation
$\asso V$ of the M\"obius Lie algebra are the gravitational masses of
Fierz's theory of gravity. Hence Fierz gravitational masses occur as
sections $f$ of the density bundle $L^1$ which lie in the kernel of
the conformal Hessian $\Hesse f=0$, see proposition
\ref{ConformalHessian}. The theory is based upon the following
kinematic sequence of differential operators: (we will use the
following short hand notations $T^0:=L^1\tens T^\dual M$ and
$C(L^1)=C^\infty(M,L^1)$ etc.)
\[
\begin{array}{*{9}{c}}
\mathrm{charge} &&
\mathrm{gauge} &&
\mathrm{potential} &&
\mathrm{kinematic\ field} &&\\
{\asso V}
&\to&
C(L^1) 
&\stackyrel{\Hesse}{\rightarrow}&
C(T^\dual\cartan T^0)
&\stackyrel{\Codazzi}{\rightarrow}&
C(\Lambda^2T^\dual\cartan T^0) 
&\stackyrel{\Fierz}{\rightarrow}&
C(\Lambda^3T^\dual\cartan T^0)\\
&&f&&A&&F&&\\
\end{array}
\]
The gauges are scalar densities of weight $+1$, the potentials are
symmetric trace-free bilinear forms of weight $-1$, the kinematic
fields are covector valued $2$-forms of weight $-2$ which are
alternating-free and trace-free. To summarize: $H_0(\asso V)=L^1$,
$H_1(\asso V)=V^\dual\cartan V$ and $H_2(\asso
V)=\Lambda^2V^\dual\cartan V^0$. The gauge operator is second order
and given by the conformal Hessian. Potential and kinematic field
operator are first order Stein Weiss operators \ref{SteinWeiss} and
hence their conformal invariance is given by Fegan's result, theorem
\ref{Fegansfirstorderconformalinvariants}. We will define them
explicitly in terms of a Weyl derivative $D$ with symmetric and
trace-free normalized Ricci curvature $\nric^D_0$: let $X,Y,Z$ and $U$
be vector fields, then
\begin{eqnarray*}
\Hesse f(X,U)
&:=& \oneover{2} ( D^2_{X,U}f + D^2_{U,X}f )
- \oneover{n} \sum_i(D_{t_i,\theta^i}^2f)\cip{X,U} 
+ f \nric^D_0(X,U),\\
\Codazzi A(X,Y,U)
&:=& D_XA(Y,U) 
+ \oneover{n-1}\sum_iD_{t_i}A(\theta^i,X)\cip{Y,U} 
- \swap(X,Y),\\
\Fierz F(X,Y,Z,U)
&:=& D_XF(Y,Z,U) 
- \oneover{n-2}\sum_iD_{t_i}F(Y,Z,\theta^i)\cip{X,U} 
+ \cyclic(X,Y,Z).
\end{eqnarray*}
It is elementary to check that this defines a complex in a conformally
flat $\Weylcurv=0$ spacetime. However, in a curved background this is
just a sequence of operators and the Weyl curvature is the obstruction
for being a complex. The dynamic sequence is given by:
\[
\begin{array}{*{7}{c}}
0&&j&&G&&\\
C(L^{-n-1})\!
&\stackyrel{\divdiv}{\ot}&\!
C(L^{-n}\tens T\cartan T^0)
&\stackyrel{\Symdiv}{\ot}&
C(L^{-n}\tens \Lambda^2T\cartan T^0)\!
&\stackyrel{\Fierzdiv}{\ot}&
C(L^{-n}\tens \Lambda^3T\cartan T^0)\\
&&
\mathrm{source} &&
\mathrm{dynamic\ field} &&
\end{array}
\]
The operators are explicitly defined by:
\begin{eqnarray*}
\Fierzdiv Q(X,Y,U)
&:=& \sum_i D_{t_i}Q(\theta^i,X,Y,U) 
- \oneover{3}\sum_i D_{t_i}Q(X,Y,U,\theta^i),\\
\Symdiv G(X,U) 
&:=& \oneover{2}(\sum_iD_{t_i}G(\theta^i,X,U) 
+ \sum_iD_{t_i}G(\theta^i,U,X)),\\
\divdiv j
&:=& \sum_{i,j}(D^2_{t_i,t_j}j(\theta^i,\theta^j) 
+ \nric^D_0(t_i,t_j)\tens j(\theta^i,\theta^j)).
\end{eqnarray*}
The constitutive relation in vacuum and in $n=4$ dimensions is given by
the conformal structure, which identifies kinematic and dynamic
fields.

\begin{remark}
Note that the conservation law for the source is second order. This is
a consequence of conformal invariance of the theory. In the original
noninvariant version Fierz in \cite{Fierz} used a first order
divergence as a candidate for a conservation law. His sources given by
$\Symdiv G$, where $G=F=\Codazzi A$ are only conserved $\divR^D\Symdiv
G=0$ if the potential $A$ satisfies an additional strong \emph{gauge
condition} $\divR^D A=0$ (which is also not conformally invariant).
\end{remark}

\begin{discussion}\label{FierzProductrulesforadjoints}
\summary{Product rules for adjoints} To see that those two sequences
are adjoint to each other we observe the following pairings and
divergence formulas. Note, that the central weights of the bundles
determine the order of operators and pairings. The operators $\Hesse$
and $\divdiv$ are second order and the required pairing is a simple
first order pairing, see proposition
\ref{Simpleconformallyinvariantfirstorderpairings}:
\begin{eqnarray*}
\innerH&\from& 
C^\infty(M,L^1)\cross C^\infty(M,L^{-n}\tens T\cartan T^0)
\to C^\infty(M,L^{-n}\tens T),\\
f \innerH j 
&:=& \Sum_i D_{t_i}f \tens j(\theta^i,\ ) 
- \Sum_i f \tens D_{t_i} j(\theta^i,\ ),\\
\divR (f \innerH j) &=& \ip{\Hesse f,j} + \ip{f,-\divdiv j}.
\end{eqnarray*}
The pairing between potentials and dynamic fields is purely tensorial:
\begin{eqnarray*}
\innerH&\from&(T^\dual\cartan T^0)
\tens (L^{-n}\tens \Lambda^2T\cartan T^0)
\to L^{-n}\tens T,\\
%A\innerH G
%&:=&\sum_{i,j} A(t_i,t_j)G(\theta^i,\ ,\theta^j),\\
-\divR (A\innerH G) &=& \ip{\Codazzi A,G} + \ip{A, \Symdiv G}.
\end{eqnarray*}
The pairing for the kinematic integrability operator is also tensorial:
\begin{eqnarray*}
\innerH&\from&
(\Lambda^2T^\dual\cartan T^0)\tens (L^{-n}\tens \Lambda^3T\cartan T^0)
\to L^{-n}\tens T,\\
%F \innerH J
%&:=& \oneover{2}\Sum_{i,j,k}
%F(t_i,t_j,t_k)J(\theta^i,\theta^j,\ ,\theta^k),\\
(-1)^2\divR(F\innerH J) &=& \ip{\Fierz F,J} + \ip{F, \Fierzdiv J}.
\end{eqnarray*}
\end{discussion}

\begin{discussion}
\summary{Mass represented by dynamic Fierz fields} To rediscover the
gravitational mass, i.e.~a Fierz twistor, represented by a dynamic
Fierz field we need the following pairings and product rules: the
pairing between gauges and dynamic Fierz fields is simple first order,
see proposition \ref{Simpleconformallyinvariantfirstorderpairings}:
\begin{eqnarray*}
\innerH&\from& 
C^\infty(M,L^1)\cross C^\infty(M,L^{-n}\tens \Lambda^2T\cartan T^0)
\to C^\infty(M,L^{-n}\tens \Lambda^2T),\\
f \innerH G 
&:=& \Sum_i (D_{t_i}f) G(\ ,\ ,\theta^i) 
- \oneover{2} f \Sum_i D_{t_i} G(\ ,\ ,\theta^i),\\
\innerH&\from& 
(T^\dual\cartan T^0) \tens (L^{-n}\tens \Lambda^2T\cartan T^0)
\to (L^{-n}\tens T),\\
\innerH&\from& 
C^\infty(M,L^1)\cross C^\infty(M,L^{-n}\tens T\cartan T^0)
\to C^\infty(M,L^{-n}\tens T),\\
\divR (f \innerH G)
&=& \Hesse f \innerH G + f \innerH \Symdiv G 
- f\oneover{2}\Sum_{i,j,k}
G(\theta^i,\theta^j,\theta^k)\Weylcurv_{t_i,t_j}t_k.
\end{eqnarray*}
Note that the third pairing needed is the first order pairing from the
above paragraph \ref{FierzProductrulesforadjoints} and that the
appearance of the Weyl curvature tensor prevents a true codimension
one divergence formula in the curved case. As was explained in
paragraph \ref{Chargerepresentedbydynamicfields} the first pairing
together with a cooriented $(n-2)$-sphere $S$ and a dynamic Fierz
field $G$ induces a linear form on the kernel of the conformal Hessian
$\kernel(\Hesse)\ni f\mapsto \int_S \ip{f\innerH G,\coorie}$. In the
affine case this linear form represents a Fierz twistor using the
inner product from definition \ref{InnerproductonassoV} on $\asso V$.

The next operators in the kinematic and dynamic sequences are true
codimension one adjoints, see definition \ref{Codimensiononeadjoints}
even in the curved case, since the pairing is purely tensorial and the
operators are first order:
\begin{eqnarray*}
\innerH&\from&(T^\dual\cartan T^0)
\tens (L^{-n}\tens \Lambda^3T\cartan T^0)
\to L^{-n}\tens \Lambda^2T,\\
-\divR (A\innerH J) &=& \Codazzi A\innerH J + A\innerH \Fierzdiv J.
\end{eqnarray*}
\end{discussion}

\begin{discussion}
\summary{Fierz particles as sources} To view a Fierz twistor along a
worldline as a pointlike source for gravity, as in paragraph
\ref{Chargedparticlesassources}, we need a pairing between Fierz
potentials and Fierz gauges to give $1$-forms, which is simple first
order:
\begin{eqnarray*}
\wedgeH
&\from& C^\infty(M,L^1)\cross C^\infty(M,T^\dual\cartan T^0)
\to C^\infty(M,T^\dual),\\
f \wedgeH A
&:=& f \tens \divR^D A - (n-1)\grad^D f \inner A.
\end{eqnarray*}
Given $f\in\kernel(\Hesse)$ along an oriented worldline $c$ we define
a distributional source $j\in{\cal D}(M,L^{-n}\tens T\cartan T^0)
=(C^\infty_o(M,T^\dual\cartan T^0))^\dual$ by evaluating it on
$\alpha\in C^\infty_o(M,T^\dual\cartan T^0)$ as
$\ipp{j,\alpha}:=\int_\R (f\wedgeH \alpha)(\dot c)\,dt$. To check that
this source is conserved we need a pairing between Fierz gauges and
Fierz gauges to give functions, which is a Laplace pairing, example
\ref{Laplacepairing}:
\begin{eqnarray*}
\wedgeH
&\from& C^\infty(M,L^1)\cross C^\infty(M,L^1)
\to C^\infty(M,\R),\\
f \wedgeH g
&:=& {n-1\over n}\left(
(\Laplace^D f)\tens g - n\cip{Df,Dg} + f \tens (\Laplace^D g) \right).
\end{eqnarray*}
To verify the product rule notice $(D^2+\nric^D)f =\Hesse f
+\oneover{n}(\Laplace^D f)\com$ and recall the contracted second
Bianchi identity, proposition \ref{ContractedBianchiidentity}: $D_X\nscal^D =
\trace_\com D\nric^D(X)$. Together this gives
\begin{eqnarray*}
D_X\Laplace^D f
%&=& \Sum_i D^3_{X,e_i,e_i}f + D_X\nscal^D \tens f + \nscal^D \tens D_Xf\\
%&=& \Sum_i R^D_{X,e_i}.Df(e_i) + D^3_{e_i,X,e_i}f + \nscal^D \tens D_Xf
%+ D_{e_i}\nric^D(X,e_i) \tens f\\
%&=& \Sum_i \Weylcurv_{X,e_i}.Df(e_i) 
%- \Lieb{\nric^D(X),e_i}.Df(e_i)  
%+ \Lieb{\nric^D(e_i),X}.Df(e_i) + \nscal^D \tens D_Xf\\
%&&+ \Sum_i D_{e_i}(D^2 + \nric^D)f(X,e_i)\\ 
%&=& (2-n)\nric^D(X,\grad^D f)
%+ \divR^D\Hesse f(X) + \oneover{n}D_X\Laplace^Df 
&=& (2-n)\nric^D(X,\grad^D f) + \divR^D\Hesse f(X) + \oneover{n}D_X\Laplace^Df,
\end{eqnarray*}
and finally
$$
\dR(f \wedgeH g) = (\Hesse f) \wedgeH g + f \wedgeH (\Hesse g).
$$
\end{discussion}

\begin{discussion}
\summary{Kinematic Fierz fields as accelerations} The Lorentz force
law in Fierz theory is based upon a pairing between Fierz gauges
(gravitational test masses) and kinematic Fierz fields to give a
$2$-form. This pairing is simple first order:
\begin{eqnarray*}
\wedgeH
&\from& C^\infty(M,L^1)\cross C^\infty(M,\Lambda^2 T^\dual\cartan T^0)
\to C^\infty(M,\Lambda^2 T^\dual),\\
(f \wedgeH F)(X,Y)
&:=& (n-2)F(X,Y,\grad^D f) - f\tens\Sum_i D_{t_i}F(X,Y,\theta^i).
\end{eqnarray*}
\end{discussion}

\begin{discussion}
\summary{Lienard Wiechert fields in Fierz's theory} Given a solution
of the Max\-well equation $F_{\mathrm{em}}=\dR A_{\mathrm{em}}$, like
the Lienard Wiechert field, the following pairing between a Fierz
twistor and a $1$-form respectively a $2$-form induces a Fierz
potential respectively a Fierz field: the pairings are simple first
order:
\begin{eqnarray*}
\wedgeH
&\from& C^\infty(M,L^1)\cross C^\infty(M,T^\dual)
\to C^\infty(M,T^\dual\cartan T^0),\\
(f \wedgeH A_{\mathrm{em}})
&:=& (-2)A_{\mathrm{em}}\cartan Df - f\tens \Killing^D A_{\mathrm{em}},\\
\wedgeH
&\from& C^\infty(M,L^1)\cross C^\infty(M,\Lambda^2 T^\dual)
\to C^\infty(M,\Lambda^2 T^\dual\cartan T^0),\\
(f \wedgeH F_{\mathrm{em}})
&:=& (-3)F_{\mathrm{em}}\cartan Df - f\tens \Twistor^D F_{\mathrm{em}},
\end{eqnarray*}
where $\cartan$ and $\Twistor^D$ are defined in definition
\ref{Firstorderoperatoronforms}.
\end{discussion}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bach's theory of gravity}
\label{SectionBachstheoryofgravity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\emph{Linearized conformal geometry} and \emph{Linearized Bach's
theory of gravity} are the same in $n=4$ dimensions. This theory is a
linearization of Bach's conformal theory of gravity which we briefly
recalled in remark \ref{Bachstheoryofgravity}. It is based upon the
adjoint representation $W=\g$, hence elements in $W$ have the
geometric interpretation as being conformal Killing fields and play
the role of gravitational masses in Bach's theory of gravity. The
first operator in the kinematic sequence is the conformal
Killing operator:
$$
\Killing=\oneover{2}\Lie\com\from
C^\infty(M,TM)\to C^\infty(M,L^2\tens\Sym_0^2T^\dual M),
$$
with $H_0(\g):=TM$ and $H_1(\g):=L^2\tens\Sym_0^2T^\dual
M$. \emph{Vector fields} play the role of gauges and the potentials
$h\in C^\infty(M,H_1(\g))$ are weightless symmetric bilinear forms and
have the interpretation of \emph{linearized conformal metrics}: if
$\com_t$ denotes a family of conformal structures on $M$ with
$\com_0:=\com$ being the background conformal structure, then
$\dot\com:=\delbydel{t}\com_t\restr{t=0}$ is trace-free with respect
to $\com_0$, since all $\com_t$ are normalized $|\deter\com_t|=1$. To
understand the next operator in the kinematic sequence ($n\geq 4$)
note that each conformal metric $\com_t$ has an associated Weyl
curvature tensor $\mathrm{W}^{\com_t}$ and the linearization of this
process $\com_t\mapsto\mathrm{W}^{\com_t}$ defines a differential
operator $\LinWeyl(\dot c) :=\delbydel{t}\mathrm{W}^{\com_t}$ with
$$
\LinWeyl\from C^\infty(M,L^2\tens\Sym_0^2T^\dual M) \to
C^\infty(M,\Lambda^2T^\dual\cartan \so(TM)),
$$
where $H_2(\g):=\Lambda^2T^\dual M\cartan \so(TM)$ is the bundle of
Weyl tensors of central weight $-2$, where the kinematic fields take
their values. This is the beginning of a complex only if the
background metric is conformally flat, otherwise the Weyl curvature is
an obstruction: if $X$ is an arbitrary vector field with (local) flow
$\Phi_t$, then
\begin{eqnarray*}
\LinWeyl\circ\Killing(X)
&=& \oneover{2} \LinWeyl(\delbydel{t} \Phi_t^\dual\com)
= \oneover{2} \delbydel{t} \mathrm{W}^{\Phi_t^\dual\com}
= \oneover{2} \delbydel{t} \Phi_t^\dual( \mathrm{W}^{\com_0} )\\
&=& \oneover{2}\Lie_X \mathrm{W}^{\com_0}
\end{eqnarray*}
If the background conformal structure $\com_0=\com$ has zero Weyl
curvature $\Weylcurv=0$ then this Lie derivative vanishes. The Weyl
curvature tensor $\mathrm{W}^{\com_t}$ on a conformal manifold
$M,\com_t$ satisfies a differential Bianchi identity
$\Bianchi_{\com_t}(\mathrm{W}^{\com_t})=0$, see propositions
\ref{FirstorderBianchiidentity},
\ref{FourdimensionalBianchioperator}. Here $\Bianchi_{\com_t}$ is a
linear conformally invariant operators which (as any invariant
operator) depends upon the conformal structure (in a nonlinear
way). Hence the kinematic integrability condition on Weyl tensors is
given by the differential Bianchi identity: for $n\geq 5$ dimensions
this is
$$
\Bianchi\from C^\infty(M,\Lambda^2T^\dual \cartan \so(TM)) \to
C^\infty(M,\Lambda^3T^\dual\cartan \so(TM)),
$$
and in $n=4$ dimension this is a second order operator:
$$
\Bianchi\from C^\infty(M,\Lambda^2T^\dual \cartan \so(TM)) \to
C^\infty(M,\Lambda^3T^\dual\cartan T^\dual M).
$$
For the composite $\Bianchi\circ\LinWeyl$ note that for all $t$ we
know $\Bianchi_{\com_t}\mathrm{W}^{\com_t}=0$ and differentiating gives:
\begin{eqnarray*}
0 &=&
\delbydel{t}(\Bianchi_{\com_t}\mathrm{W}^{\com_t})\\
&=& \delbydel{t} \Bianchi_{\com_t}\mathrm{W}^{\com_0} +
\delbydel{t} \Bianchi_{\com_0}\mathrm{W}^{\com_t}\\
&=&
(\delbydel{t}\Bianchi_{\com_t})\mathrm{W}^{\com_0} +
\Bianchi_{\com_0}\circ \LinWeyl(\dot \com).
\end{eqnarray*}
Hence we have $\Bianchi\circ\LinWeyl (\dot\com)= -
(\delbydel{t}\Bianchi_{\com_t})\Weylcurv$, which is zero in the flat
case. The beginning of the kinematic sequence:
\[
\begin{array}{*{9}{c}}
\mathrm{Killing\ fields} &&
\mathrm{vector\ fields} &&
\mathrm{lin.\ conf.\ metrics} &&
\mathrm{Weyl\ fields} &&\\
\so(\asso V)
&\to&
C(T)
&\stackyrel{\Killing}{\longrightarrow}&
C(L^2\tens\Sym_0^2T^\dual)
&\stackyrel{\LinWeyl}{\longrightarrow}&
C(\Lambda^2T^\dual\cartan\so(T))
&\stackyrel{\Bianchi}{\longrightarrow}&,
\end{array}
\]
has and adjoint providing the end of the dynamic sequence:
\[
\begin{array}{*{7}{c}}
C(L^{-n-2}\tens T)
&\stackyrel{\divR}{\ot}&
C(L^{-n-2}\tens \Sym_0^2T)
&\stackyrel{\Bach}{\ot}&
C(L^{-n}\tens \Lambda^2T\cartan\so(T)),&&\\
&&
\mathrm{source} &&
\mathrm{dynamic\ field} &&
\end{array}
\]
where the adjoint of $\Killing$ is a first order divergence operator
$\divR$ acting on symmetric bilinear forms of central weight $-n$ and
the operator adjoint to $\LinWeyl$ is second order and called the
\emph{Bach operator}, see \ref{Bachoperator}, acting on
Weyl tensors of central weight $2-n$. In $n=4$ dimensions the
constitutive relation in vacuum between kinematic and dynamic fields
is simply the conformal structure, since both fields are Weyl tensors
of central weight $-2$.

For convenience we will write down the linearized Weyl operator which
is simple second order, proposition
\ref{Simplesecondorderdifferentialoperators}, and associated to
$E:=V^\dual\cartan V$, $F:=\Lambda^2 V^\dual\cartan V$, $G:=\Lambda^2
V^\dual\cartan\so(V)$: the aim is to make the projection $\pi_W\from
V^\dual\tens V^\dual \tens E\to G$ explicit. For this we write $\pi_W$
as a composition $\pi_W:=\pi_L\circ(\id_{V^\dual}\tens \pi_C)$ with
$\pi_C\from V^\dual \tens E\to F$ and $\pi_L\from V^\dual \tens F\to
G$. The Stein Weiss operator of these projections are called
\emph{Codazzi operator} and \emph{Lanczos operator} and are given by
($X$, $Y$, $U$, $V$ are vector fields):
\begin{eqnarray*}
\Codazzi^D h(X,Y,U)
&:=& \pi_C\circ Dh (X,Y,U)\\
&=& D_Xh(Y,U) 
+ \oneover{n-1}\sum_iD_{t_i}h(\theta^i,X)\cip{Y,U} 
- \swap(X,Y),\\
\Lanczos^D A(X,Y,U,V)
&:=& \pi_L\circ DA (X,Y,U,V)\\
&=& D_XA(U,V,Y) - D_YA(U,V,X) + D_UA(X,Y,V) - D_VA(X,Y,U)\\
&&- {2\over n-2} \Big( \Symdiv^D A(Y,V)\cip{X,U}  - \Symdiv^D A(X,V)\cip{Y,U}\\
&&\quad\quad- \Symdiv^D A(Y,U)\cip{X,V} + \Symdiv^D A(X,U)\cip{Y,V} \Big),\\
\Symdiv^D A(X,U)
&:=& \oneover{2} ( \trace DL(\ ,X,U) + \trace DL(\ ,U,X) ),\\
\LinWeyl h
&:=& \pi_L(\id\tens \pi_C)(D^2h + \nric^D\tens\, h).
\end{eqnarray*}
The Bach operator is adjoint to this and therefore given by:
\begin{eqnarray*}
\Bach^D W(Y,U) &:=& \Sum_{i,j} (D^2_{t_i,t_j} +
\nric^D(t_i,t_j))W(\theta^i,Y,U,\theta^j) -\swap(Y,U).
\end{eqnarray*}

\begin{discussion}\label{BachProductrulesforadjoints}
\summary{Product rules for adjoints} To see that those two sequences
are adjoint to each other we remark that first order Stein Weiss
operators like $\Killing$ always have an obvious divergence formula,
see remark \ref{AdjointSteinWeiss}. For convenience we will write out
the tensorial pairing between vector fields and Bach sources:
\begin{eqnarray*}
\innerH&\from&T \tens (L^{-n-2}\tens T\cartan T)
\to L^{-n}\tens T,\\
f\innerH j 
&:=&\sum_{i} \cip{f,t_i}j(\theta^i,\ ),\\
\divR (f\innerH j) &=& \ip{\Killing f,j} + \ip{f,\divR j}.
\end{eqnarray*}
The second order operators
$\LinWeyl$ and $\Bach$ need the following simple first order pairing:
\begin{eqnarray*}
\innerH&\from&
C(M,L^2\tens T^\dual\cartan T^\dual)\cross
C(M,L^{-n}\tens \Lambda^2T\cartan \so(T)) \to
C(M,L^{-n}\tens T),\\
h \innerH W
&:=& \oneover{2}\Sum_{i,j,k}
h(t_j,t_k)D_{t_i}W(\theta^j,\theta^k)\theta^i
- D_{t_i}h(t_j,t_k)W(\theta^j,\theta^k)\theta^i,\\
-\divR (h \innerH W)
&=& \ip{\LinWeyl h, W} + \ip{h,-\Bach W}.
\end{eqnarray*}
\end{discussion}

\begin{discussion}
\summary{Mass represented by dynamic Bach fields} To rediscover the
gravitational mass, a Killing field, represented by a dynamic Bach
field, a Weyl tensor, we need the following pairings and product
rules: the pairing between gauges and dynamic Bach fields is simple
first order:
\begin{eqnarray*}
\innerH&\from& 
C^\infty(M,T)\cross C^\infty(M,L^{-n}\tens \Lambda^2T\cartan\so(T))
\to C^\infty(M,L^{-n}\tens \Lambda^2T),\\
f \innerH W 
&:=& \Sum_{i,j} D_{t_i}f(\theta^j)\ip{W(\ ,\ ,t_j),\theta^i} 
-  2f(\theta^j)\ip{D_{t_i}W(\ ,\ ,t_j),\theta^i},\\
\innerH&\from& 
C^\infty(M,L^2\tens T^\dual\cartan T^\dual)
\cross C^\infty(M,\Lambda^2T^\dual\cartan\so(T))
\to C^\infty(M,L^{-n}\tens T),\\
\innerH&\from& 
T\tens (\Lambda^2 T^\dual\cartan\so(T)) \to (L^{-n}\tens T),\\ 
\divR (f \innerH W)
&=& \Killing f \innerH W + f \innerH (-\Bach W) + \ip{W\tens K,\Weylcurv}. 
\end{eqnarray*}
Note that the second pairing is the first order pairing from the above
paragraph \ref{BachProductrulesforadjoints}. The appearance of the
Weyl curvature tensor prevents a true codimension one divergence
formula in the curved case. As was explained in paragraph
\ref{Chargerepresentedbydynamicfields} the first pairing together with
a cooriented $(n-2)$-sphere $S$ and a dynamic Bach Weyl field $W$
induces a linear form on the kernel of the conformal Killing equation
$\kernel(\Killing)\ni f\mapsto \int_S \ip{f\innerH W,\coorie}$. In the
affine case this linear form represents a Killing field using the
Killing form on $\so(\asso V)$.
\end{discussion}

\begin{discussion}
\summary{Bach particles as sources} To view a Killing field along a
worldline as a pointlike source for gravity we need a pairing between
vector fields and linearized metrics to give $1$-forms:
$$
\wedgeH\from 
C^\infty(M,T)\cross C^\infty(M,L^2\tens T^\dual\cartan T^\dual) \to
C^\infty(M,T^\dual).
$$
This pairing is second order and not simple. Its existence follows
from the general theory, theorem \ref{Conformalexteriorpairings},
applied to $\so(\asso V)\tens\so(\asso V)\to \R$. To check that this
source is conserved we need the Branson pairing, example
\ref{Bransonpairing}, with $k=1$.
\end{discussion}

\begin{discussion}
\summary{Kinematic Bach fields as accelerations} The Lorentz force law
in Bach's theory is based upon a pairing between Killing fields and
kinematic Bach fields to give a $2$-form. This pairing is simple
first order:
\begin{eqnarray*}
\wedgeH
&\from& C^\infty(M,T)\cross C^\infty(M,\Lambda^2T^\dual\cartan\so(T))
\to C^\infty(M,\Lambda^2 T^\dual),\\
(f \wedgeH W)(X,Y) &:=& (n-2)F(X,Y,\grad^D f) - f\tens\Sum_i
D_{t_i}F(X,Y,\theta^i).
\end{eqnarray*}
\end{discussion}

\begin{discussion}
\summary{Lienard Wiechert fields in Bach's theory} Given a solution of
the Max\-well equation $F=\dR A$, like the Lienard Wiechert field, a
tensorial pairing between a Killing field and a
$1$-form induces a Bach potential:
\begin{eqnarray*}
\wedgeH
&\from& T \tens T^\dual
\to L^{2}\tens T^\dual\cartan T^\dual,\\
(f \wedgeH A)
&:=& f\cartan A.
\end{eqnarray*}
\end{discussion}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Penrose's theory of gravity}
\label{Penrosestheoryofgravity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Twistor fields induced by elements in the representation on trivectors
$\Lambda^3\asso V$ of the M\"obius Lie algebra $\so(\asso V)$ are the
gravitational masses of Penrose's theory of gravity. Hence Penrose
gravitational masses are $2$-forms $f$ of central weight $+1$ in the
kernel of the Twistor operator $\Twistor f=0$, see definition
\ref{Firstorderoperatoronforms}. The theory is based upon the
following kinematic sequence:
\[
\begin{array}{*{8}{c}}
\mathrm{charge} &&
\mathrm{gauge} &&
\mathrm{potential} &&
\mathrm{kinematic\ field} &\\
\Lambda^3{\asso V}
&\to&
C(\Lambda^2T^0\tens L^1) 
&\stackyrel{\Twistor}{\rightarrow}&
C(T^\dual\cartan \Lambda^2T^0\tens L^1)
&\stackyrel{\Lanczos}{\rightarrow}&
C(\Lambda^2T^\dual\cartan \Lambda^2T^0\tens L^1)
&\stackyrel{}{\rightarrow}\\
&&f&&A&&F&\\ 
\end{array}
\]
The kinematic fields are Weyl tensors of central weight $-1$. The two
operators above are first order Stein Weiss operators, see definition
\ref{SteinWeiss}, and were already defined in the last section
\ref{SectionBachstheoryofgravity}. It is a beginning of a complex in
the flat case. They have natural adjoints:
\[
\begin{array}{*{6}{c}}
&&j&&G&\\
C(L^{-n-1}\tens \Lambda^2T^0)
&\stackyrel{\Skewdiv}{\ot}&
C(L^{-n-1}\tens T\cartan \Lambda^2T^0)
&\stackyrel{\Lanczosdiv}{\ot}&
C(L^{-n-1}\tens \Lambda^2T\cartan \Lambda^2T^0)
&\stackyrel{}{\ot}\\
&&
\mathrm{source} &&
\mathrm{dynamic\ field} &
\end{array}
\]
Note the the dynamic field is a Weyl tensor of central weight $-3$,
whereas the kinematic field is a Weyl tensor of central weight
$-1$. Hence the constitutive relation in vacuum is more involved,
since we can't use the conformal metric to identify kinematic and
dynamic fields as in electromagnetism or Fierz's and Bach's
theory. Theorem \ref{ConformalBGGcomplex} yields in $n=4$ dimensions
an additional third order operator from potential to fields, since
$H_2(\Lambda^3{\asso V})$ splits into a direct sum, see example
\ref{Penroseshomology}:
$$
H_2(\Lambda^3{\asso V})
= \Lambda^2 V^\dual \cartan \Lambda^2V^0 \tens (L^1\dsum L^{-1}).
$$
Hence in $n=4$ dimensions there is an invariant third order
differential operator
$$
\UltraLanczos\from
C^\infty(M,T^\dual\cartan \Lambda^2T^0\tens L^1)
\to C^\infty(M,\Lambda^2T^\dual\cartan \Lambda^2T^0\tens L^{-1}),
$$
which annihilates the image of $\Twistor$ in the flat case,
i.e.~$\UltraLanczos\circ\Twistor=0$. This allows to suggest that
kinematic and dynamic Weyl field $F$ and $G$ come from the same
kinematic potential $A\in C^\infty(M,T^\dual\cartan \Lambda^2T^0\tens
L^1)$ using
\begin{eqnarray*}
F&:=& \Lanczos A,\\
%\in C^\infty(M,\Lambda^2T^\dual\cartan \Lambda^2T^0\tens L^{+1}),
G&:=& \UltraLanczos A,
%\in C^\infty(M,\Lambda^2T^\dual\cartan \Lambda^2T^0\tens L^{-1}),
\end{eqnarray*}
which can be interpreted as a conformally invariant constitutive
relation in $n=4$ dimensions, since it does not depend upon the choice
of the potential. This third order operator is related to the
kinematic integrability condition for $F$, which is also third order,
since by example \ref{Penroseshomology}: $H_3(\Lambda^3{\asso V}) =
\Lambda^3 V^\dual \cartan \Lambda^2V^0 \tens L^{-1}$, which has
central weight $-4$.

\begin{discussion}
\summary{Mass represented by dynamic Penrose Weyl fields} To
rediscover the gravitational mass, i.e.~a Penrose twistor, represented
by a dynamic Penrose Weyl field we just need a zero order pairing:
\begin{eqnarray*}
\innerH&\from& 
(\Lambda^2T^0\tens L^1) \tens L^{-1-n}\tens \Lambda^2T\cartan \Lambda^2T^0)
\to L^{-n}\tens \Lambda^2T,\\
f \innerH G 
&:=& \oneover{2}  \Sum_{i,j} f(t_i,t_j) G(\ ,\ ,\theta^i,\theta^j),\\
\divR (f \innerH G)
&=& \Twistor f \innerH G + f \innerH \Lanczosdiv G .
\end{eqnarray*}
This divergence formula and related product rules based upon zero
order pairings were first observed by Penrose
\cite{PenroseRindler}. As was explained in paragraph
\ref{Chargerepresentedbydynamicfields} this pairing together with a
cooriented $(n-2)$-sphere $S$ and a dynamic Penrose Weyl field $G$
induces a linear form on the kernel of the twistor operator
$\kernel(\Twistor)\ni f\mapsto \int_S \ip{f\innerH G,\coorie}$. In the
affine case this linear form represents a twistor using the inner
product on $\Lambda^3{\asso V}$.
\end{discussion}

\begin{discussion}
\summary{Penrose particles as sources} To view a Penrose twistor along
a worldline as a pointlike source for gravity, as in paragraph
\ref{Chargedparticlesassources}, we need a pairing between Penrose
gauges and Penrose potentials to give $1$-forms, which is second
order, but not simple. We refer to the general theory, theorem
\ref{Conformalexteriorpairings}, applied to $\Lambda^3{\asso V}\tens
\Lambda^3{\asso V} \to \R$:
\begin{eqnarray*}
\wedgeH
&\from& C^\infty(M,\Lambda^2T^0\tens L^1)
\cross C^\infty(M,T^\dual\cartan \Lambda^2T^0\tens L^1)
\to C^\infty(M,T^\dual).
\end{eqnarray*}
Given $f\in\kernel(\Twistor)$ along an oriented worldline $c$ we
define a distributional source $j\in{\cal D}(M,L^{-n-1}\tens T\cartan
\Lambda^2T^0) =(C^\infty_o(M,T^\dual\cartan \Lambda^2T^0\tens
L^1)^\dual$ by evaluating it on $\alpha$ as $\ipp{j,\alpha}:=\int_\R
(f\wedgeH \alpha)(\dot c)\,dt$. The product rule which ensures
conservation of this source involves a pairing between Penrose gauges
and Penrose gauges to give functions, which is a Branson pairing,
example \ref{Bransonpairing}.
\end{discussion}

\begin{discussion}\label{KinematicPenroseWeylfieldsasaccelerations}
\summary{Kinematic Penrose Weyl fields as accelerations} The Lorentz
force law in Penrose's theory is based upon a pairing between Penrose
gauges (gravitational test masses) and kinematic Penrose Weyl fields
to give a $2$-form. This pairing is second order and not simple. We
refer to the general theory, theorem \ref{Conformalexteriorpairings},
applied to $\Lambda^3{\asso V}\tens \Lambda^3{\asso V} \to \R$:
\begin{eqnarray*}
\wedgeH
&\from& C^\infty(M,\Lambda^2T^0\tens L^1)
\cross C^\infty(M,\Lambda^2 T^\dual\cartan \Lambda^2T^0\tens L^1)
\to C^\infty(M,\Lambda^2 T^\dual).
\end{eqnarray*}
\end{discussion}

\begin{discussion}
\summary{Lienard Wiechert fields in Penrose's theory} Given a solution
of the Maxwell equation $F=\dR A$, like the Lienard Wiechert field, a
tensorial pairing between a Penrose twistor and a $1$-form
respectively a $2$-form induces a Penrose potential respectively a
kinematic Penrose field:
\begin{eqnarray*}
\wedgeH
&\from& (\Lambda^2T^0\tens L^1) \tens T^\dual
\to (T^\dual\cartan \Lambda^2T^0\tens L^1),\\
\wedgeH
&\from& (\Lambda^2T^0\tens L^1)
\tens \Lambda^2 T^\dual
\to (\Lambda^2 T^\dual \tens \Lambda^2T^0\tens L^1).
\end{eqnarray*}
\end{discussion}

\begin{remark}
\summary{Schwarzschild Weyl field} Penrose's linear theory of gravity
provides a direct link to Newton's theory of gravity via the Weyl
tensor of the Schwarzschild solution. The Schwarzschild geometry in
Einstein's theory is static and spherically symmetric. It's Riemann
curvature tensor only has a Weyl curvature part. Linearizing the
Schwarzschild solution (by the mass parameter) with Minkowski
background we obtain a tensor with the symmetries of a Weyl tensor in
Minkowski space. It is divergence-free by the contracted second
Bianchi identity \ref{ContractedBianchiidentity}. On the other hand,
let $c(t):=c_0 + tN$ be the straight worldline in the affine space
representing the pointlike gravitational source. The Schwarzschild
Weyl field is (up to scale) uniquely determined to be the dynamic
Penrose Weyl field $G_{\mathrm{Schw}}$ which is static and spherically
symmetric and satisfies $\divH (G_{\mathrm{Schw}})=0$ away from the
worldline $c$:
$$
G_{\mathrm{Schw}}
= {M\over r^3}\, (\vec r \wedge N) \cartan (\vec r \wedge N),
$$
where $M\in \R$ is a parameter interpreted as gravitational mass of
the source, $r$ is the luminosity distance to the source, $\vec r :=
\grad^D r$ is the spacelike radial vector field and $D$ is the affine
derivative. The Schwarzschild Weyl field falls off like $1/r^3$ in
agreement with the central weight of $G_{\mathrm{Schw}}$ being
$-3$. It has a potential $A_{\mathrm{Schw}}$ and a kinematic field
$F_{\mathrm{Schw}}$ given by
\begin{eqnarray*}
A_{\mathrm{Schw}}
&=& M (\vec r \wedge N) \cartan N,\\
F_{\mathrm{Schw}}
&=& \Lanczos A_{\mathrm{Schw}} 
= {M\over r}\, (\vec r \wedge N) \cartan (\vec r \wedge N),\\
G_{\mathrm{Schw}}
&=& \UltraLanczos A_{\mathrm{Schw}}.
\end{eqnarray*}
Note that $c$ is in particular a conformal geodesic and along a
conformal geodesic any Penrose twistor in $\Lambda^3{\asso V}$ induces
a constant real number along $c$ according to proposition
\ref{Twistorconservation}. The inner product on $\Lambda^3{\asso V}$
turns this linear form on $\Lambda^3{\asso V}$ into a Penrose twistor
$f_{\mathrm{Schw}}$ induced by $c$. This twistor is up to scale the
unique static and spherically bivector field in the kernel of the
twistor operator $\Twistor$ given by the linear polynomial
$$
f_{\mathrm{Schw}} := r\, (\vec r \wedge N).
$$
Obviously it is this twistor which can be used to raise the Coulomb
potential and field to the Schwarzschild potential and kinematic
field: $A_{\mathrm{Schw}}= A_{\mathrm{Coul}} \cartan (M
f_{\mathrm{Schw}})$ with $A_{\mathrm{Coul}} = {1\over r}\, N$ and
$F_{\mathrm{Schw}} =F_{\mathrm{Coul}} \cartan (M f_{\mathrm{Schw}})$
with $F_{\mathrm{Coul}} = {1\over r^2}\, \vec r \wedge N$. To
calculate the predicted force of $F_{\mathrm{Schw}}$ on a test
particle we assume a multiple of $f_{\mathrm{Schw}}$ to be the
gravitational mass of the test particle, like $m f_{\mathrm{Schw}}$,
with $m\in \R$. Although we haven't calculated the relevant second
order pairing $(m f_{\mathrm{Schw}})\wedgeH F_{\mathrm{Schw}}$ we know
that the resulting closed $2$-form is static and spherically
symmetric, hence a multiple of $F_{\mathrm{Coul}}$:
$$
(m f_{\mathrm{Schw}})\wedgeH F_{\mathrm{Schw}}
= {mM\over r^2}\, \vec r \wedge N.
$$
(The second order pairing is in this case only first order, since
$f_{\mathrm{Schw}}$ is linear, which also explains why the force falls
off like $1/r^2$.) Hence the general Lorentz force law from
\ref{KinematicPenroseWeylfieldsasaccelerations} specializes to the
Newtonian force law in the spherically symmetric case. Paragraph
\ref{Perihelionadvance} applies also to this gravitational force law
and predicts a perihelion advance in this linear theory of gravity.
\end{remark} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Bernstein Gelfand Gelfand theory}
\label{BGGtheory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this chapter we develop the Bernstein Gelfand Gelfand resolution
together with its coproduct. This chapter is algebraic in flavour and
the link to differential geometry is given in the first section
\ref{Vermamodulesandtheholonomicjetoperator}. For a Lie group $G$ with
closed subgroup $P\subset G$, we define an invariant jet operator, see
\ref{Jetoperator}, acting on sections of a homogeneous bundle over the
homogeneous space $G/P$. This jet operator encodes the information of
all derivatives of a section at a point into an algebraic object, a
linear form on the \emph{induced module}. This jet operator allows to
translate differential operators into homomorphisms and differential
pairings into coproducts between induced modules. From section
\ref{Homogeneousparabolicgeometry} on we deal with a semisimple $G$
and a parabolic $P$. Parabolic subgroups contain a further subgroup
$L\subset P\subset G$ which can be thought of as the structure group
of the reduced frame bundle of $G/P$. For instance for the conformal
sphere $G/P$ we have $L:=\CO(V)$ and $P:=\CO(V)\semidirect V^\dual$,
see section \ref{Projectivelightcone}. Note that in general a
$P$-representation leads to vector bundle over $G/P$ which is not
associated to the $L$-reduction of the linear frame bundle of
$G/P$. Only those $P$-representations on which $V^\dual$ acts
trivially lead to tensor bundles. The most substantial piece of
representation theory that we will need is relative Lie algebra
homology. The elementary definitions and properties are presented in
section \ref{EilenbergsrelativeLiealgebrahomology} with detailed
proofs. If $\p\subset \g$ denote the Lie algebras with $V:=\g/\p$,
then the chains of the homology theory are $\Lambda^k V\tens W$, where
$W$ is a $G$-representation. The homologies $H_k(W)$ are examples of
$P$-representations on which $V^\dual$ acts trivially. In section
\ref{SectionTwisteddeRhamresolution} we twist the exterior deRham complex by
a representation $W$ of $G$. The resulting complex is too big to be
geometrically relevant, since $V^\dual$ acts nontrivially on
$\Lambda^k V^\dual\tens W$. In section \ref{ProjectiononVermacochains}
we construct a projection $S$ onto the homologies and the BGG complex
between sections in the homology occurs as homotopy equivalent to the
twisted deRham complex between sections of the chains. Adjoint
operators can also be studied purely algebraically, which we will do
in section \ref{Adjointhomomorphisms}.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Verma modules and the holonomic jet operator}
\label{Vermamodulesandtheholonomicjetoperator}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $G$ denote a (real) Lie group with closed subgroup $P\subset G$
and Lie algebras $\g\subset\p$. We are interested in invariant
differential operators and pairings on the following manifold:

\begin{definition}
\summary{Homogeneous model} The quotient $G/P$ where $P$ acts as
subgroup on $G$ from the right is called a \emph{Kleinian} or
\emph{homogeneous model}.
\end{definition}

The differential operators and pairings act between \emph{homogeneous
vector bundles}: any $P$-representation $E$ from the right leads to a
bundle $G \cross_P E:=G\cross E/\sim$ where pairs $(g,e)\sim(gp,e.p)$
in the orbit of $p\in P$ are identified. Sections of that bundle can
be viewed as $P$-equivariant functions on $G$ with values in $E$:
$C^\infty(G,E)^P:=\setof{ e\in C^\infty(G,E)\property
e(gp)=e(g).p}$. On these bundles we have various group action which we
need to distinguish:

\begin{definition}\label{regularrightaction}
\summary{Regular right $G$-actions} The map $R_h\from C^\infty(G,E)\to
C^\infty(G,E)$ for $h\in G$ defined on functions $e$ by right
multiplication $R_he(g) := e(gh)$ is called the \emph{regular right
action}. This is a $G$-action from the left:
$$
R\from G \cross C^\infty(G,E)\to C^\infty(G,E).
$$
\end{definition}

\begin{definition}\label{inducedleftaction}
\summary{Induced left $P$-action}
A function $e$ is $P$-equivariant, if $R_pe=e.p$ for all $p\in P$. Hence 
$e$ is $P$-equivariant, if it is invariant under the following 
\emph{induced left $P$-action}:
$$
P\cross C^\infty(G,E)\to C^\infty(G,E),
$$
defined by $p.e := R_pe.p^{-1}$, i.e.~$(p.e)(g):= e(gp).p^{-1}$.
\end{definition}

\begin{definition}\label{regularleftaction}
\summary{Regular left $G$-action} The map $L_h\from C^\infty(G,E)\to
C^\infty(G,E)$ for $h\in G$ defined on functions $e$ by left
multiplication $L_he(g) := e(hg)$ is called the \emph{regular left
action}. This is a $G$-action from the right:
$$
L\from C^\infty(G,E) \cross G \to C^\infty(G,E).
$$
\end{definition}

\begin{remark}
Regular right and left action commute, hence if $E$ is a right
$P$-representation, and a section $e$ is $P$-equivariant, then so is
$L_h(e)$. I.e.~we have a right $G$-action on sections of the
homogeneous bundle:
$$
L\from C^\infty(G,E)^P\cross G \to C^\infty(G,E)^P.
$$
\end{remark}

Associated to a Lie algebra $\g$ is the universal enveloping algebra
$\U(\g)$, which is associative and has a unit (see Appendix). The
algebra $\U(\g)$ can be interpreted as the algebra of scalar
differential operators on $G$ which are equivariant under the regular
left $G$-action. This is how we will use it in proposition
\ref{Jetoperator}, when we define a universal jet operator on $G/P$,
which is the aim of the next paragraphs.

The Lie algebra $\g$ acts upon $\U(\g)$ by multiplication from the
left, from the right and by conjugation:
\begin{eqnarray*}
l\from \g\tens\U(\g)\to\U(\g) &;& l_X(U)=XU,\\
r\from \g\tens \U(\g)\to\U(\g) &;& r_X(U)=-UX,\\
\ad\from  \g\tens\U(\g)\to\U(\g) &;& \ad_X(U)=(l_X-r_X)(U)=XU-UX.
\end{eqnarray*}
In what follows it is the left action which is \emph{the} action of
$\g$ on $\U(\g)$ and other actions will be mentioned explicitly.

\begin{definition}\label{InducedmodulesorVermamodules}
\summary{Induced modules or Verma modules} If $\p$ is a subalgebra of
$\g$ and $E^\dual$ is a left $\p$-module, then $\U(\g)\tens E^\dual$
contains an \emph{induced left $\U(\g)$ ideal} generated by
$$
I(\g,\p,E^\dual)
:= \generatedby{X\tens \eta - 1 \tens X.\eta
\property X\in\p, \eta\in E^\dual}.
$$
The quotient
$$
\Verma(\g,\p,E^\dual)
:= \U(\g) \tens_{\U(\p)} E^\dual
:= \U(\g) \tens E^\dual/I(\g,\p,E^\dual),
$$
is called the \emph{induced left $\g$-module} by the left $\p$-module
$E^\dual$. The action of $\g$ on $\U(\g)$ by multiplication from the
left clearly leaves the ideal invariant and descends to a left
$\g$-action on $\Verma(\g,\p,E^\dual)$. 
\end{definition}

\begin{definition}
\summary{$P$-actions on Verma modules} Let $P\subset G$ be Lie groups 
with Lie algebras $\p\subset \g$. The group $G$ acts upon $\U(\g)$ by the 
adjoint action, which corresponds to the action of $\g$ by conjugation on 
$\U(\g)$. If $E^\dual$ is a left $P$-representation, then $P$ as subgroup 
acts upon $\U(\g)$ by the adjoint action and hence on the tensor product:
\begin{center}
$\Ind\from P\cross\U(\g)\tens E^\dual \to \U(\g)\tens E^\dual$,\\
$\Ind_p(U\tens\eta) := \Ad_p(U)\tens p.\eta$.
\end{center}
This action leaves the induced ideal $I(\g,\p,E^\dual)$ invariant, since 
it maps generators to generators:
$\Ind_p(X\tens\eta - 1\tens X.\eta)
= (\Ad_p(X)\tens p.\eta - 1\tens p.X.\eta)
= (\Ad_p(X)\tens p.\eta - 1\tens \Ad_p(X).p.\eta)$.
Hence the action descends to the quotient
$$
P\cross\Verma(\g,\p,E^\dual)\to\Verma(\g,\p,E^\dual).
$$
\end{definition}

\begin{remark}
Infinitesimally we get a left $\p$-action
\begin{center}
$\ind\from \p\tens\U(\g)\tens E^\dual \to \U(\g)\tens E^\dual$,\\
$\ind_X(U\tens\eta) := \ad_X(U)\tens \eta + U\tens X.\eta$,
\end{center}
leaving the ideal $I(\g,\p,E^\dual)$ invariant:
$\ind_X (Y\tens\eta - 1\tens Y.\eta)
= (\ad_X(Y)\tens \eta + Y\tens X.\eta - 1\tens X.Y.\eta)
= (\Lieb{X,Y}\tens\eta - 1\tens\Lieb{X,Y}.\eta )
  + (Y\tens X.\eta - 1\tens Y.X.\eta)$.
The corresponding $\p$-action on $\Verma(\g,\p,E^\dual)$ is 
indeed the restriction of the $\g$-action by left multiplication, since
\begin{eqnarray*}
\ind_X (U\tens_{\U(\p)}\eta)
&=& (XU-UX)\tens_{\U(\p)}\eta + U\tens_{\U(\p)}X.\eta\\
&=& XU\tens_{\U(\p)}\eta.
\end{eqnarray*}
\end{remark}

Following Weingart \cite{Weingart} we will use the induced module
$\Verma(\g,\p,E^\dual)$ by a right $P$-representation $E$ to define a
\emph{jet operator} for sections of the homogeneous bundle $G\cross_P
E$ over $G/P$: the left action on functions $R\from G\cross
C^\infty(G,E)\to C^\infty(G,E)$; $R_he(g):=e(gh)$, induces a left
representation of the Lie algebra $\g$ and hence a left action $R_X$
by elements of the universal enveloping algebra $X\in\U(\g)$:
$$
R\from \U(\g) \tens C^\infty(G,E) \to C^\infty(G,E).
$$

\begin{proposition}\label{Jetoperator}
\summary{Jet operator} The linear differential operator:
$$
\jet^{\infty} \from 
C^\infty(G,E) \to C^\infty(G,(\U(\g)\tens E^\dual)^\dual)
$$
given by
$$
\ip{\jet^\infty(e)(g),U\tens \eta}
:= \ip{(R_Ue)(g),\eta}
= (R_U\ip{e,\eta})(g)
$$
is $G$-equivariant under the regular action and $P$-equivariant under
the induced left action. Its restriction to $P$-invariant sections
vanishes on the induced ideal $I(\g,\p,E^\dual)$:
$$
\jet^{\infty} \from 
C^\infty(G,E)^P \to C^\infty(G,\Verma(\g,\p,E^\dual)^\dual)^P.
$$
\end{proposition}

\proof The operator $\jet^\infty$ is $G$-equivariant since regular
left and right multiplication commute: $L_h\circ R_U=R_U\circ L_h$ for
$h\in G$. To show $P$-equivariance note for $p\in P$:
\begin{eqnarray*}
\ip{p.(\jet^\infty(e))(g),U \tens \eta}
&=& \ip{(\jet^\infty(e))(gp).p^{-1},U \tens \eta}\\
&=& \ip{(\jet^\infty(e))(gp),\Ad_{p^{-1}}U \tens p^{-1}.\eta}\\
&=& \ip{(R_{\Ad_{p^{-1}}U}e)(gp), p^{-1}.\eta}\\
&=& \ip{R_U(R_p e)(g), p^{-1}.\eta}\\
&=& \ip{R_U((R_p e).p^{-1})(g), \eta}\\
&=& \ip{R_U(p.e)(g), \eta}\\
&=& \ip{\jet^\infty(p.e)(g),U\tens\eta}.
\end{eqnarray*}
A function $e\in C^\infty(G,E)$ is $P$-equivariant, if $R_pe=e.p$ for
all $p\in P$, which infinitesimally means $R_Xe=e.X$ for $X\in
\p$. That shows $\ip{\jet^\infty(e)(g),I(\g,\p,E^\dual)}=0$. \qed

Using this operator, we will present next the result, independently due
to Baston Eastwood Rice \cite{BastonEastwood}, \cite{EastwoodRice},
Collingwood Shelton \cite{CollingwoodShelton} and Soergel
\cite{Soergel}, that $G$-equi\-var\-i\-ant linear differential operators are
in one to one correspondence with $\g$-equivariant homomorphisms
between the induced Verma modules:

\begin{discussion}\label{Correspondencebetweenoperatorsandhomomorphisms}
\summary{Correspondence between operators and homomorphisms} Let $E$
and $F$ be two right $P$-representations. The above jet operator
translates $\g$-equivariant homomorphisms between the induced Verma
modules
$$
\Nabla\from\Verma(\g,\p,F^\dual)\to\Verma(\g,\p,E^\dual)
$$
into a $G$-equivariant differential operator 
$$
\Nabla\from C^\infty(G,E)^P\to C^\infty(G,F)^P
$$ 
as follows: if $e\in C^\infty(G,E)$ is a $P$-equivariant section,
$\phi\in F^\dual$ an element of the dual space and $g\in G$ a point,
then
$$
\ip{(\Nabla e)(g),\phi} 
= \ip{{\jet^\infty(e)(g)},{\Nabla(1\tens_{\U(\p)}\phi)}}.
$$
This is a one to one correspondence.
\end{discussion}

Hence the geometric question of finding $G$-equivariant differential
operators between sections of homogeneous bundles has been translated
into the algebraic question of finding homomorphisms between Verma
modules. Similarly, if $E_1$ and $E_2$ are two right $P$
representations, then a $G$-equivariant differential pairing
$$
\vee \from C^\infty(G,E_1)^P \cross C^\infty(G,E_2)^P \to C^\infty(G,F)^P,
$$ 
corresponds to a $\g$-equivariant coproduct:
$$
\vee \from \Verma(\g,\p,F^\dual)
\to\Verma(\g,\p,E_1^\dual) \tens \Verma(\g,\p,E_2^\dual).
$$
We will finish this section with some general remarks about
$\g$-homomorphisms between Verma modules:

\begin{remark}
Let $W^\dual$ be some (finite or infinite dimensional) left
$\g$-representation and $E^\dual$ a left
$\p$-representation. Clearly, any $\g$-homomorphism $W^\dual\to
\U(\g)\tens E^\dual$ induces a $\g$-homomorphism
$W^\dual\to\Verma(\g,\p,E^\dual)$ by projection into the quotient.
\end{remark}

\begin{proposition}
\summary{Frobenius reciprocity} Let $E^\dual$ be a left
$\p$-representation and $W^\dual$ a left $\g$-representation. Any
$\p$-equivariant linear map $\phi_0\from E^\dual\to W^\dual$ induces a
$\g$-equivariant homomorphism $\phi\from \Verma(\g,\p,E^\dual)\to
W^\dual$ via $\phi(U\tens_{\U(\p)}\eta):=U.(\phi_0(\eta))$, with
$U\in\U(\g)$ and $\eta\in E^\dual$. This defines an isomorphism:
$$
\Hom_\p(E^\dual,W^\dual) \stackrel{\isomorph}{\longrightarrow}
\Hom_\g(\Verma(\g,\p,E^\dual),W^\dual);\quad
\phi_0\mapsto\phi.
$$
\end{proposition}

\proof Any linear map $\phi_0\from E^\dual\to W^\dual$ induces a
$\g$-homomorphism $\phi\from \U(\g)\tens E^\dual\to W^\dual$ via
$\phi(U\tens\eta):=U.(\phi_0(\eta))$. If $\phi_0$ is $\p$-equivariant
then $\phi_0(X.\eta)=X.(\phi_0(\eta))$ for all $X\in\p$. In that case
$\phi$ vanishes on the generators of the induced ideal
$I(\g,\p,E^\dual)$ since $\phi(X\tens \eta-1\tens
X.\eta)=X.(\phi(1\tens \eta))-\phi(1\tens
X.\eta)=X.(\phi_0(\eta))-\phi_0(X.\eta)$. Hence $\phi$ descends to the
quotient $\phi\from\Verma(\g,\p,E^\dual)\to W^\dual$. On the other
hand, given a $\g$-homomorphism $\phi\from\Verma(\g,\p,E^\dual)\to
W^\dual$, there is an induced $\phi_0\from E^\dual\to W^\dual$ defined
by $\phi_0(\eta) =\phi(1\tens_{\U(\p)}\eta)$ and $\phi_0$ satisfies
$\phi_0(X.\eta) =\phi(1\tens_{\U(\p)}X.\eta)
=\phi(X\tens_{\U(\p)}\eta) =X.(\phi(1\tens_{\U(\p)}\eta))
=X.(\phi_0(\eta))$.\qed

Let $E^\dual$ and $F^\dual$ be two left
$\p$-re\-pre\-sen\-ta\-tions. In the sequel we will construct
$\g$-homo\-mor\-phisms between Verma modules like
$\Verma(\g,\p,F^\dual)\to\Verma(\g,\p,E^\dual)$ which lift to
$\g$-homomorphism between $\U(\g)\tens F^\dual\to\U(\g)\tens
E^\dual$. In general a $\g$-homomorphism $\phi\from \U(\g)\tens
F^\dual \to \U(\g)\tens E^\dual$ induces a homomorphism between the
corresponding Verma modules, iff $\phi$ maps the induced ideal into
the induced ideal: $\phi(I(\g,\p,F^\dual))\subset
I(\g,\p,E^\dual)$. To evaluate $\phi$ on the ideal we summaries the
relevant left $\p$-actions on $\U(\g)\tens E^\dual$ as follows: for
$X\in\p$, $U\in\U(\g)$ and $\eta\in E^\dual$ we have
\begin{eqnarray*}
l_X(U\tens\eta)&:=&XU\tens\eta,\\
r_X(U\tens\eta)&:=&-UX\tens\eta,\\
\ad_X(U\tens\eta)&:=&XU\tens\eta - UX\tens\eta,\\
\ind_X(U\tens\eta)&:=&XU\tens\eta - UX\tens\eta + U\tens X.\eta,\\
(\ind-l)_X(U\tens\eta)&:=& - UX\tens\eta + U\tens X.\eta.
\end{eqnarray*}
Notice that the last action maps onto the ideal $I(\g,\p,E^\dual)$. We
will often use the following criterion:

\begin{proposition}\label{Criterion}
A $\g$-homomorphism $\phi\from\U(\g)\tens F^\dual\to \U(\g)\tens
E^\dual$ descends to a homomorphism between the corresponding Verma
modules iff its commutator with the induced $\p$-actions maps into the
induced ideal: $\image \Lieb{\phi,\ind_X} \subset I(\g,\p,E^\dual)$
with $X\in\p$.
\end{proposition}

\proof The action of $(\ind-l)_X$ maps onto the induced ideal, hence
$\phi$ descends iff $\phi\circ (\ind-l)_X$ takes values in the
ideal. Since $\phi$ commutes with the left $l_X$ action we have:
\begin{eqnarray*}
\phi\circ (\ind-l)_X
&=& \Lieb{\phi,(\ind-l)_X} + (\ind-l)_X\circ\phi\\
&=& \Lieb{\phi,\ind_X} + (\ind-l)_X\circ\phi,
\end{eqnarray*}
where the second summand already is in the ideal.\qed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Homogeneous parabolic geometry}
\label{Homogeneousparabolicgeometry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We will now focus on the class of parabolic homogeneous spaces $G/P$
which will be characterized algebraically on the level of the Lie
algebras $\p\subset \g$. We will use the following definition which
emphasizes exactly the properties which we will need in the sequel:
 
\begin{definition}\label{Parabolicgeometry}
\summary{Parabolic geometry} Let $\g$ be a finite dimensional
semisimple Lie algebra, which splits as vector space into a direct sum
$\g=V\dsum\l\dsum V^\dual$, where $V$, $\l$, $V^\dual$ are
subalgebras, $V^\dual$ and $V$ are dual to each other (via the Killing
form of $\g$), and we have the following properties of the Lie
bracket: $\Lieb{V,\l}\subset V$, $\Lieb{V^\dual,\l}\subset
V^\dual$. Denote by $\p:=\l\dsum V^\dual$ the complementary subalgebra
to $V$. Such a Lie algebra $\g=V\dsum\l\dsum V^\dual$ characterizes a
so called \emph{parabolic geometry}. If $V$ is Abelian, then this is
called the \emph{Abelian} or \emph{almost Hermitian} case. If $G$
denotes a Lie group with Lie algebra $\g$ (e.g.~$G=\Aut(\g)$) then
$P:=\setof{g\in G\property \Ad_g(\p)=\p}$ is a closed subgroup with
Lie algebra $\p$ and we refer to $G/P$ as the \emph{homogeneous
model of parabolic geometry}.
\end{definition}

\begin{remark}
In the above situation $\l$ is reductive, $V$ and $V^\dual$ are
nilpotent and $\p$ is a parabolic subalgebra. Alternatively, if $\g$
is semisimple and $\p\subset\g$ a parabolic subalgebra, then $\g$
splits into a direct sum as explained above. Hence a parabolic
geometry is characterized by a semisimple $\g$ and a choice of a
parabolic $\p\subset\g$. In the Abelian case $\Centre(\l)$, the centre
of $\l$, is one dimensional. For completeness we mention the dual
parabolic subalgebra, defined by $\p^\dual:=V\dsum\l$.
\end{remark}

\begin{example}
\summary {Conformal geometry} Let $V$ be an $n$-dimensional conformal
vector space. The M\"obius Lie algebra $\g=V\dsum\co(V)\dsum V^\dual$
with $\p:=\co(V)\dsum V^\dual$ is a special case of a parabolic
geometry. In view of the isomorphism $\g=\so(\asso V)$ with $\asso
V:=L^{1}\dsum V^0\dsum L^{-1}$ we can take as M\"obius group
$G:=\O(\asso V)$ and as subgroup $P=\CO(V)\semidirect V^\dual$ (see
also section \ref{Projectivelightcone}). For positive definite signature we
obtain the sphere $S^n=G/P$ as homogenous model of parabolic geometry.
\end{example}

\begin{example}
\summary{Projective geometry} Let $V$ be an $n$-dimensional vector
space. The Lie algebra of projective geometry is $\g=V\dsum\gl(V)\dsum
V^\dual$ with $\p=\gl(V)\dsum V^\dual$, where the Lie bracket on
$(a,A,\alpha), (b,B,\beta)\in\g$ is given by
\begin{eqnarray*}
\Lieb{a,b} &=& 0,\\
\Lieb{A,b} &=& Ab,\\
\Lieb{\alpha,b} &=&
   -\alpha(b)\id - \alpha\tens b,\\
\Lieb{A,B} &=& A\circ B - B\circ A,\\
\Lieb{\alpha,B} &=& \alpha\circ B,\\
\Lieb{\alpha,\beta} &=& 0.
\end{eqnarray*}
We have a Lie algebra isomorphism $\psi\from\g\to\sll(\asso V)$ where
$\asso V:= L^{-n/(n+1)}\tens V \dsum L^{-n/(n+1)}$ is
$(n+1)$-dimensional and 
$$
\psi(a,A,\alpha)(v,t)
:= (at -\oneover{n+1}\trace(A)v +Av,
-\oneover{n+1}\trace(A)t +\alpha(v)),
$$
with $v\in L^{-n/(n+1)}\tens V$ and $t\in L^{-n/(n+1)}$. The central
element $\id_V\in\gl(V)$ acts as
$\psi(\id_V)(v,t)=(1/(n+1)v,-n/(n+1)t)$ which explains the choice of
weights in the definition. $\asso V$ is equipped with a canonical
density. As Lie group we can take the Lie group $G:=\SL(\asso V)$ leaving
this density invariant and $P:=\GL(V)\semidirect V^\dual$. As
homogeneous model we obtain $n$-dimensional projective space $G/P=\R
P^n$.
\end{example}
%
\hidden{
\begin{example}
\summary{Lagrange-Gra{\ss}mannian geometry} Let $U$ be an $n$-dimensional
vector space. The Lie algebra of Lagrange Gra{\ss}mann geometry is
$\g=\Sym^2U\dsum\gl(U)\dsum \Sym^2U^\dual$ with $\p=\gl(U)\dsum
\Sym^2U^\dual$, where the Lie bracket on $(a,A,\alpha),
(b,B,\beta)\in\g$ is given by
\begin{eqnarray*}
\Lieb{a,b} &=& 0,\\
\Lieb{A,b} &=& A.b,\\
\Lieb{\alpha,b} &=& -\Sum_{i=1}^n \alpha(t_i)\tens b(\theta^i),\\
\Lieb{A,B} &=& A\circ B - B\circ A,\\
\Lieb{\alpha,B} &=& \alpha.B,\\
\Lieb{\alpha,\beta} &=& 0,
\end{eqnarray*}
where $A.b$ and $\alpha.B$ denote the natural left respectively right
actions of $\gl(U)$. We have a Lie algebra isomorphism
$\psi\from\g\to\sp(\asso V,\omega)$ where $\asso V:= U\dsum U^\dual$
is a $(2n)$-dimensional symplectic vector space with symplectic form
given by $\omega((q,p),(q',p')):=p(q')-p'(q)$ with $q\in U$ and $p\in
U^\dual$. It is given by
$$
\psi(a,A,\alpha)(q,p)
:= (p(a)+Aq, -p\circ A + \alpha(q)).
$$
The central element $\id_U\in\gl(U)$ acts as
$\psi(\id_U)(q,p)=(q,-p)$. As Lie group we can take the symplectic
group $G:=\Sp(\asso V,\omega)$.
\end{example}
}
\begin{construction}
\summary{Penrose's twistor fields} Let $W$ be a right
$G$-representation. This clearly induces a right $\g$-action on
$W$. Elements of $W$ are called \emph{parabolic twistors}. The space
of coinvariants $W_{V^\dual}:=W/WV^\dual$ obviously defines a right
$P$-representation. Each twistor $w\in W$ induces (in a linear way) a
section of the homogeneous bundle $G\cross_P W_{V^\dual}$, a so called
\emph{twistor field} by mapping $g\in G$ to $[wg]_{V^\dual}\in
W_{V^\dual}$. This gives a $G$-equivariant linear inclusion:
$$ 
\iota_H \from W\to C^\infty(G,W_{V^\dual})^P,
$$
i.e.~$\iota_H(wh)=L_h(\iota_H(w))$ for all $h\in G$.
\end{construction}

The aim of the next sections is to identify the twistor fields as
solution of a $G$-equivariant linear differential operator. For this we 
assume $W$ to be finite dimensional. In fact
the above inclusion $\iota_H$ is the beginning of a $G$-equivariant
differential complex:

\begin{theorem}
\summary{Parabolic Bernstein Gelfand Gelfand complex} If we denote by
$H_k(V^\dual,W)$ the relative Lie algebra homology spaces defined in
the next section in paragraph \ref{VdualhomologywithvaluesinW}, then
$H_k(V^\dual,W)$ are right $P$-representations and there is a locally
exact complex
$$ 
0 {\to} 
W \stackrel{\iota_H}{\rightarrow}
C^\infty(G,H_0(V^\dual,W))^P \stackrel{\dH}{\rightarrow}
C^\infty(G,H_1(V^\dual,W))^P \stackrel{\dH}{\rightarrow} 
C^\infty(G,H_2(V^\dual,W))^P \stackrel{\dH}{\rightarrow}
\ldots\quad,
$$
of $G$-equivariant linear differential operators.
\end{theorem}

In particular we have $H_0(V^\dual,W)=W_{V^\dual}$. Such a complex can
be viewed as a variation of the deRham complex of alternating
multilinear forms. So we denoted the resulting differential operators
by $\dH$ where the subscript $H$ stands for homology. The first
differential operator $C^\infty(G,H_0(V^\dual,W))^P \to
C^\infty(G,H_1(V^\dual,W))^P$ will be called the \emph{twistor
operator}. On the deRham complex we have the wedge product on forms
and the exterior derivative satisfies the Leibniz rule. Similarly we
have the following:

\begin{theorem}
If $W_1$, $W_2$ and $W_3$ are three finite dimensional
$G$-representations from the right and $F\from W_1\tens W_2 \to W_3$,
is a linear (nontrivial) $G$-equivariant map, then there is a
$G$-equivariant bilinear differential pairing
$$
\wedgeH \from C^\infty(G,H_k(V^\dual,W_1))^P
\tens C^\infty(G,H_l(V^\dual,W_2))^P
\to C^\infty(G,H_{k+l}(V^\dual,W_3))^P. 
$$
For $k=l=0$ this pairing is an extension of $F$ since it satisfies
$$
\wedgeH \circ (\iota_H\tens \iota_H) = \iota_H\circ F.
$$
More generally for $s\in C^\infty(G,H_k(V^\dual,W_1))^P$ and $t\in
C^\infty(G,H_l(V^\dual,W_2))^P$ the following Leibniz rule holds:
$$
\dH (s \wedgeH t) 
= (\dH s) \wedgeH t + (-1)^k s \wedgeH (\dH t).
$$
\end{theorem}

\begin{examples}
Such pairings between $G$-representations are as follows: the
$G$-equivariant contraction $W\tens W^\dual \to \R$, the trivial
multiplication $W\tens \R \to W$, the Lie algebra action $\g\tens W
\to W$ and its dual $W \tens W^\dual \to \g$.
\end{examples}

We will prove the above results not on the level of differential
operators acting on sections (differential geometric picture). Instead
we will work in the dual (algebraic) picture with homomorphisms
between induced modules, see definition
\ref{InducedmodulesorVermamodules}. In the parabolic context, when
$\g$ is semisimple and $\p$ is parabolic, the induced module
$\Verma(\g,\p,E^\dual)$ is also called a \emph{parabolic Verma
module}.

\begin{remark}\label{BorelVermamodules}
\summary{Universal lowest weight modules} The case when $\lie{b}$ is a
\emph{Borel subalgebra} of a semisimple $\g$ containing a \emph{Cartan
subalgebra} $\lie{h}\subset\lie{b}\subset\g$ is a special (non
Abelian) parabolic geometry $\g=V\dsum\l\dsum V^\dual$ with
$\l=\lie{h}$, $\p=\lie{b}=\lie{h}\dsum \lie{n}^-$ and
$\lie{n}^-=V^\dual$. Let $W^\dual$ be a left $\g$-representation. An
element $\lambda\in\lie{h}^\dual$ is called a \emph{weight of
$W^\dual$} if the corresponding \emph{weight space}
$W^\dual_\lambda=\setof{\omega\in W^\dual\property
H.\omega=\lambda(H)\omega, \forall H\in\lie{h}}$ is
nontrivial. Nonzero elements of nontrivial weight spaces are called
\emph{weight vectors}. A weight vector $\omega\in W^\dual_\lambda$ is
called \emph{lowest weight vector} if $\lie{n}^-\subset\g$ acts
trivially on it: $\lie{n}^-.\omega=0$. A representation $W^\dual$ is
called \emph{lowest weight module} if it is generated by a lowest
weight vector $\omega$, i.e.~$W^\dual=\U(\g)\omega$. In this context
Verma defined in \cite{Verma} to each \emph{weight}
$\lambda\in\lie{h}^\dual$ a \emph{universal lowest weight module} as
quotient $\U(\g)/I(\g,\lie{b},\lambda)$, where $I(\g,\lie{b},\lambda)$
is the left $\U(\g)$ ideal generated by $\lie{n}^-$ and
$\generatedby{X-\lambda(X)\property X\in \lie{h}}$. This defines a
\emph{lowest weight module} which is universal in the sense that it
maps onto all other lowest weight modules of weight $\lambda$. Indeed
all finite dimensional irreducible $\g$-representations occur as
quotients. Lepowsky in \cite{LepowskyBGG} generalized this notion to
the above parabolic Verma modules.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Eilenberg's relative Lie algebra homology}
\label{EilenbergsrelativeLiealgebrahomology}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In case of a parabolic geometry $\p\subset\g$ see definition
\ref{Parabolicgeometry}, we will consider a $\p$-equivariant relative
Lie algebra homology theory with values in a right $\g$-representation
$W$: the boundary operator is a map between
$$
\delta_{V^\dual}\from
\Lambda^k V^\dual\tens W\to\Lambda^{k-1} V^\dual\tens W,
$$
where the subscript $V^\dual$ denotes the Lie algebra which
effectively acts. The homology spaces $H_k(V^\dual,W)$ will define the
right $\p$-representations relevant in the sequence of differential
operators (geometric picture). Completely dual to the above homology
theory is a cohomology theory with values in $W^\dual$:
$$
d_{V^\dual}=\delta_{V^\dual}^\dual \from\Lambda^k V\tens W^\dual
\to\Lambda^{k+1} V\tens W^\dual.
$$
This coboundary operator will define left $\p$-representations
$H^k(V^\dual,W^\dual) =H_k(V^\dual,W)^\dual$ which occur when the
differential operators are expressed in terms of Verma module
homomorphisms (algebraic picture).

In addition we like to mention another boundary operator
$$
\delta_V\from\Lambda^k V\tens W^\dual\to\Lambda^{k-1} V\tens W^\dual
$$
which will appear in the next section as zero order part of the
twisted deRham homomorphism. Therefore we define it here, although its
$\p^\dual$-equivariant homology theory is not relevant in our
context. For completeness we also like to mention its dual coboundary
operator, which is the zero order part of the twisted deRham operator
(geometric picture):
$$
d_V=\delta_V^\dual\from
\Lambda^k V^\dual\tens W\to\Lambda^{k+1} V^\dual\tens W.
$$

\begin{definition}\label{VdualhomologywithvaluesinW}
\summary{$V^\dual$ homology with values in $W$} Let $W$ be a right 
representation of $\g$. We denote the right action
of $X\in\g$ on $w\in W$ by $w.X\in W$. The \emph{$k$-chains} are
defined by
$$
C_k(V^\dual,W) := \Lambda^k V^\dual \tens W.
$$
Note that $\p$ acts upon $V^\dual$ by the adjoint representation (from
the right) and therefore $\p$ acts on $\Lambda^k V^\dual$ by the
product rule: let $t_i$, $\theta^i$ be a dual basis of $V$, then we have 
for $X \in \p$ and $\alpha \in
\Lambda^k V^\dual$:
$$
\alpha.X 
:= \sum_i \Lieb{\theta^i,X} \wedge t_i \inner \alpha.
$$
In the Abelian case the action of $V^\dual\subset\p$ on $\Lambda^k
V^\dual$ is trivial. Clearly $\p$ as subalgebra of $\g$ acts also on
$W$ from the right and hence $C_k(V^\dual,W)$ is a right
$\p$-representation $C_k(V^\dual,W)\tens\p\to C_k(V^\dual,W)$: for
$w\in W$ we have: $(\alpha\tens w).X := (\alpha.X)\tens w +
\alpha\tens (w.X)$.  Next we define the boundary operator
$\delta_{V^\dual}$ where the subscript denotes the Lie algebra which
effectively acts upon $W$ in the following definition:
$\delta_{V^\dual}\from C_k(V^\dual,W) \to C_{k-1}(V^\dual,W)$;
$$
\delta_{V^\dual}( \alpha \tens w )
:= \oneover{2} \sum_i (t_i \inner \alpha).\theta^i \tens w 
   + \sum_i (t_i \inner \alpha) \tens w.\theta^i.
$$
For $k=0,1$ this definition means $\delta_{V^\dual}(w) = 0$ and
$\delta_{V^\dual}(\alpha \tens w) = w.\alpha$.
\end{definition}

\begin{proposition}
This defines a complex $\delta_{V^\dual}\circ \delta_{V^\dual} = 0$,
which is equivariant under the right action of $X\in\p$, i.e.
$\Lieb{{.X},\delta_{V^\dual}} := .X \circ \delta_{V^\dual} -
\delta_{V^\dual}\circ .X = 0$. Hence the kernel, image and homology of
$\delta_{V^\dual}$ give right $\p$-representations:
\begin{eqnarray*}
Z_k(V^\dual,W)
&:=& \kernel \delta_{V^\dual}\from C_k(V^\dual,W)\to C_{k-1}(V^\dual,W),\\
B_k(V^\dual,W)
&:=& \image \delta_{V^\dual}\from C_{k+1}(V^\dual,W)\to C_k(V^\dual,W),\\
H_k(V^\dual,W)
&:=& Z_k(V^\dual,W)/B_k(V^\dual,W).
\end{eqnarray*}
For $\gamma \in V^\dual$ and $c \in C_k(V^\dual,W)$ we have Cartan's 
identity:
$$
\delta_{V^\dual}( \gamma \wedge c ) 
+ \gamma \wedge (\delta_{V^\dual} c) 
= c.\gamma.
$$ 
Hence, $V^\dual$ acts trivially on $H_k(V^\dual,W)$, such that
$H_k(V^\dual,W)$ can equally well be viewed as a right
$\l$-representation extending trivially to a $\p$-representation.
\end{proposition}

\proof To do these calculation we remark the following formulas: the
right adjoint action of $V^\dual$ on $V^\dual$ is clearly given by
$\theta.\gamma = \Lieb{\theta,\gamma}$. Hence for the right coadjoint
action of $V^\dual$ on $V$ we have $v.\gamma =\Sum_k t_k
\ip{v.\gamma,\theta^k} =\Sum_k t_k \ip{v,\gamma.\theta^k} =\Sum_k t_k
\ip{v,\Lieb{\gamma,\theta^k}}$, with $v\in V$. The identity on $V$ is
invariant under this action $0= \id_V.\gamma = \Sum_i (\theta^i \tens
t_i).\gamma$. To show $\delta_{V^\dual}^2=0$ note that the square of
the first summand in the definition of $\delta_{V^\dual}$ has to be
zero separately, since this is the only summand in case of a trivial
$\g$-representation $W$. This follows from the Jacobi identity in
$V^\dual$:
\begin{eqnarray*}
\Sum_{i,j} (t_j\inner(t_i\inner\alpha).\theta^i).\theta^j
&=& \Sum_{i,j} (t_j\inner t_i\inner\alpha).\theta^i.\theta^j
- (t_j.\theta^i\inner t_i\inner\alpha).\theta^j\\
&=& \oneover{2} \Sum_{i,j} (t_j\inner t_i\inner\alpha).\Lieb{\theta^i,\theta^j}
- \Sum_{i,k}(t_k\inner t_i\inner\alpha).\Lieb{\theta^i,\theta^k}\\
&=& \oneover{2}\Sum_{i,j}(t_i\inner t_j\inner\alpha).\Lieb{\theta^i,\theta^j}\\
&=& \oneover{2}\Sum_{i,j,k} \Lieb{\theta^k,\Lieb{\theta^i,\theta^j}}
\wedge (t_k\inner t_i\inner t_j\inner\alpha)
%&=& \oneover{6}\Sum_{i,j,k} 
%(\Lieb{\theta^k,\Lieb{\theta^i,\theta^j}}
%+\Lieb{\theta^i,\Lieb{\theta^j,\theta^k}}
%+\Lieb{\theta^j,\Lieb{\theta^k,\theta^i}})
%\wedge(t_k\inner t_i\inner t_j\inner\alpha)
= 0.
\end{eqnarray*}
In general we have
\begin{eqnarray*}
\delta_{V^\dual}(\delta_{V^\dual}(\alpha\tens w))
&=& \oneover{4}\Sum_{i,j}(t_j\inner(t_i\inner\alpha).\theta^i).\theta^j \tens w
  + \oneover{2}\Sum_{i,j}t_j\inner (t_i\inner\alpha).\theta^i\tens w.\theta^j\\
& &\quad
  + \oneover{2}\Sum_{i,j}(t_j\inner t_i\inner\alpha).\theta^j\tens w.\theta^i
  + \Sum_{i,j}(t_j\inner t_i\inner\alpha)\tens w.\theta^i.\theta^j\\
&=& 0 
  + \oneover{2}\Sum_{i,j}(t_j\inner t_i\inner\alpha).\theta^i\tens w.\theta^j
  - \oneover{2}\Sum_{i,j}(t_j.\theta^i \inner t_i\inner\alpha)\tens w.\theta^j\\
& &\quad
  + \oneover{2}\Sum_{i,j}(t_j\inner t_i\inner\alpha).\theta^j\tens w.\theta^i
  + \oneover{2}\Sum_{i,j}(t_j\inner t_i\inner\alpha)\tens w.\Lieb{\theta^i,\theta^j}\\
&=&-\oneover{2}\Sum_{i,k}(t_k\inner t_i\inner\alpha)
\tens w.\Lieb{\theta^i,\theta^k}
+\oneover{2}\Sum_{i,j}(t_j\inner t_i\inner\alpha)
\tens w.\Lieb{\theta^i,\theta^j}
= 0.
\end{eqnarray*}
For the equivariance under $X\in\p$ note once more $v.X=\Sum_k
t_k\ip{v,\Lieb{X,\theta^k}}$. The following calculation shows
$\p$-equivariance in case of a trivial $\g$-representation $W$:
\begin{eqnarray*}
(\Sum_i (t_i\inner\alpha).\theta^i).X - \Sum_i (t_i\inner(\alpha.X)).\theta^i
&=& \Sum_i (t_i\inner\alpha).\theta^i.X - (t_i\inner\alpha).X.\theta^i
      + (t_i.X \inner\alpha).\theta^i\\
&=& \Sum_i (t_i\inner\alpha).\Lieb{\theta^i,X} 
      + \Sum_k (t_k \inner\alpha).\Lieb{X,\theta^k}
= 0.
\end{eqnarray*}
In general we have
\begin{eqnarray*}
\lefteqn{\Lieb{.X,\delta_{V^\dual}}(\alpha\tens w)}&&\\
&=& (\delta_{V^\dual}(\alpha\tens w)).X 
- \delta_{V^\dual}(\alpha.X\tens w) 
- \delta_{V^\dual}(\alpha\tens w.X)\\
&=& \oneover{2} \Sum_i (t_i \inner \alpha).\theta^i.X \tens w 
+ \oneover{2} \Sum_i (t_i \inner \alpha).\theta^i \tens w.X 
+ \Sum_i (t_i \inner \alpha).X \tens w.\theta^i\\
& &\quad
+ \Sum_i (t_i \inner \alpha) \tens w.\theta^i.X
- \oneover{2} \Sum_i (t_i \inner \alpha.X).\theta^i \tens w 
- \oneover{2} \Sum_i (t_i \inner \alpha).\theta^i \tens w.X \\
& & \quad
- \Sum_i (t_i \inner \alpha.X) \tens w.\theta^i
- \Sum_i (t_i \inner \alpha) \tens w.X.\theta^i\\
&=& 0 + \Sum_i (t_i.X \inner \alpha) \tens w.\theta^i
+ \Sum_i (t_i \inner \alpha) \tens w.\theta^i.X
- \Sum_i (t_i \inner \alpha) \tens w.X.\theta^i\\
&=&  \Sum_k (t_k \inner \alpha) \tens w.\Lieb{X,\theta^k}
+ \Sum_i (t_i \inner \alpha) \tens w.\Lieb{\theta^i,X}
= 0.
\end{eqnarray*}
Cartan's identity is a straight forward calculation: $c=\alpha\tens w$,
\begin{eqnarray*}
\lefteqn{\delta_{V^\dual}( \gamma \wedge c ) + \gamma \wedge (\delta_{V^\dual} c)}&&\\ 
&=& \oneover{2} \Sum_i (t_i\inner(\gamma\wedge\alpha)).\theta^i\tens w
+ \Sum_i t_i\inner(\gamma\wedge\alpha)\tens w.\theta^i\\
& &\quad
+ \oneover{2} \Sum_i \gamma\wedge (t_i\inner\alpha).\theta^i\tens w
+ \Sum_i \gamma\wedge (t_i\inner\alpha)\tens w.\theta^i\\
&=& \oneover{2} \alpha.\gamma\tens w 
- \oneover{2} \Sum_i (\gamma\wedge t_i\inner\alpha).\theta^i\tens w
+ \alpha\tens w.\gamma
- \Sum_i \gamma \wedge t_i\inner\alpha \tens w.\theta^i\\
& &\quad
+ \oneover{2} \Sum_i \gamma\wedge (t_i\inner\alpha).\theta^i\tens w
+ \Sum_i \gamma\wedge (t_i\inner\alpha)\tens w.\theta^i\\
&=& \oneover{2} \alpha.\gamma\tens w 
- \oneover{2} \Sum_i \gamma.\theta^i \wedge t_i\inner\alpha\tens w
+ \alpha\tens w.\gamma\\
&=& \alpha.\gamma\tens w + \alpha\tens w.\gamma.\qed 
\end{eqnarray*}    

\begin{remark}
If $G$ is a (real) Lie group with (complexified) Lie algebra $\g$ and
if $W$ is indeed a right $G$-representation, then all the above
constructed $\p$-representation are also $P$-representation with
$P=\setof{g\in G\property \Ad_g(\p)=\p}$.
\end{remark}

\begin{definition}
The zero homology $H_0(V^\dual,W)$ is also called the \emph{space of
coinvariants} of $W$ with respect to $V^\dual$, since in that case
$Z_0(V^\dual,W)=W$ and $B_0(V^\dual,W)=\image(W\tens V^\dual \to
W)=W.V^\dual$, i.e.~$H_0(V^\dual,W)=W/W.V^\dual=:W_{V^\dual}$.
\end{definition}

\begin{example}\label{Maxwellshomology}
\summary{Maxwell's homology}
The homology for the trivial representation $W=\R$ in the Abelian
case gives back the usual multilinear forms:
$$
H_k(V^\dual,\R) = \Lambda^k V^\dual.
$$
\end{example}

\begin{example}\label{Fierzshomology}
\summary{Fierz's homology}
In case of $V$ being a conformal vector space, the conformal Lie
algebra $\g$ is given by $\g = V \dsum \co(V) \dsum V^\dual$, which is
semisimple, since we have an isomorphism $\g=\so(\asso V)$ with $\asso
V := L^1 \dsum V^0 \dsum L^{-1}$. In particular $\asso V$ itself is a
$\g$-representation:
\begin{eqnarray*}
H_0(V^\dual,\asso V) &=& L^1,\\
H_k(V^\dual,\asso V) &=& \Lambda^k V^\dual \cartan V^0,\\
H_n(V^\dual,\asso V) &=& \Lambda^n V^\dual \tens L^{-1}.
\end{eqnarray*}
Here elements in the Cartan product $\Lambda^k V^\dual \cartan
V^\dual$ are tensors in $\Lambda^k V^\dual \tens
V^\dual$ which are alternating-free and trace-free.
\end{example}

\begin{example}\label{Bachshomology}
\summary{Linearized conformal geometry, Bach's homology} The
adjoint representation $W =\g =V\dsum\co(V)\dsum V^\dual$ in the
conformal case gives:
\begin{eqnarray*}
H_0(V^\dual,\g) &=& V,\\
H_1(V^\dual,\g) &=& V^\dual \cartan V,\\
H_k(V^\dual,\g) &=& \Lambda^k V^\dual \cartan \so(V),\\
H_{n-1}(V^\dual,\g) &=& \Lambda^{n-1} V^\dual \cartan V^\dual,\\
H_n(V^\dual,\g) &=& \Lambda^n V^\dual \tens V^\dual.
\end{eqnarray*}
Elements in the homology for $k=0,1,2$ have immediate geometric
interpretations as vectors, linearized conformal metrics and
conformal Weyl curvature tensors.
\end{example}

\begin{example}\label{Penroseshomology}
\summary{Penrose's homology} The representation on trivectors
$W = \Lambda^3{\asso V}$ in the conformal case gives for $n>4$:
\begin{eqnarray*}
H_0(V^\dual,W) &=& \Lambda^2 V^0\tens L^1,\\
H_1(V^\dual,W) &=& V^\dual \cartan \Lambda^2 V^0\tens L^1,\\
H_2(V^\dual,W) &=& \Lambda^2V^\dual \cartan \Lambda^2 V^0 \tens L^1,\\
H_k(V^\dual,W) &=& \Lambda^k V^\dual \cartan \Lambda^3 V^0,\\
H_{n-2}(V^\dual,W) &=& \Lambda^{n-2} V^\dual \cartan \Lambda^2V^0\tens L^{-1},\\
H_{n-1}(V^\dual,W) &=& \Lambda^{n-1} V^\dual \cartan \Lambda^2V^0\tens L^{-1},\\
H_n(V^\dual,\g) &=& \Lambda^n V^\dual \tens \Lambda^2V^0\tens L^{-1}.
\end{eqnarray*}
For $n=4$ the middle dimensional homology becomes:
$$
H_2(V^\dual,W) = \Lambda^2V^\dual \cartan \Lambda^2 V^0 \tens (L^1\dsum L^{-1}).
$$
\end{example}

\begin{example}\label{Spinorhomology}
\summary{Spinor homology} In the even dimensional conformal case
$n=2l$ let $\Spinor^+$ and $\Spinor^-$ be the two (weightless)
(fundamental) spinor representations for $\co(V)$ (on which the centre
acts trivially). The spinor representation of central weight $w$ will
be denoted by $\Spinor^{+,w}$ and $\Spinor^{-,w}$. Then (by
periodicity) ${\asso\Spinor}^+ := \Spinor^{+,{1\over 2}} \dsum
\Spinor^{-,-{1\over 2}}$ and ${\asso\Spinor}^- := \Spinor^{-,{1\over
2}} \dsum \Spinor^{+,-{1\over 2}}$ are the two spinor representations
for $\so({\asso V})$. For the homology we find:
$$
H_k(V^\dual,{\asso\Spinor}^+) = \Lambda^k V^\dual \cartan \Spinor^{+,{1\over 2}}. 
$$
The elements in the Cartan product $\Lambda^k V^\dual \cartan \Spinor$
are elements in the tensors product which are in the kernel of
the Clifford multiplication.
\end{example}

\begin{example}
\summary{Linearized projective geometry} The adjoint representation $W
=\g =V\dsum\gl(V)\dsum V^\dual$ in the projective case gives:
\begin{eqnarray*}
H_0(V^\dual,\g) &=& V,\\
H_1(V^\dual,\g) &=& \Sym^2 V^\dual \cartan V,\\
H_k(V^\dual,\g) &=& \Lambda^k V^\dual \cartan \sll(V),\\
H_{n-1}(V^\dual,\g) &=& \Lambda^{n-1} V^\dual \cartan V^\dual \cartan V,\\
H_n(V^\dual,\g) &=& \Lambda^n V^\dual \tens V^\dual.
\end{eqnarray*}
Elements in the Cartan product are in the kernel of the $\gl(V)$
equivariant alternation or contraction. Elements in the homology for
$k=0,1,2$ have immediate geometric interpretations as vectors,
linearized torsion-free derivatives (up to projective equivalence) and
projective Weyl curvature tensors.
\end{example}

\begin{discussion}\label{VdualcohomologywithvaluesinWdual}
\summary{$V^\dual$ cohomology with values in $W^\dual$} Secondly we
describe the $V^\dual$ cohomology with values in $W^\dual$ and here we
like to view $W^\dual$ as a left $\g$-representation. It is then
completely dual to the above homology theory: the \emph{$k$-cochains}
are defined by
$$
C^k(V^\dual,W^\dual) := \Lambda^k V \tens W^\dual = (C_k(V^\dual,W))^\dual.
$$
The left action of $X\in\p$ on $v\in V$ is given by projecting the
adjoint action onto $V$: $\Lieb{X,v}_V=X.v=\Sum_k
\ip{\theta^k,X.v}t_k= \Sum_k\ip{\theta^k.X,v}t_k =
\Sum_k\ip{\Lieb{\theta^k,X},v}t_k$. This extends as derivation on
$\Lambda^k V$, hence the left action of $X\in\p$ on $a\tens\omega\in
C^k(V^\dual,W^\dual)$ is therefore given by:
\begin{eqnarray*}
X.(a\tens\omega)
&:=& \sum_{i} (\Lieb{X,t_i}_V\wedge \theta^i \inner a)\tens\omega
   + a\tens X.\omega\\
&=& \sum_{j} t_j \wedge (\Lieb{\theta^j,X} \inner a)\tens\omega
   + a\tens X.\omega.
\end{eqnarray*}
We define the coboundary operator $d_{V^\dual}\from
C^k(V^\dual,W^\dual)\to C^{k+1}(V^\dual,W^\dual)$ to be dual to the
boundary operator $d_{V^\dual}=\delta_{V^\dual}^\dual$:
$$
d_{V^\dual}( a \tens \omega )
:= \oneover{2} \sum_i (t_i \wedge \theta^i.a) \tens \omega
   + \sum_i (t_i \wedge a) \tens \theta^i.\omega.
$$
Therefore it satisfies $d_{V^\dual}\circ d_{V^\dual}= 0$ and it is
equivariant under the action of $X\in\p$, i.e.
$\Lieb{{X.},d_{V^\dual}} = 0$. Hence the kernel, image and cohomology
of $d_{V^\dual}$ give left $\p$-representations:
\begin{eqnarray*}
Z^k(V^\dual,W^\dual)
&:=& \kernel d_{V^\dual}\from C^k(V^\dual,W^\dual)\to C^{k+1}(V^\dual,W^\dual),\\
B^k(V^\dual,W^\dual)
&:=& \image d_{V^\dual}\from C^{k-1}(V^\dual,W^\dual)\to C^k(V^\dual,W^\dual),\\
H^k(V^\dual,W^\dual) 
&:=& Z^k(V^\dual,W^\dual)/B^k(V^\dual,W^\dual).
\end{eqnarray*}
The zero cohomology $H^0(V^\dual,W^\dual)$ is also called the
\emph{space of invariants} of $W^\dual$ with respect to $V^\dual$,
since in that case $B^0(V^\dual,W^\dual)=0$ and $H^0(V^\dual,W^\dual)
= Z^0(V^\dual,W^\dual) = W^{\dual V^\dual}$ with $W^{\dual V^\dual}
=\setof{\omega\in W^\dual\property \gamma.\omega=0\ \forall\gamma\in
V^\dual}$. Again $V^\dual$ acts trivially on $H^k(V^\dual,W^\dual)$
since Cartan's identity holds for $\gamma \in V^\dual$ and $c \in
C^k(V^\dual,W^\dual)$:
$$
d_{V^\dual}( \gamma \inner c ) 
+ \gamma \inner (d_{V^\dual} c) 
= \gamma.c.
$$ 
\end{discussion}

\begin{proposition}\label{Poincareduality}
\summary{Poincar\'{e} duality} For $\dim V=n$ there is a canonical
isomorphism $\inner\from \Lambda^kV^\dual\tens \Lambda^nV\to
\Lambda^{n-k}V$ which induces isomorphisms $C_k(V^\dual,W) \tens
\Lambda^nV \isomorph C^{n-k}(V^\dual,W)$ and
\begin{eqnarray*}
H_k(V^\dual,W) \tens \Lambda^nV
&\isomorph& H^{n-k}(V^\dual,W)\\
&=& H_{n-k}(V^\dual,W^\dual)^\dual.
\end{eqnarray*}
\end{proposition}

\proof Let $v\in \Lambda^nV$ be nontrivial and $\theta\in
V^\dual$. For the action of $\theta$ on the volume element $v$ note
$v.\theta =\trace_{V^\dual}(\alpha\mapsto \alpha.\theta)v =\Sum_k
\ip{\theta^k.\theta,t_k}v =\ip{\theta,\Sum_k t_k.\theta^k}v$. Then we
find that $\inner v$ intertwines $d_{V^\dual}$ and
$\delta_{V^\dual}$:
\begin{eqnarray*}
\lefteqn{(\delta_{V^\dual}(\alpha\tens w))\inner v}&&\\
&=&\oneover{2}\sum_i( (t_i\inner\alpha).\theta^i\tens w
+ 2t_i\inner\alpha \tens w.\theta^i)\inner v\\
&=&\oneover{2}\sum_i( t_i.\theta^i\inner\alpha\tens w
+ t_i\inner\alpha.\theta^i\tens w
+ 2t_i\inner\alpha \tens w.\theta^i)\inner v\\
&=&\oneover{2}(-1)^k \sum_i
( t_i.\theta^i\wedge (\alpha\inner v)\tens w
+ t_i\wedge (\alpha.\theta^i\inner v)\tens w
+ 2t_i\wedge (\alpha\inner v) \tens w.\theta^i)\\
%&=&(-1)^k \oneover{2}\sum_i( t_i.\theta^i\wedge (\alpha\inner v)\tens w
%+ t_i\wedge (\alpha\inner v).\theta^i\tens w
%- t_i\wedge (\alpha\inner v.\theta^i)\tens w
%+ 2t_i\wedge (\alpha\inner v) \tens w.\theta^i)\\
&=&\oneover{2}(-1)^k \sum_i(
t_i\wedge (\alpha\inner v).\theta^i\tens w
+ 2t_i\wedge (\alpha\inner v) \tens w.\theta^i)\\
&=&\oneover{2}(-1)^{k+1} \sum_i(
t_i\wedge \theta^i.(\alpha\inner v)\tens w
+ 2t_i\wedge (\alpha\inner v) \tens \theta^i.w)\\
&=&(-1)^{k+1} d_{V^\dual}((\alpha\tens w)\inner v).\qed
\end{eqnarray*}

\begin{discussion}\label{VhomologywithvaluesinWdual}
\summary{$V$ homology with values in $W^\dual$} Thirdly we discuss the
$V$ homology with values in $W^\dual$ viewed as a left
$\g$-representation: the \emph{$k$-chains} are defined by
$$
C_k(V,W^\dual) := \Lambda^k V \tens W^\dual = C^k(V^\dual,W^\dual)
$$
Note, that $\p^\dual$ acts upon $V$ by the adjoint representation and
therefore $\p^\dual$ acts on $C_k(V,W^\dual)$: for $Y \in \p^\dual$
and $a\in \Lambda^k V$, $\omega\in W^\dual$:
$$
Y.(a\tens \omega) 
:= \sum_i \Lieb{Y,t_i} \wedge \theta^i \inner a \tens \omega
+ a \tens Y. \omega.
$$
The boundary operator $\delta_V\from C_k(V,W^\dual) \to
C_{k-1}(V,W^\dual)$ is defined to be
$$
\delta_V( a \tens \omega )
:= - \oneover{2} \sum_i t_i.(\theta^i \inner a) \tens \omega
   - \sum_i (\theta^i \inner a) \tens t_i.\omega.
$$
It satisfies $\delta_V\circ \delta_V = 0$ and is equivariant under the
action of $Y\in\p^\dual$, i.e.  $\Lieb{{Y.},\delta_V} = 0$.

\begin{remark}
The commutator on the spaces $C^k(V^\dual,W^\dual) =C_k(V,W^\dual)
=\Lambda^kV\tens W^\dual$ of $\delta_V$ with the left $X\in\p$ action
is given by
$$
\Lieb{X.,\delta_V}
= - \sum_{i} \Lieb{X,t_i}_\p.(\theta^i\inner\tens\id_{W^\dual}).
$$
To check this we recall that the left action of $X\in\p$ on $v\in V$
is given by projecting the adjoint action onto $V$, see paragraph
\ref{VdualcohomologywithvaluesinWdual}. The extention as derivation on
$a\in \Lambda^k V$ is then given by $X.a =\sum_i\Lieb{X,t_i}_V\wedge
\theta^i\inner a =\sum_k t_k\wedge\Lieb{\theta^k,X}\inner a$.  Notice
the following commutator rules on $\Lambda^kV$ for $t\in V$ and
$\theta\in V^\dual$:
\begin{eqnarray*}
\Lieb{X.,t\,\wedge} 
&=& \Lieb{X,t}_V \wedge\ ,\\
\Lieb{X.,\theta\inner} 
&=& \Lieb{X,\theta}\inner\ .
\end{eqnarray*}
Then we get:
\begin{eqnarray*}
\Lieb{X.,\delta_V}
&=& \Lieb{X., -\oneover{2}\sum_{i,j}
\Lieb{t_i,t_j}\wedge\theta^j\inner\theta^i\inner \tens\id_{W^\dual}
- \sum_i \theta^i \inner \tens t_i.}\\
&=& -\oneover{2}\sum_{i,j}
\Lieb{X,\Lieb{t_i,t_j}}_V\wedge\theta^j\inner\theta^i\inner \tens\id_{W^\dual}
-\oneover{2}\sum_{i,j}
\Lieb{t_i,t_j}\wedge\Lieb{X,\theta^j}\inner\theta^i\inner \tens\id_{W^\dual}\\
&&
-\oneover{2}\sum_{i,j}
\Lieb{t_i,t_j}\wedge\theta^j\inner\Lieb{X,\theta^i}\inner \tens\id_{W^\dual}
- \sum_i \Lieb{X,\theta^i} \inner \tens t_i.
- \sum_i \theta^i \inner \tens \Lieb{X,t_i}.\\
&=& -\sum_{i,j}
\Lieb{\Lieb{X,t_i},t_j}_V\wedge\theta^j\inner\theta^i\inner \tens\id_{W^\dual}
-\sum_{i,j}
\Lieb{t_i,t_j}\wedge\Lieb{X,\theta^j}\inner\theta^i\inner \tens\id_{W^\dual}\\
&&
- \sum_i \Lieb{X,\theta^i} \inner \tens t_i.
- \sum_i \theta^i \inner \tens \Lieb{X,t_i}.\\
&=& - \sum_{i,j} \Lieb{\Lieb{X,t_i}_\p,t_j}_V
\wedge\theta^j\inner\theta^i\inner \tens\id_{W^\dual}
- \sum_i \Lieb{X,\theta^i} \inner \tens t_i.
- \sum_i \theta^i \inner \tens \Lieb{X,t_i}.\\
&=& - \sum_{i} \Lieb{X,t_i}_\p.(\theta^i\inner )\tens\id_{W^\dual}
- \sum_i \Lieb{X,\theta^i} \inner \tens t_i.
- \sum_i \theta^i \inner \tens \Lieb{X,t_i}.\\
&=& - \sum_{i} \Lieb{X,t_i}_\p.(\theta^i\inner \tens\id_{W^\dual}).
\end{eqnarray*}
\end{remark}
%
%Again Cartan's identity holds for $v \in V$ and $c \in
%C_k(V,W^\dual)$:
%$$
%\delta_{V}( v \wedge c ) + v \wedge (\delta_{V} c) = v.c.
%$$
\end{discussion}
%
%\begin{discussion}
%\summary{$V$ cohomology with values in $W$} Lastly we describe the
%$V$ cohomology with values in $W$. Again $W$ is a right $\g$
%-representation. The \emph{$k$-cochains} are defined by
%$$
%C^k(V,W) := \Lambda^k V^\dual \tens W = C_k(V^\dual,W).
%$$
%Note that $\p^\dual$ acts upon $V^\dual$ from the right by projecting the Lie
%bracket into $V^\dual$: for $Y \in \p^\dual$ and $\alpha \in
%\Lambda^k V^\dual$ and $w\in W$, we have
%$$
%(\alpha\tens w).Y
%:= \sum_i \Lieb{Y,\theta^i}_{V^\dual} \wedge (t_i \inner \alpha) \tens w
%  + \alpha\tens Y.w.
%$$
%Next we define the coboundary operator $d_V\from C^k(V,W) \to
%C^{k+1}(V,W)$ to be the dual to the above boundary operator
%$d_V=\delta^\dual_V$,
%$$
%d_V( \alpha \tens w )
%:= - \oneover{2} \sum_i (\theta^i \wedge t_i.\alpha) \tens w
%     - \sum_i (\theta^i \wedge \alpha) \tens t_i.w.
%$$
%This therefore defines a complex $d_V\circ d_V = 0$, which is
%equivariant under the action of $Y\in\p^\dual$, i.e. $\Lieb{{Y.},d_V}
%= 0$. Cartan's identity holds for $v \in V$ and $c \in C^k(V,W)$:
%$$
%d_{V}( v \inner c ) + v \inner (d_{V} c) = v.c.
%$$ 
%\end{discussion}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Twisted deRham resolution}
\label{SectionTwisteddeRhamresolution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\p\subset \g$ be again a Cartan geometry with $\g=V\dsum\l\dsum
V^\dual$ and $\p=\l\dsum V^\dual$. Let $t_i$, $\theta^i$ be a dual
basis of $V$. The exterior deRham operator on differential forms is
given in terms of a $\g$-equivariant homomorphism between Verma
modules by the following:

\begin{definition}
We define a $\g$-homomorphism
$\deltadR\from
\U(\g)\tens \Lambda^kV
\to \U(\g)\tens \Lambda^{k-1}V$ generated on
$a\in\Lambda^kV$ by
\begin{eqnarray*}
\deltadR( 1\tens a )
&:=& \sum_i t_i \tens\theta^i \inner a
   + \oneover{2} \sum_{i,j} 1 \tens \Lieb{t_i,t_j} \wedge \theta^i \inner 
   \theta^j \inner a.
\end{eqnarray*}
\end{definition}

For low degrees $k=1,2$ the above definition of $\deltadR$ means
\begin{eqnarray*}
\deltadR(a) 
&=& a \tens 1,\\
\deltadR(a\wedge b) 
&=& a\tens b - b\tens a - 1 \tens \Lieb{a,b}.
\end{eqnarray*}
In general we notice that the deRham homomorphism and the boundary
operator $\delta_V$ of the $V$ homology with values in the trivial
$\R$, from paragraph\ref{VhomologywithvaluesinWdual}, are related as:
\begin{eqnarray*}
\deltadR( 1\tens a )
&=& \sum_i t_i \tens\theta^i \inner a
   - \oneover{2} \sum_{i,j} 1 \tens t_i.(\theta^i \inner a)\\
&=& \sum_i t_i \tens\theta^i \inner a + \delta_V a.
\end{eqnarray*}

\begin{proposition}\label{deRhamresolution}
\summary{deRham resolution} The homomorphism $\deltadR$ maps the
induced ideal $I(\g,\p,\Lambda^kV)$ to $I(\g,\p,\Lambda^{k-1}V)$ and
thus induces a $\g$-homomorphism
$$
\deltadR \from \Verma(\g,\p,\Lambda^kV) \to \Verma(\g,\p,\Lambda^{k-1}V).
$$ 
This leads to the deRham complex $\deltadR\circ\deltadR = 0$. In zero
degree $k=0$ we can replace the trivial $\deltadR=0$ by
$\evaluation\from\U(\g)\tens \R\to\R$ (with $\R$ as trivial
$\g$-representation) given by $1\tens 1\mapsto 1$ and still obtain a
complex:
$$
0\gets\R\stackrel{\evaluation}{\ot}
\Verma(\g,\p,\R)\stackrel{\deltadR}{\ot}\Verma(\g,\p,V)
\stackrel{\deltadR}{\ot}\Verma(\g,\p,\Lambda^2V)
\ot\ldots\quad.
$$
\end{proposition}

\proof The homomorphism $\deltadR$ maps the induced ideal
$I(\g,\p,\Lambda^kV)$ into $I(\g,\p,\Lambda^{k-1}V)$ since for the
commutator with the induced action of $X\in\p$ we find:
\begin{eqnarray*}
\Lieb{\deltadR,\ind_X}
&=& \Lieb{\sum_i r_{t_i}\tens\theta^i\inner + 1\tens \delta_V,
-r_X\tens\id_{C^k}+1\tens X.}\\
&=& \Lieb{\sum_i r_{t_i}\tens\theta^i\inner, -r_X\tens\id_{C^k}}
+ \Lieb{\sum_i r_{t_i}\tens\theta^i\inner,1\tens X.}
+ \Lieb{1\tens \delta_V,1\tens X.}\\
&=& \sum_i r_{\Lieb{t_i,X}}\tens\theta^i\inner
+ \sum_i r_{t_i}\tens \Lieb{\theta^i,X}\inner
+ 1\tens\Lieb{\delta_V,X.}\\
&=& \sum_i -r_{\Lieb{X,t_i}_\p}\tens\theta^i\inner
+ 1\tens\Lieb{X,t_i}_\p.(\theta^i\inner).\\
\end{eqnarray*}
Hence $\deltadR\from
\Verma(\g,\p,\Lambda^kV)\to\Verma(\g,\p,\Lambda^{k-1}V)$ is well
defined. For the composite we have:
\begin{eqnarray*}
\deltadR\circ \deltadR
&=& \deltadR\circ \sum_i t_i \tens  \theta^i \inner\ 
    + \oneover{2} \deltadR \circ \sum_{i,j} 1\tens
      \Lieb{t_i,t_j} \wedge \theta^i \inner \theta^j \inner \\
&=& \sum_{i,j} t_it_j \tens  \theta^j\inner \theta^i \inner\  
 + \oneover{2} \sum_{i,j,k} 
 t_i\tens \Lieb{t_j,t_k}\wedge\theta^j\inner\theta^k\inner\theta^i\inner\\
&&
 + \oneover{2} \sum_{i,j,k} t_k\tens
 \theta^k\inner\ (\Lieb{t_i,t_j}\wedge\theta^i\inner\theta^j\inner\ )\\
&&
 + \oneover{4} \sum_{i,j,k,l} 1\tens
 \Lieb{t_k,t_l}\wedge\theta^k\inner\theta^l\inner\ 
 (\Lieb{t_i,t_j}\wedge\theta^i\inner\theta^j\inner\ )\\
&=& \oneover{2} \sum_{i,j,k} 
 t_i\tens \Lieb{t_j,t_k}\wedge\theta^j\inner\theta^k\inner\theta^i\inner\ 
 - \oneover{2} \sum_{i,j,k} t_k\tens
 \Lieb{t_i,t_j}\wedge\theta^k\inner\theta^i\inner\theta^j\inner\\
&&
 + \oneover{4} \sum_{i,j,k,l} 1\tens
 \Lieb{t_k,\Lieb{t_i,t_j}}\wedge\theta^k\inner\theta^i\inner\theta^j\inner\\
&&
 - \oneover{4} \sum_{i,j,k,l} 1\tens
 \Lieb{t_k,t_l}\wedge\theta^k\inner\ 
 (\Lieb{t_i,t_j}\wedge\theta^l\inner\theta^i\inner\theta^j\inner\ )\\
&=& - \oneover{4} \sum_{i,j,k,l} 1\tens
 \Lieb{t_k,t_l}\wedge\Lieb{t_i,t_j}\wedge
 \theta^k\inner\theta^l\inner\theta^i\inner\theta^j\inner
= 0.
\end{eqnarray*}
For the last statement note $\evaluation(\deltadR(1\tens a))
=\evaluation(a\tens 1) = a.1 = 0$.\qed

\begin{remark}
If $G$ is a Lie group with Lie algebra $\g$ and $P:=\setof{g\in
G\property \Ad_g(\p)=\p}$ the (closed) sub group with Lie algebra
$\p$, then the above homomorphism $\deltadR$ induces the exterior
derivative as differential operator on $G/P$: indeed the Verma module
homomorphism $\deltadR$ corresponds to a differential operator like
$$
\dR \from C^\infty(G,\Lambda^{k-1}V^\dual)^P \to
C^\infty(G,\Lambda^kV^\dual)^P,
$$
which on a $(k-1)$-form $F\in C^\infty(G,\Lambda^{k-1}V^\dual)^P$ can
be evaluated on a $k$-multivector $a\in \Lambda^{k}V$ at $g\in G$ to
give
\begin{eqnarray*}
\ip{(\dR F)(g),a} 
&=& \ip{{\jet^\infty(F)(g)},{\deltadR(1\tens_{\U(\p)}a)}}\\
&=& \ip{{\jet^\infty(F)(g)},
\sum_i t_i \tens_{\U(\p)} \theta^i \inner a
+ \oneover{2} \sum_{i,j} 1 \tens_{\U(\p)} 
\Lieb{t_i,t_j} \wedge \theta^i \inner \theta^j \inner a}\\
&=& \sum_i R_{t_i} \ip{\theta^i\wedge F,a}(g)
+\oneover{2}\sum_{i,j}
\ip{\theta^j\wedge\theta^i\wedge\Lieb{t_i,t_j}\inner F,a}(g).\\
\end{eqnarray*}
\end{remark}

\begin{remark}
A complex of homomorphisms is called a \emph{resolution}, if it is
exact. From local exactness of the deRham complex, the Poincar\'{e}
lemma, we deduce with the above remark that the deRham homomorphisms
of proposition \ref{deRhamresolution} form a resolution of $\R$.
\end{remark}
 
\begin{discussion}\label{Hopfmap}
\summary{Hopf map} The universal enveloping algebra $\U(\g)$ of a Lie
algebra $\g$ is an associative algebra, hence $\U(\g)\tens\U(\g)$
carries an obvious algebra structure. Moreover there is a coproduct,
the \emph{Hopf map}:
$$
\coproduct\from \U(\g)\to\U(\g)\tens\U(\g),
$$
defined to be an algebra homomorphism induced by
$$
1\mapsto 1\tens 1\ \hbox{and}\ 
X\mapsto X\tens 1 + 1\tens X,
$$
for $X\in\g$. Similarly if $W_1^\dual$ and $W_2^\dual$ are left
$\g$-representations then their tensor product $W_1^\dual\tens
W_2^\dual$ is a left $\U(\g)\tens\U(\g)$ module, hence $\U(\g)$ acts
upon $W_1^\dual\tens W_2^\dual$ through $\coproduct$.
\end{discussion}

\begin{proposition}
\summary{Co-Leibniz rule} The wedge coproduct $\cowedge^{kl} \from
\Lambda^{k+l}V\to \Lambda^kV \tens \Lambda^lV$ defined on
$c\in\Lambda^{k+l}V$ with the help of $\alpha\in\Lambda^kV^\dual$ and
$\beta\in\Lambda^lV^\dual$ by:
$$
\ip{{\cowedge^{kl}}\,c,{\alpha\tens \beta}}
:= \ip{c,\alpha\wedge\beta},
$$
is $\p$-equivariant. This induces $\g$-equivariant pairings
$$
\cowedge^{kl}\from
\U(\g)\tens\Lambda^{k+l}V \to 
(\U(\g)\tens \Lambda^kV) \tens (\U(\g)\tens \Lambda^lV),\quad
\cowedge^{kl} (U\tens c) := \coproduct U \tens \cowedge^{kl} c,
$$
and
$$
\cowedge^{kl}\from
\Verma(\g,\p,\Lambda^{k+l}V)\to
\Verma(\g,\p,\Lambda^kV) \tens \Verma(\g,\p,\Lambda^lV).
$$
This pairing is compatible with the deRham homomorphisms in the sense 
that the following co-Leibniz rule holds:
$$
\cowedge^{kl} \circ \deltadR 
= (\deltadR \tens \id) \circ \cowedge^{k+1,l} 
+(-1)^k (\id \tens \deltadR) \circ \cowedge^{k,l+1}.
$$ 
\end{proposition}

\proof The $\p$-equivariance of $\cowedge^{kl}$ is clear. To verify the 
Co-Leibniz rule note that the associativity of the wedge product with 
$\theta\in V^\dual$ means
$\cowedge^{kl}\circ\theta\inner
=(\theta\inner\tens\id)\circ\cowedge^{k+1,l}
=(-1)^k(\id\tens\theta\inner)\circ\cowedge^{k,l+1}$.
Similar the interior multiplication by $t\in V$ is a derivation hence
$\cowedge^{kl}\circ t\,\wedge
=(t\,\wedge\tens\id)\circ\cowedge^{k-1,l}
+ (-1)^k(\id\tens t\,\wedge)\circ\cowedge^{k,l-1}$. With this in hands 
we find:
\begin{eqnarray*}
\cowedge^{kl} \circ \deltadR
&=&\Sum_i (t_i\tens 1 + 1\tens t_i)\tens \cowedge^{kl}\circ\theta^i\inner
+ \oneover{2}\sum_{i,j} (1\tens 1) \tens 
\cowedge^{kl}\circ\Lieb{t_i,t_j}\wedge\theta^i\inner\theta^j\inner\\
&=&\Sum_i (t_i\tens 1)\tens (\theta^i\inner\tens\id)\circ\cowedge^{k+1,l}
+ (-1)^k\Sum_i(1\tens t_i) \tens
(\id\tens\theta^i\inner)\circ\cowedge^{k,l+1}\\
&&
+ \oneover{2}\sum_{i,j} (1\tens 1) \tens 
(\Lieb{t_i,t_j}\wedge\theta^i\inner\theta^j\inner\tens\id)\circ
\cowedge^{k+1,l}\\
&&
+\oneover{2} (-1)^k \sum_{i,j} (1\tens 1) \tens 
(\id\tens\Lieb{t_i,t_j}\wedge\theta^i\inner\theta^j\inner)
\circ\cowedge^{k,l+1}\\
&=& (\deltadR \tens \id) \circ \cowedge^{k+1,l} 
+(-1)^k(\id\tens\deltadR)\circ\cowedge^{k,l+1}.\qed
\end{eqnarray*}

In what follows we will twist the deRham homomorphism and the wedge
coproduct by a left $\g$-representation $W^\dual$. This is easily
done after appreciating the following proposition: if $W^\dual$ is a
left $\g$-representation, then $\g$ acts on $(\U(\g)\tens E^\dual)
\tens W^\dual$ by the product rule. Also $W^\dual$ is a left
$\p$-representation by restriction and therefore $(E^\dual \tens
W^\dual)$ is a left $\p$-representation.

\begin{proposition}
There is an isomorphism between $\g$-representations:
$$
\Psi\from\U(\g)\tens (E^\dual\tens W^\dual)
\stackrel{\isomorph}{\longrightarrow}
(\U(\g) \tens E^\dual) \tens W^\dual,
$$
induced by
$1\tens(\eta\tens\omega)\mapsto(1\tens\eta)\tens\omega$. For $X\in\g$
this means $\Psi(X\tens(\eta\tens\omega))= (X\tens\eta)\tens\omega
+(1\tens\eta)\tens X.\omega$ and hence
$\Psi^{-1}((X\tens\eta)\tens\omega) =X\tens (\eta\tens\omega) - 1\tens
(\eta\tens X.\omega)$. This isomorphism maps $I(\g,\p,(E^\dual\tens
W^\dual))$ onto $I(\g,\p,E^\dual)\tens W^\dual$ and hence descends to
an isomorphism
$$
\Psi\from\Verma(\g,\p,E^\dual\tens W^\dual)
\stackrel{\isomorph}{\longrightarrow}
\Verma(\g,\p,E^\dual) \tens W^\dual.
$$
\end{proposition}

\begin{definition}
\summary{Twisted deRham homomorphism} If $W^\dual$ is a left
$\g$-representation, then $\deltadR\tens \id_{W^\dual}$ leads with
the above isomorphism to the so called \emph{twisted deRham
homomorphism} between the Verma modules induced by
$C^k(V^\dual,W^\dual):=\Lambda^kV\tens W^\dual$:
$$
\delta:=\Psi^{-1}\circ(\deltadR\tens\id_{W^\dual})\circ\Psi
\ \from\ \U(\g)\tens (\Lambda^kV\tens W^\dual) \to \U(\g)\tens
(\Lambda^{k-1}V\tens W^\dual).
$$
\end{definition}

\begin{proposition}
The twisted deRham homomorphism is induced by the $\g$-homomorphism
defined on $a\tens\omega\in \Lambda^kV\tens W^\dual$ by
$$
\delta( a \tens \omega ) 
= \sum_i t_i \tens \theta^i \inner a \tens \omega
+ 1 \tens \delta_V (a \tens \omega),
$$
where $\delta_V$ is the boundary operator of paragraph
\ref{VhomologywithvaluesinWdual}.  The ideal $I(\g,\p,(\Lambda^kV\tens
W^\dual))$ is mapped into $I(\g,\p,(\Lambda^{k-1}V\tens W^\dual))$
hence we have and induced map
$$
\delta\from\Verma(\g,\p,C^k(V^\dual,W^\dual)) \to
\Verma(\g,\p,C^{k-1}(V^\dual,W^\dual)).
$$
\end{proposition}

\proof
We simply have to unravel the definitions:
\begin{eqnarray*}
\delta(a\tens\omega) 
&=& \Psi^{-1}\circ (\deltadR\tens\id_{W^\dual})\circ \Psi(a\tens\omega)\\
&=& \Psi^{-1}\circ (\deltadR\tens\id_{W^\dual})(a \tens\omega)\\
&=& \Psi^{-1}(\deltadR a \tens\omega)\\
&=& \Psi^{-1}(\sum_i t_i\tens\theta^i\inner a\tens\omega
+ \oneover{2} \sum_{i,j} 1 \tens 
\Lieb{t_i,t_j}\wedge\theta^i\inner\theta^j\inner a\tens\omega)\\
&=& \sum_i (t_i\tens\theta^i\inner a\tens\omega
- 1\tens\theta^i\inner a\tens t_i.\omega)
+ \oneover{2} \sum_{i,j} 1 \tens 
\Lieb{t_i,t_j}\wedge\theta^i\inner\theta^j\inner a\tens\omega\\
&=&\sum_i t_i \tens \theta^i \inner a \tens \omega
+ 1 \tens \delta_V (a \tens \omega).
\end{eqnarray*}
For the commutator with the induced action of $X\in\p$ we have:
$$
\Lieb{\delta,\ind_X}
= \sum_i -r_{\Lieb{X,t_i}_\p}\tens\theta^i\inner
+ 1\tens\Lieb{X,t_i}_\p.(\theta^i\inner).\qed
$$

\begin{remark}
Note that $C^k(V^\dual,W^\dual)=C_k(V,W^\dual)$ and that in this
formula the $\p^\dual$-equi\-var\-iant boundary operator $\delta_V$
of the $V$ homology with values in $W^\dual$ occurs as zero order
part: the twisted deRham homomorphism can therefore be viewed as a
$\p$-equivariant extension of $\delta_V\from C^k(V^\dual,W^\dual)\to
C^{k-1}(V^\dual,W^\dual)$ to $\delta\from C^k(V^\dual,W^\dual)\to
\Verma(\g,\p,C^{k-1}(V^\dual,W^\dual))$
\end{remark}

\begin{proposition}\label{TwisteddeRhamresolution}
\summary{Twisted deRham resolution}
The twisted deRham homomorphisms define a complex
$\delta\circ\delta=0$. In degree zero $k=0$ we can replace the trivial
homomorphism $\delta=0$ by the $\g$-equivariant evaluation 
$\evaluation\from\Verma(W^\dual)\to W^\dual$ generated by
$1\tens_{\U(\p)}\omega \mapsto \omega$ and still obtain a complex:
$$
0\gets W^\dual\stackrel{\evaluation}{\ot}
\Verma(\g,\p,W^\dual)\stackrel{\delta}{\ot}
\Verma(\g,\p,V\tens W^\dual)\stackrel{\delta}{\ot}
\Verma(\g,\p,\Lambda^2V\tens W^\dual) \ot\ldots\quad.
$$
\end{proposition}

\begin{proposition}\label{TwistedCoLeibnizrule}
\summary{Twisted Co-Leibniz rule} Let $W^\dual_1$, $W^\dual_2$ and
$W^\dual_3$ be three $\g$-rep\-re\-senta\-tions and
$F^\dual\from W^\dual_3\to W^\dual_1\tens W^\dual_2$ be a (non
trivial) $\g$-equivariant homomorphism. Then $F^\dual$ together with
the wedge coproduct $\cowedge^{kl} \from \Lambda^{k+l}V\to
\Lambda^kV \tens \Lambda^lV$ defines a $\p$-equivariant coproduct on
the level of the cochains
$\cowedge^{kl}\tens F^\dual \from
C^{k+l}(V^\dual,W^\dual_3) \to C^k(V^\dual,W^\dual_1) \tens
C^l(V^\dual,W^\dual_2)$ and thus induces a $\g$-equivariant pairing:
$$
\cowedge^{kl}\from 
\Verma(\g,\p,C^{k+l}(V^\dual,W^\dual_3))
\to \Verma(\g,\p,C^k(V^\dual,W^\dual_1))
\tens \Verma(\g,\p,C^l(V^\dual,W^\dual_2)).
$$
For $k=l=0$ this coproduct satisfies
$$
F^\dual\circ \evaluation
= (\evaluation\tens\evaluation) \circ \cowedge^{00},
$$
and for general $k,l$ the following Co-Leibniz rule holds for the
twisted deRham operators:
$$
(\cowedge^{kl}\tens F^\dual) \circ \delta 
= (\delta \tens \id) \circ (\cowedge^{k+1,l}\tens F^\dual) 
  +(-1)^k (\id \tens \delta) \circ (\cowedge^{k,l+1}\tens F^\dual).
$$ 
\end{proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Projection on Verma cochains}
\label{ProjectiononVermacochains}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\par From now on we assume that the left $\g$-representation $W^\dual$ is a
finite dimensional representation. In that case Kostant's Hodge theory
provides a splitting of $C^k(V^\dual,W^\dual) :=\Lambda^kV\tens
W^\dual$ (the chains of the $V^\dual$ cohomology with values in
$W^\dual$) into a direct sum, where one summand is isomorphic to
$H^k(V^\dual,W^\dual)$. This splitting is not $\p$-invariant, but only
$\l$-invariant. The basic ingredient is the so called quabla operator
defined by the following anti commutator:
$$
{\Quabla_\l}
:= d_{V^\dual} \circ \delta_V + \delta_V \circ d_{V^\dual}\quad
\from\quad C^k(V^\dual,W^\dual)\to C^k(V^\dual,W^\dual).
$$
Since $d_{V^\dual}$ is $\p$-equivariant and $\delta_V$ is $\p^\dual$
equivariant, the quabla operator is only $\l=\p\cap\p^\dual$
equivariant. Clearly quabla commutes with $d_{V^\dual}$, in particular
quabla maps the image $B^k(V^\dual,W^\dual):= \image(d_{V^\dual})$ to
itself.

\begin{theorem}\label{Kostantsinversion}
\summary{Kostant's inversion} For a finite dimensional $\g$-module
$W^\dual$, Kostant's qua\-bla operator ${\Quabla_\l}=d_{V^\dual} \delta_V
+ \delta_V d_{V^\dual}$ is invertible on the image of the coboundary
operator $\image(d_{V^\dual})=B^k(V^\dual,W^\dual)$.
\end{theorem}

\proof In \cite{Kostant} Kostant demonstrates that $\Quabla_\l$ is
diagonalizable. He calculates the eigenvalues of $\Quabla_\l$ which
are nonzero on the image of $d_{V^\dual}$. \qed

\begin{corollary}
The above proposition allows to define a homomorphism like
$$
Q_\l := {\Quabla_\l}^{-1}\circ d_{V^\dual}\ 
\from\ C^{k-1}(V^\dual,W^\dual)\to C^k(V^\dual,W^\dual),
$$
which maps into $B^k(V^\dual,W^\dual)$. Furthermore we define a map
$$
S_\l = \id - \delta_V \circ Q_\l - Q_\l \circ \delta_V\ 
\from\ C^k(V^\dual,W^\dual)\to C^k(V^\dual,W^\dual),
$$
which satisfies the following properties:
\begin{eqnarray*}
d_{V^\dual} \circ S_\l &=& 0,\\
S_\l \circ d_{V^\dual} &=& 0,\\
\project\circ S \circ \represent &=& \id \from
H^{k}(V^\dual,W^\dual) \to
H^{k}(V^\dual,W^\dual),
\end{eqnarray*}
where $\project\from Z^k(V^\dual,W^\dual)\to H^k(V^\dual,W^\dual)$
denotes the $\p$-equivariant projection from the kernel of
$d_{V^\dual}$ to cohomology and $\represent$ means the choice of a
representative of the cohomology class. Furthermore we have
\begin{eqnarray*}
S_\l\circ S_\l &=& S_\l,\\
\delta_V \circ S_\l &=& S_\l \circ \delta_V.
\end{eqnarray*}
\end{corollary}

\proof The properties have simple verifications:
\begin{eqnarray*}
d_{V^\dual}  S_\l
&=& d_{V^\dual} - d_{V^\dual}\delta_V{\Quabla_\l}^{-1}d_{V^\dual}
= d_{V^\dual} - {\Quabla_\l}{{\Quabla_\l}}^{-1}d_{V^\dual}
= 0,\\
S_\l d_{V^\dual}
&=& d_{V^\dual} - {\Quabla_\l}^{-1}d_{V^\dual}\delta_Vd_{V^\dual}
= d_{V^\dual} - {\Quabla_\l}^{-1}{\Quabla_\l} d_{V^\dual}
= 0.
\end{eqnarray*}
For $z\in Z^k(V^\dual,W^\dual)$ (i.e.~$d_{V^\dual} z = 0$) we have
$\project(S_\l(z))= \project(z -\delta_VQ_\l(z)-Q_\l\delta_V(z))
=\project(z)$, since $Q_\l(z)={\Quabla_\l}^{-1}d_{V^\dual}z=0$ and
$\image(Q_\l)\subset\image(d_{V^\dual})$. For the projection property
note:
\begin{eqnarray*}
Q_\l\delta_V Q_l
&=& {\Quabla_\l}^{-1}d_{V^\dual}\delta_V{\Quabla_\l}^{-1}d_{V^\dual}
= {\Quabla_\l}^{-1}{\Quabla_\l}{\Quabla_\l}^{-1}d_{V^\dual}
= Q_\l,
\end{eqnarray*}
which gives:
\begin{eqnarray*}
S_\l S_\l
&=&(\id - \delta_V  Q_\l - Q_\l \delta_V)^2\\
&=&\id - \delta_V  Q_\l - Q_\l \delta_V
- \delta_V Q_\l + (\delta_V Q_\l)^2 
- \delta_V Q_\l Q_\l \delta_V
- Q_\l \delta_V + Q_\l\delta_V\delta_V Q_\l
+ (Q_\l \delta_V)^2\\
&=&S_\l,\\
\delta_V S_\l
&=& \delta_V - \delta_V Q_\l \delta_V
= S_\l\delta_V.\qed
\end{eqnarray*}

\begin{corollary}
The projection $S_\l$ provides a splitting
$$
C^k(V^\dual,W^\dual)
=\Lambda^kV\tens W^\dual 
= \kernel S_\l\dsum\image S_\l,
$$
with $\image(S_\l)\subset\kernel(d_{V^\dual})$ and the natural
projection $\project\from\image(S_\l) \to H^k(V^\dual,W^\dual)$ into
cohomology is an isomorphism.
\end{corollary}

The next aim is to modify Kostant's $\l$-equivariant projection from
cochains to cohomology into a $\g$-equivariant projection from the
Verma module induced by the cochains to the Verma module induced by
cohomology. In analogy we will construct a map
$$
S\from\Verma(\g,\p,C^k(V^\dual,W^\dual))
 \to\Verma(\g,\p,C^k(V^\dual,W^\dual)),
$$ 
with the following properties: $S$ maps to the kernel of the
coboundary operator $d_{V^\dual}$, it vanishes on the image of
$d_{V^\dual}$, it is an extension of the canonical projection from the
kernel of $d_{V^\dual}$ onto the cohomology of $d_{V^\dual}$, and it
commutes with the deRham homomorphism $\delta$.

\begin{remark}
Recall that given a linear map between $\p$-representations
$\phi_0\from F^\dual\to E^\dual$ we have an induced $\g$-homomorphism
$\phi:=\id\tens\phi_0\from\U(\g)\tens F^\dual\to\U(\g)\tens
E^\dual$. This descends to the corresponding Verma modules iff
$\phi_0$ is $\p$-equivariant. Therefore the boundary operator
$d_{V^\dual}\from C^k(V^\dual,W^\dual)\to C^{k+1}(V^\dual,W^\dual)$
gives a $\g$-homomorphism between Verma modules, whereas the
coboundary operator $\delta_V\from C^k(V^\dual,W^\dual)\to
C^{k-1}(V^\dual,W^\dual)$ does not since $\delta_V$ is only
$\l$-equivariant. Therefore Kostant's quabla operator
${\Quabla_\l}\from C^k(V^\dual,W^\dual) \to C^k(V^\dual,W^\dual)$
induces a $\g$-equivariant homomorphism:
${\Quabla_\l}\from\U(\g)\tens C^k(V^\dual,W^\dual) \to \U(\g)\tens
C^k(V^\dual,W^\dual)$, but this homomorphism does not descend to the
Verma module.
\end{remark}

\begin{definition}
The anti commutator of the twisted deRham homomorphism $\delta$ and
the coboundary operator $d_{V^\dual}$ defines a quabla operator
$$
\Quabla := d_{V^\dual} \circ \delta + \delta \circ d_{V^\dual}\quad
\from\quad \U(\g)\tens C^k(V^\dual,W^\dual)
\to \U(\g)\tens C^k(V^\dual,W^\dual),
$$
which descends to the corresponding Verma modules. Clearly $\Quabla$
commutes with $d_{V^\dual}$, so it maps the image of $d_{V^\dual}$ to
itself:
$$
\Quabla\from
\U(\g)\tens B^k(V^\dual,W^\dual)
\to \U(\g)\tens B^k(V^\dual,W^\dual).
$$
\end{definition}

\begin{theorem}
The difference between the two quabla operators
$A_\l:=\Quabla-{\Quabla_\l}$
commutes with $d_{V^\dual}$, so also $A_\l$ maps $\U(\g)\tens
B^k(V^\dual,W^\dual)$ to itself.  $A_\l$ is given by the action of
$V^\dual$ on cochains as:
$$
A_\l
= \Quabla - {\Quabla_\l}
= \sum_i r_{t_i} \tens \theta^i.
$$
For a finite dimensional $W^\dual$ the composite
$({\Quabla_\l}^{-1}\circ A_\l)^m(1\tens c)$ vanishes for a large
$m\in\N$ depending on $c\in B^k(V^\dual,W^\dual)$ and ${\Quabla}^{-1}$
defined by the following Neumann series
$$
{\Quabla}^{-1} 
:= \Sum_{m=0} (-1)^m ({\Quabla_\l}^{-1}\circ A_\l)^m \circ {\Quabla_\l}^{-1},
$$
gives a two sided inverse of $\Quabla$ which leaves the ideal
$I(\g,\p,B^k(V^\dual,W^\dual))$ invariant and hence descends to an
endomorphism of the corresponding Verma module
$\Verma(\g,\p,B^k(V^\dual,W^\dual))$.
\end{theorem}

\proof We will use the symbol $\anticom{,}$ for the anti commutator of
two maps. The difference $A_\l$ is given by the following:
\begin{eqnarray*}
A_\l
&=& \anticom{d_{V^\dual},\delta} - \anticom{d_{V^\dual},\delta_V}\\
&=& \anticom{d_{V^\dual}, (\delta-\delta_V)}\\
&=& \Sum_i t_i\tens 
   \anticom{d_{V^\dual},(\theta^i\inner \tens\id_{W^\dual})}\\
&=& \sum_i t_i\tens \theta^i.\quad,
\end{eqnarray*}
where we used Cartan's identity in the last equality. The action of
$V^\dual$ on $C^k(V^\dual,W^\dual)$ lowers the $\Centre(\l)$ weight in
$C^k(V^\dual,W^\dual)$. Now ${\Quabla_\l}^{-1}$ restricted to
$B^k(V^\dual,W^\dual)=\image d_{V\dual}$ is $\l$-equivariant, and
hence when applied leaves the weight invariant. Consequently
${\Quabla_\l}^{-1}\circ A_\l$ also lowers the weight. Since
$\Lambda^kV\tens W^\dual$ is a finite sum of lowest weight
modules it follows that $({\Quabla_\l}^{-1}\circ A_\l)^m(1\tens c)$
vanishes for large enough $m\in\N$ for all $c\in
B^k(V^\dual,W^\dual)$. Obviously ${\Quabla}^{-1}$ maps $\U(\g)\tens
B^k(V^\dual,W^\dual)$ to itself (since so do ${\Quabla_\l}^{-1}$ and
$A_\l$). Then its clear that restricted to $\U(\g)\tens
B^k(V^\dual,W^\dual)$ we find $\Quabla^{-1} \circ \Quabla=\id$ and
$\Quabla \circ \Quabla^{-1}=\id$. We recall the commutators between
the relevant operators and the induced $\p$-action:
\begin{eqnarray*}
\Lieb{d_{V^\dual},\ind_X}
&=& 0,\\
\Lieb{\delta,\ind_X}
&=& \sum_i (\ind-l)_{\Lieb{X,t_i}_\p}\circ (1\tens\theta^i\inner),\\
\Lieb{\Quabla,\ind_X}
&=&\Lieb{\anticom{d_{V^\dual},\delta},\ind_X}
= \anticom{d_{V^\dual},\Lieb{\delta,\ind_X}}\\
&=& \anticom{d_{V^\dual},
\sum_i (\ind-l)_{\Lieb{X,t_i}_\p}\circ(1\tens\theta^i\inner)}\\
&=& \sum_i (\ind-l)_{\Lieb{X,t_i}_\p}\circ
(1\tens \anticom{d_{V^\dual},\theta^i\inner})\\
&=& \sum_i (\ind-l)_{\Lieb{X,t_i}_\p}(1\tens\theta^i.),\\
\Lieb{\Quabla^{-1},\ind_X}
&=& -\Quabla^{-1}\Lieb{\Quabla,\ind_X}\Quabla^{-1}. 
\end{eqnarray*} From the last two equations we deduce:
\begin{eqnarray*}
\Quabla^{-1}(\ind-l)_X
&=& (\ind-l)_X\Quabla^{-1}
+ \sum_i \Quabla^{-1}(\ind-l)_{\Lieb{t_i,X}_\p}
(1\tens\theta^i.)\Quabla^{-1}.
\end{eqnarray*}
We now iterate this equation to obtain:
\begin{eqnarray*}
\lefteqn{\Quabla^{-1}(\ind-l)_X}&&\\
&=& (\ind-l)_X\Quabla^{-1}\\
&&+ \sum_{m=1}^{M-1} \ \sum_{i_1,\ldots, i_m}
(\ind-l)_{\Lieb{t_{i_m},\ldots\Lieb{t_{i_1},X}_\p\ldots}_\p}
\Quabla^{-1}(1\tens\theta^{i_m}.)\ldots
\Quabla^{-1}(1\tens\theta^{i_1}.)
\Quabla^{-1}\\
&&+\sum_{i_1,\ldots, i_M}
\Quabla^{-1}
(\ind-l)_{\Lieb{t_{i_M},\ldots\Lieb{t_{i_1},X}_\p\ldots}_\p}
(1\tens\theta^{i_M}.)\Quabla^{-1}\ldots
(1\tens\theta^{i_1}.)\Quabla^{-1}.
\end{eqnarray*}
For a large $M$ the last sum is zero, since $\p$ has a highest
$\Centre(\l)$ weight with respect to the adjoint $V$ action. Hence the 
commutator with the induced $\p$-action is:
$$
\Lieb{\Quabla^{-1},\ind_X}
= \sum_{m=1}\ \sum_{i_1,\ldots, i_m}
(\ind-l)_{\Lieb{t_{i_m},\ldots\Lieb{t_{i_1},X}_\p\ldots}_\p}
\Quabla^{-1}(1\tens\theta^{i_m}.)\ldots
\Quabla^{-1}(1\tens\theta^{i_1}.)
\Quabla^{-1}\ ,
$$
which maps into the ideal $I(\g,\p,B^k(V^\dual,W^\dual))$.\qed

\begin{corollary}\label{Projectionmap}
\summary{Projection map} From above we have a well defined
$\g$-equivariant homomorphism
$$
Q:={\Quabla}^{-1} \circ d_{V^\dual}
\from \Verma(\g,\p,C^{k-1}(V^\dual,W^\dual))
\to \Verma(\g,\p,C^{k}(V^\dual,W^\dual)).
$$
The homomorphism $S$ from
$\Verma(\g,\p,C^k(V^\dual,W^\dual))$ to itself defined by
$$
S := \id - \delta \circ Q - Q \circ \delta,
$$
satisfies the following properties:
\begin{eqnarray*}
d_{V^\dual} \circ S &=& 0,\\
S \circ d_{V^\dual} &=& 0,\\
\project\circ S\circ \represent &=& \id\ \hbox{on\ }
\Verma(\g,\p,H^{k}(V^\dual,W^\dual)),
\end{eqnarray*}
where $\project\from Z^k(V^\dual,W^\dual)\to H^k(V^\dual,W^\dual)$
denotes the $\p$-equivariant projection from the kernel of
$d_{V^\dual}$ to cohomology and $\represent$ means the choice of a
representative of the cohomology class. Furthermore we have
\begin{eqnarray*}
S\circ S &=& S,\\
\delta \circ S &=& S\circ \delta,\\
\evaluation\circ S&=&\evaluation\ \hbox{in degree $k=0$}.
\end{eqnarray*}
\end{corollary}
\proof The calculations are the same as in the presented version
of Kostant's Hodge theory for $S_\l$. For $d_{V^\dual} z=0$ we know
$Sz=z-Q\circ\delta(z)$ and $Q$ maps into
$\Verma(\g,\p,B^k(V^\dual,W^\dual))$, hence
$\project(Sz)=\project(z)$. For $k=0$ note $S=\id-\delta\circ Q$ and
$\evaluation\circ\delta=0$, hence $\evaluation\circ
S=\evaluation$. \qed

\begin{corollary}
The projection $S$ provides a splitting
$$
\Verma(\g,\p,C^k(V^\dual,W^\dual))
= \kernel S\dsum\image S,
$$
with $\image(S)\subset \Verma(\g,\p,Z^k(V^\dual,W^\dual))$ and the
natural projection
$$
\project\from \image(S) \to
\Verma(\g,\p,H^k(V^\dual,W^\dual))
$$ is an isomorphism onto cohomology.
\end{corollary}

\begin{remark}
The projection $S$ can be viewed as a chain map from the twisted deRham
complex to itself, since $\delta\circ S=S\circ \delta$. From the
definition of $S$ the map $Q$ is a chain homotopy between $S$ and
$\id$ such that the induced map of $S$ in the twisted deRham homology
is the identity.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bernstein Gelfand Gelfand resolution}
\label{BernsteinGelfandGelfandresolution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Again $W^\dual$ is a finite dimensional module for a semisimple $\g$
and $\p\subset \g$ is a parabolic subalgebra with complement
$V\dsum\p=\g$, see definition \ref{Parabolicgeometry}. In this context
we have constructed a projection $S\from \Verma(\g,\p,\Lambda^kV\tens
W^\dual)\to \Verma(\g,\p,\Lambda^kV\tens W^\dual)$, see corollary
\ref{Projectionmap}, between the Verma modules, see definition
\ref{InducedmodulesorVermamodules}, induced by the twisted forms
$\Lambda^kV\tens W^\dual$. On these forms $C^k(V^\dual,W^\dual)
:=\Lambda^kV\tens W^\dual$ acts the $\p$-equivariant coboundary
operator $d_{V^\dual}$, with cohomology spaces denoted by
$H^k(V^\dual,W^\dual)$, see paragraph
\ref{VdualcohomologywithvaluesinWdual}. The two additional properties
$d_{V^\dual}\circ S=0$ and $S\circ d_{V^\dual}=0$ allow to define two
$\p$-equivariant homomorphisms
\begin{eqnarray*}
\project\circ S &\from& \Verma(\g,\p,C^k(V^\dual,W^\dual))
\to \Verma(\g,\p,H^k(V^\dual,W^\dual)), \hbox{\ and}\\
S\circ\represent &\from& \Verma(\g,\p,H^k(V^\dual,W^\dual))
\to \Verma(\g,\p,C^k(V^\dual,W^\dual)),
\end{eqnarray*}
where $\project\from Z^k(V^\dual,W^\dual)\to H^k(V^\dual,W^\dual)$
denotes the $\p$-equivariant projection from the kernel of
$d_{V^\dual}$ to cohomology and $\represent$ means the choice of a
representative of the cohomology class.

In the dual geometric picture {\v Cap, Slov\'ak and Sou\v cek} in
\cite{CapSlovakSoucekIV} constructed the homomorphism $\project\circ S$ even in
curved rather then homogeneous parabolic geometry. Their construction
of this map is given by an inductive process. Our construction
involves a Neumann series. Moreover to construct bilinear pairings we
need the full $S$ rather then just $\project\circ S$.

\begin{definition}\label{BGGhomomorphisms}
\summary{Bernstein Gelfand Gelfand homomorphisms} From the twisted
deRham resolution, see proposition \ref{TwisteddeRhamresolution},
$$
0\ot
W^\dual \stackrel{\evaluation}{\ot}
\Verma(\g,\p,W^\dual) \stackrel{\delta}{\ot}
\Verma(\g,\p,V\tens W^\dual) \stackrel{\delta}{\ot}
\Verma(\g,\p,\Lambda^2V\tens W^\dual) \stackrel{\delta}{\ot}\ldots,
$$
we can now define $\g$-homomorphisms between the induced Verma modules
corresponding to the cohomology spaces:
$$
\delta_H 
:= \project \circ S \circ \delta \circ S \circ \represent
\from
\Verma(\g,\p,H^k(V^\dual,W^\dual))
\to \Verma(\g,\p,H^{k-1}(V^\dual,W^\dual)).
$$
The zero cohomology $H^0(V^\dual,W^\dual)={W^\dual}^{V^\dual}$ is the
space of invariants in $W^\dual$ with respect to $V^\dual$. This is a
$\p$-invariant subspace, so that the inclusion
$H^0(V^\dual,W^\dual)\inclusion W^\dual$ is $\p$-equivariant, inducing
a $\g$-equivariant homomorphism:
$$
\evaluation_H\from \Verma(\g,\p,H^0(V^\dual,W^\dual))\to W^\dual,
$$
which could also be viewed as
$\evaluation_H=\evaluation\circ\represent$.
\end{definition}

\begin{theorem}\label{ParabolicBGGresolution}
\summary{Parabolic Bernstein Gelfand Gelfand resolution} The resulting
sequence
$$
0\ot
W^\dual \stackrel{\evaluation_H}{\longleftarrow}
\Verma(\g,\p,H^0(V^\dual,W^\dual)) \stackrel{\delta_H}{\ot}
\Verma(\g,\p,H^1(V^\dual,W^\dual)) \stackrel{\delta_H}{\ot}
\ldots,
$$
defines a complex $\delta_H\circ\delta_H=0$,
$\evaluation_H\circ\delta_H =0$ and $\evaluation_H$ is surjective. The
two homomorphisms $S\circ\represent$ and $\project\circ S$ are chain
maps going back and forth between the twisted deRham complex and the
Bernstein Gelfand Gelfand complex (on $W^\dual$ the chain map is given
by the identity). These chain maps induce an isomorphism between the
homologies of the two complexes. Since twisted deRham is a resolution,
so is the BGG complex.
\end{theorem}

\proof Unraveling the definitions and using $S^2=S$,
$\Lieb{S,\delta}=0$, $\evaluation\circ S=\evaluation$ and
$\evaluation\circ\delta=0$ gives:
\begin{eqnarray*}
\delta_H\circ\delta_H
&=& (\project\circ S\circ\delta\circ S\circ\represent)\circ
(\project\circ S\circ\delta\circ S\circ\represent)\\
&=& (\project\circ S\circ\delta\circ S\circ\delta\circ S\circ\represent)\\
&=& (\project\circ S\circ\delta\circ\delta\circ S\circ\represent)
=0,\\
\evaluation_H\circ \delta_H
&=& (\evaluation\circ\represent)\circ
(\project\circ S\circ\delta\circ S\circ\represent)\\
&=& \evaluation\circ S\circ\delta\circ S\circ\represent\\
&=& \evaluation\circ \delta\circ S\circ\represent
=0.
\end{eqnarray*}
The evaluation $\evaluation\from\Verma(\g,\p,W^\dual)\to W^\dual$ is
surjective and $\evaluation_H\circ\project\circ S=\evaluation$, hence
so is $\evaluation_H$. The two homomorphisms $S\circ\represent$ and
$\project\circ S$ are chain maps since
\begin{eqnarray*}
\delta_H\circ(\project\circ S)
&=&(\project\circ S\circ\delta\circ S\circ\represent)\circ(\project\circ S)\\
&=&\project\circ S\circ\delta\circ S\\
&=&(\project\circ S)\circ\delta,\\
\evaluation_H\circ(\project\circ S)
&=&\id\circ \evaluation,\\
(S\circ\represent)\circ\delta_H
&=& (S\circ\represent)\circ(\project\circ S\circ\delta\circ
S\circ\represent)\\
&=& S\circ \delta\circ S\circ\represent\\
&=& \delta\circ (S\circ\represent),\\
\id\circ \evaluation_H
&=& \evaluation\circ(S\circ\represent).
\end{eqnarray*}
These chain maps induce an isomorphism between the homologies of the
two complexes, since $(\project\circ S)\circ(S\circ\represent)
=\project\circ S\circ\represent=\id$.\qed

\begin{conjecture}
The complex constructed above is equivalent to the resolution of
$W^\dual$ in terms of Verma modules, which was originally constructed
by Verma \cite{Verma} and Bernstein Gelfand Gelfand
\cite{BGGStructure} in the case where $\p\subset\g$ is a Borel
subalgebra and later was generalized by Lepowsky \cite{LepowskyBGG} to
the case where $\p$ is a parabolic subalgebra.
\end{conjecture}

\begin{theorem}\label{CoLeibnizrule}
\summary{Co-Leibniz rule} Let $W^\dual_1$, $W^\dual_2$ and $W^\dual_3$
be three $\g$-representations which are finite sums of lowest weight
modules. Let $F^\dual\from W^\dual_3\to W^\dual_1\tens W^\dual_2$ be a
(nontrivial) $\g$-equivariant homomorphism. Then $F^\dual$ together
with the wedge coproduct $\cowedge^{kl} \from \Lambda^{k+l}V\to
\Lambda^kV \tens \Lambda^lV$ and the projection $S$ define a
$\g$-equivariant coproduct on the Verma modules from cohomology
$$
\cowedge_H^{kl}
\from
\Verma(\g,\p,H^{k+l}(V^\dual,W^\dual_3))
\to \Verma(\g,\p,H^k(V^\dual,W^\dual_1)) 
\tens \Verma(\g,\p,H^l(V^\dual,W^\dual_2)),
$$
given by
$$
\cowedge_H^{kl}
:= (\project\circ S \tens \project\circ S)
\circ(\cowedge^{kl}\tens F^\dual)
\circ (S\circ\represent).
$$
This is an extension of $F^\dual$ in the sense that for $k=l=0$ this
coproduct satisfies
$$
F^\dual\circ \evaluation_H
= (\evaluation_H\tens\evaluation_H) \circ \cowedge_H^{00}.
$$
For general $k$, $l$ the coproduct satisfies the following Co-Leibniz
rule with the Bernstein Gelfand Gelfand homomorphisms:
$$
\cowedge_H^{kl} \circ \delta_H 
= (\delta_H \tens \id) \circ \cowedge_H^{k+1,l} 
  +(-1)^k (\id \tens \delta_H) \circ \cowedge_H^{k,l+1}.
$$ 
\end{theorem}

\proof The result demonstrates the flexibility of using $S$, since it
follows directly from the Co-Leibniz rule of the twisted deRham
complex:
\begin{eqnarray*}
(\evaluation_H\tens\evaluation_H) \circ \cowedge_H^{00}
&=& (\evaluation\tens\evaluation)\circ(S\tens S)\circ
    (\cowedge^{00}\tens F^\dual)\circ(S\circ\represent)\\
&=& (\evaluation\tens\evaluation)\circ(\cowedge^{00}\tens F^\dual)\circ
(S\circ\represent)\\
&=& F^\dual\circ \evaluation\circ (S\circ\represent)\\
&=& F^\dual\circ \evaluation\circ \represent\\
&=& F^\dual\circ \evaluation_H.
\end{eqnarray*}
Similarly:
\begin{eqnarray*}
\lefteqn{
(\delta_H \tens \id)
\cowedge_H^{k+1,l} 
+(-1)^k (\id \tens \delta_H)
\cowedge_H^{k,l+1}}&&\\
&=& (\project S\delta S\,\represent\tens\id)
(\project S\tens\project S)\cowedge^{k+1,l}(S\,\represent)\\
&&
+(-1)^k (\id\tens \project S\delta S\,\represent)
(\project S\tens\project S)\cowedge^{k,l+1}(S\,\represent)\\
&=& (\project S\tens\project S)(\delta \tens\id)
\cowedge^{k+1,l}(S\,\represent)\\
&&
+(-1)^k (\project S\tens\project S)(\id\tens\delta )
\cowedge^{k,l+1}(S\,\represent)\\
&=& (\project S\tens\project S)
\cowedge^{k,l}\delta
(S\,\represent)\\
&=& (\project S\tens\project S)
\cowedge^{k,l}S\,\represent\, \project S\delta
(S\,\represent)\\
&=& \cowedge_H^{k,l}\delta_H.\qed
\end{eqnarray*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Adjoint homomorphisms}
\label{Adjointhomomorphisms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For a left $\p$-representation $E^\dual$ we have a $\p$-equivariant
map $\Lambda^nV\to E^\dual \tens (E\tens \Lambda^nV)$ induced by the
identity: if $e_m$, $\eta^m$ is a dual basis of $E$ then $\Lambda^nV
\ni a\mapsto \Sum_m \eta^m \tens (e_m\tens a)$. The coproduct
$\coproduct\from \U(\g)\to \U(\g)\tens \U(\g)$ then induces a
coproduct
$$
\coproduct\tens\id_{E\tens\Lambda^nV}\from
\U(\g)\tens \Lambda^nV\to
(\U(\g)\tens E^\dual) \tens (\U(\g)\tens (E\tens \Lambda^nV)),
$$
which descends to the corresponding Verma modules.

\begin{definition}
Let $E^\dual$, $F^\dual$ be left $\p$-representations and
$$
\phi\from \U(\g)\tens F^\dual \to \U(\g)\tens E^\dual
$$ be a $\g$-homomorphism. A $\g$-homomorphism
$$
\phi^\dual\from \U(\g)\tens (E\tens \Lambda^nV)
\to \U(\g)\tens (F\tens \Lambda^nV)
$$
is called an \emph{adjoint to $\phi$} if there is a $\g$-equivariant
coproduct
$$
X_\phi\from
\U(\g)\tens \Lambda^{n-1}V\to
(\U(\g)\tens E^\dual) \tens (\U(\g)\tens (F\tens \Lambda^nV)),
$$
such that with the deRham homomorphism
$$
\deltadR
\from \U(\g)\tens \Lambda^nV
\to \U(\g)\tens \Lambda^{n-1}V
$$
the following codivergence formula holds:
$$
X_\phi\circ\deltadR
= (\phi\tens\id)\circ(\coproduct\tens\id_{F\tens\Lambda^nV})
-(\id\tens\phi^\dual)\circ(\coproduct\tens\id_{E\tens\Lambda^nV}).
$$
A similar definitions holds for $\g$-homomorphisms descending to the
corresponding Verma modules.
\end{definition}

\begin{remark}
In view of the definition of adjoint differential operators in
paragraph \ref{Adjoints} we remark that adjoint Verma module
homomorphisms have corresponding differential operators on the
homogeneous space $G/P$ which are adjoint after one of them is twisted
by the bundle of pseudoscalars $L^{-n}\tens\Lambda^nV$. The Lie
algebra $\p$ acts trivially on $L^{-n}\tens\Lambda^nV$, so we used
$\Lambda^nV$ in the above algebraic definition rather then $L^n$.
\end{remark}

\begin{example}
A \emph{derivative} $D\from \U(\g)\tens (E^\dual\tens V)\to
\U(\g)\tens E^\dual$ is defined to be generated by
$$
D
:= \Sum_i r_{t_i}\tens\id_{E^\dual}\tens\ip{\theta^i,\ }.
$$
It has an adjoint homomorphism $D^\dual\from \U(\g)\tens
(E\tens\Lambda^nV)\to \U(\g)\tens (E\tens V^\dual\tens \Lambda^nV)$
given by
$$
D^\dual
:= -\Sum_i r_{t_i}\tens (\id_E\tens\theta^i\tens \id_{\Lambda^nV})
-\Sum_i 1\tens(\id_E\tens\theta^i\tens t_i\wedge \delta_V).
$$
The coproduct $X_D$ is generated by the wedge product
$\Lambda^{n-1}V\mapsto (E^\dual)\tens (E\tens V^\dual\tens
\Lambda^nV)$, as $X_D :=\coproduct\circ \sum_i 1\tens (\id_E \tens
\theta^i\tens t_i\wedge)$. For the composite $X_D\circ\deltadR$ note
$t\wedge(\theta\inner)=\theta(t)$ on $\Lambda^nV$ and
$\coproduct\circ r_t =(r_t\tens 1)\coproduct + (1\tens
r_t)\coproduct$, hence:
\begin{eqnarray*}
X_D\circ\deltadR
&=& \Sum_{i,j}
\coproduct\circ
(1\tens \id_E\tens\theta^i\tens t_i\wedge)\circ
(r_{t_j}\tens\theta^j\inner + \delta_V)\\
&=& \Sum_{i}
(r_{t_i}\tens 1)\coproduct(\id_{E}\tens\theta^i\tens\id_{\Lambda^nV})
+ (1\tens r_{t_i})\coproduct(\id_{E}\tens\theta^i\tens\id_{\Lambda^nV})\\
&&
+ \Sum_{i}(1\tens \theta^i\tens t_i\wedge \delta_V)\coproduct\id_E\\
&=& (D\tens 1)\coproduct\tens\id_{E\tens V^\dual\tens\Lambda^nV} 
- (1\tens D^\dual)\coproduct\tens\id_{E\tens\Lambda^nV}.
\end{eqnarray*}
\end{example}

\begin{example}
\summary{Adjoints of deRham homomorphisms} Notice once more the
canonical isomorphism $\inner\from\Lambda^kV^\dual\tens \Lambda^nV \to
\Lambda^{n-k}V$. Consequently, for the identity
$\id_{\Lambda^kV^\dual\tens\Lambda^nV}$ we have with the cowedge
product: $\inner\circ \id_{\Lambda^kV^\dual\tens\Lambda^nV} =
\cowedge^{k,n-k} \from \Lambda^nV\to \Lambda^kV\tens
\Lambda^{n-k}V$. The Co-Leibniz rule shows that the adjoint of the
deRham homomorphism
\begin{eqnarray*}
\deltadR &\from&
\Verma(\g,\p,\Lambda^kV)\to \Verma(\g,\p,\Lambda^{k-1}V)\\
{\deltadR}^\dual&\from&
\Verma(\g,\p,\Lambda^{k-1}V^\dual\tens\Lambda^nV)
\to \Verma(\g,\p,\Lambda^kV^\dual\tens\Lambda^nV)
\end{eqnarray*}
satisfies $\inner{\deltadR}^\dual = (-1)^k\deltadR\circ\inner$ with the
deRham homomorphism acting on degree $(n-k+1)$ since
\begin{eqnarray*}
\cowedge^{k-1,n-k}\circ\deltadR
&=& (\deltadR \tens \id) \circ \cowedge^{k,n-k} 
  +(-1)^{k-1} (\id \tens \deltadR) \circ \cowedge^{k-1,n-k+1}.
\end{eqnarray*}
\end{example}

\begin{example}
For the twisted deRham homomorphism we have $F^\dual=\Lambda^kV\tens
W^\dual$ and $F\tens\Lambda^nV=\Lambda^{n-k}\tens W$. The
twisted Co-Leibniz rule applied to the three $\g$-representations
$\R\to W^\dual\tens W$ shows that $\delta$ on $\Lambda^kV\tens
W^\dual$ is adjoint to $(-1)^k\delta$ on $\Lambda^{n-k}V\tens W$.
\end{example}

\begin{example}
\summary{Adjoints of Bernstein Gelfand Gelfand homomorphisms}
Poincar\'{e} duality applied to $F^\dual=H^k(V^\dual,W^\dual)$ gives
$F\tens\Lambda^nV=H^{n-k}(V^\dual,W)$. The Co-Leibniz rule applied to
the three $\g$-representations $\R\to W^\dual\tens W$ shows that
$\delta_H$ on $H^k(V^\dual,W^\dual)$ is adjoint to $(-1)^k\delta_H$ on
$H^{n-k}(V^\dual,W)$.
\end{example}

\begin{remark}
Let $G^\dual$ be another left $\p$-representations and
$$
\psi\from \U(\g)\tens G^\dual\to \U(\g) \tens F^\dual
$$
another $\g$-homomorphism with adjoint
$$
\psi^\dual\from \U(\g)\tens (F\tens \Lambda^nV)
\to \U(\g)\tens (G\tens \Lambda^nV),
$$
then $\phi\circ\psi$ has an adjoint given by
$\psi^\dual\circ\phi^\dual$ with the coproduct
$$
X_{\phi\psi}\from
\U(\g)\tens \Lambda^{n-1}V\to
(\U(\g)\tens E^\dual) \tens (\U(\g)\tens (G\tens \Lambda^nV)),
$$
given by
$$
X_{\phi\psi}
:= (\phi\tens\id)\circ X_\psi 
+ (\id\tens \psi^\dual)\circ X_\phi.
$$
\end{remark}

\begin{remark}
The above remark indicates that the projection
$$
S\from\Verma(\g,\p,C^{k}(V^\dual,W^\dual)) \to
\Verma(\g,\p,C^{k}(V^\dual,W^\dual)),
$$
as composite of operators which all have adjoints, also has an
adjoint. A candidate for such an adjoint is the projection $S$ induced
by $W$ in complementary degree:
$$
S\from\Verma(\g,\p,C^{n-k}(V^\dual,W)) \to
\Verma(\g,\p,C^{n-k}(V^\dual,W)).
$$
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Appendix: Elementary representation theory}
\label{AppendixElementaryrepresentationtheory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this appendix we summarize basic notions of the representations 
theory of Lie algebras. It is taken from \cite{FultonHarris}.

\begin{definition}
\summary{Lie algebras} Let $\F$ be a field (we are interested in the
cases $\F=\R$ and $\F=\C$). A finite dimensional vector space $\g$
over $\F$ with a skew symmetric bilinear multiplication $\g \tens \g
\to \g$, denoted on $X,Y \in \g$ by $X \tens Y\mapsto \Lieb{X,Y}$ is
called a \emph{Lie algebra}, if it satisfies the \emph{Jacobi
identity}: for all $X,Y,Z \in \g$ we have $\Lieb{X,\Lieb{Y,Z}} +
\Lieb{Y,\Lieb{Z,X}} + \Lieb{Z,\Lieb{X,Y}} = 0$.
\end{definition}

If $\p \subset \g$ is a subspace, such that $X,Y \in \p$ implies
$\Lieb{X,Y} \in \p$, then $\p$ is called a \emph{subalgebra} of
$\g$. A linear map $\phi \from \g \to \lie{h}$ between two Lie
algebras is called a \emph{Lie algebra homomorphism}, if
$\phi(\Lieb{X,Y})=\Lieb{\phi(X),\phi(Y)}$ for all $X,Y \in \g$ (this
condition is nonlinear in $\phi$ similar to the orthogonality
condition for a linear transformation).

\begin{example}
The space of endomorphisms $\End(V)=\gl(V)$ of a $n$-dimensional
vector space $V$ over the field $\F$ carries a natural Lie algebra
structure by $\Lieb{A,B} := A \circ B - B \circ A$ and is called the
\emph{general linear Lie algebra}. It has dimension $\dim\gl(V)=n^2$
\end{example}

Lie algebras which are isomorphic to subalgebras of some
$\gl(V)$ are called \emph{linear Lie algebras}. Lie algebras with
trivial Lie bracket are called \emph{Abelian Lie algebras}. One
dimensional Lie algebras are necessarily Abelian.

\begin{definition}
\summary{Modules} If $\g$ denotes a Lie algebra over $\F$, a vector
space $W$ (finite or infinite dimensional) is said to carry a
\emph{right $\g$-module structure} or $W$ is called a \emph{right
$\g$-representation space}, if there is a linear map $\rep\from
W\tens\g\to W$ denoted by $\rep(w\tens X)=:w.X$ such that for $Y\in
\g$ we have $w.\Lieb{X,Y} = w.X.Y-w.Y.X$.
\end{definition}

A $\g$-representation from the left $\rep\from\g\tens W\to W$ will be
denoted by $\rep(X\tens w)=X.w$ and satisfies $\Lieb{X,Y}.w = X.Y.w -
Y.X.w$. Every left action has an associated right action defined by
$w.X=-X.w$ and vice versa. If $W$ is a right module then $W^\dual$ is
a left module defined (without signs) by $\ip{X.\omega,w}
=\ip{\omega,w.X}$ for $\omega\in W^\dual$ and all $w\in W$.

\begin{example}
Let $V$ be a vector space over $\R$ or $\C$ and $\g\subset\gl(V)$ a
Lie sub algebra. Clearly, $V$ is a $\g$-representation, also its dual
$V^\dual$ and all tensor products $L^k\tens V^i \tens {V^\dual}^j$
including densities $L^k$ for $i,j\in\N$ and $k\in\R$.
\end{example}

\begin{example}
\summary{Adjoint representation} For any Lie algebra $\g$ the Jacobi
identity shows, that the Lie bracket itself induces a left
representation on $\g$ itself, called \emph{adjoint representation}
$\ad\from\g\tens \g \to \g$, $\ad(X)Y:=\Lieb{X,Y}$. The kernel of the
adjoint representation is given by the centre $\kernel\ad =
\Centre(\g)$. The \emph{coadjoint representation} $\coad \from
\g\tens\g^\dual\to\g^\dual$ is given by: $\coad := -\ad^\dual$.
\end{example}

\begin{definition}
\summary{Enveloping algebra} Let $\g$ be a Lie algebra over the field
$\F$. An \emph{enveloping algebra} of $\g$ is a pair $(\U,i)$
consisting of an associative algebra $\U$ over $\F$ with $1$ and a
linear map $i\from\g\to\U$ satisfying $i(\Lieb{X,Y}) =
i(X)i(Y)-i(Y)i(X)$ for all $X,Y \in \g$.
\end{definition}

The \emph{universal enveloping algebra} $\U(\g)$ for $\g$ satisfies
the following property: for any enveloping algebra $\mathfrak{A},j$
there exists a unique homomorphism of algebras
$\phi\from\U(\g)\to\mathfrak{A}$ (sending $1$ to $1$) such that $\phi\circ
i = j$. The universal enveloping algebra $\U(\g)$ can be realized as
$\U(\g):=T(\g)/I$, where $T(\g)$ is the graded tensor algebra and $I$
is the two sided $T(\g)$ ideal generated by $\generatedby{{X \tens Y -
Y \tens X - \Lieb{X,Y}}\property{X,Y \in \g}}$. This ideal respects
the filtration of $T(\g)$ which induces a filtration of the quotient
$\U(\g)=T(\g)/I$. If $W^\dual$ is a left $\g$-representation, then
$\U(\g)$ acts as algebra on $W^\dual$: for $XY\in\U(\g)$ we define
$(XY).\omega:=X.Y.\omega$ with $X,Y\in\g$ and $\omega\in W^\dual$.

\begin{definition}
\summary{Invariant submodules} A subspace $W'\subset W$ of a
$\g$-representation $\rep \from \g \to \gl(W)$ is called
$\g$-invariant, if $w' \in W'$ and $X \in \g$ implies $\rep(X)(w')
\in W'$. A vector $w \in W$ is called $\g$-fix or $\g$-invariant, if
for all $X \in \g$ we have $\rep(X)(w)=0$. The set of $\g$-invariant
vectors forms a $\g$-invariant subspace denoted by $W^{\g} \subset W$.
\end{definition}

\begin{application}
\summary{Linear Lie algebras defined by invariant tensors}
Let $V$ be a vector space over $\F$ and $T\in V^i \tens {V^\dual}^j$ a 
nonzero tensor. Then $\gl(V)$ acts on $T$ as an element of the 
representation space $V^i \tens {V^\dual}^j$, hence $T$ induces a linear 
map $\gl(V)\to V^i \tens {V^\dual}^j$. Define the kernel of that map to 
be $\g(V,T):=\setof{A\in\gl(V)\property T.A=0}$. This clearly 
defines a Lie subalgebra. The representation space $V$ is then called 
the \emph{defining representation} $\g(V,T)\to\gl(V)$.
\end{application}

Let $V$ be a $n$-dimensional vector space over $\F$. The following Lie
algebras defined by invariant tensors are called \emph{classical Lie
algebras}:

\begin{definition}
\summary{Special linear Lie algebra} Let $\alpha\in\Lambda^nV^\dual$
denote a nonzero volume form, then define $\sll(V,\alpha):=\setof{
A\in\gl(V)\property 0=\alpha.A=\alpha(A\ ,\ldots,) +\ldots +
\alpha(\ldots,A\ ) = \trace(A) \alpha}$. This space of trace-free
endomorphisms is clearly independent of the choice of $\alpha$ hence
$\sll(V):=\sll(V,\alpha)$ is called the \emph{special linear Lie
algebra} and has dimension $\dim\sll(V)=n^2-1$.
\end{definition}

\begin{definition}
\summary{Symplectic Lie algebra} If $n$ is even and
$\omega\in\Lambda^2 V^\dual$ denotes a nondegenerated bilinear skew
form then $\sp(V,\omega):=\setof{ A\in\gl(V)\property\omega.A =
\omega(A\ ,\ )+\omega(\ ,A\ ) = 0}$ is called \emph{symplectic Lie
algebra} and has dimension $\dim\sp(V,\omega)=n(n+1)/2$. Obviously,
for $n=2$ we have $\sll(V)=\sp(V)$.
\end{definition}

\begin{definition}
\summary{Orthogonal Lie algebra} If $g\in\Sym^2 V^\dual$ denotes a
nondegenerate bilinear symmetric form then $\so(V,g)
:=\setof{A\in\gl(V)\property g.A=g(A\ ,\ )+g(\ ,A\ )=0}$ is called the
\emph{orthogonal Lie algebra} and has dimension
$\dim\so(V,g)=n(n-1)/2$.
\end{definition}

\begin{remark}
The special linear group $\sll(n)$ is $(n^2-1)$-dimensional. On $\sll(n)$ 
the following symmetric bilinear form $g$ is invariant and non
degenerated: $g(A,B):=\trace A\circ B$. Hence the adjoint representation 
induces a map $\ad\from\sll(n)\to\so(\sll(n),g)$ which defines an 
isomorphism for $n=2$ when $\sll(2)$ is three dimensional.
\end{remark}

\begin{remark}
Let $V$ be a $n=2l$ even dimensional vector space with nondegenerated 
skew bilinear form $\omega\in\Lambda^2 V^\dual$. The space of trace-free 
bivectors (with respect to $\omega$) $\Lambda^2_0V$ forms a $((n(n-1)-2)/2)$-
dimensional space, which carries a nondegenerated symmetric bilinear 
form $g$ induced by:
$$
g(v_1\wedge v_2,w_1\wedge w_2)
   := \omega(v_1,w_1)\omega(v_2,w_2)
      -\omega(v_1,w_2)\omega(v_2,w_1).$$
The defining representation $V$ of $\sp(n)$ clearly leaves $g$-invariant. 
For $n=4$ the space $\Lambda^2_0V$ is five dimensional and this defines 
an isomorphism $\sp(V,\omega)=\so(\Lambda^2_0V,g)$.
\end{remark}

\begin{remark}
Let $V$ be a $n=2l$ even dimensional vector space with nonzero volume
form $\alpha\in\Lambda^nV^\dual$. The wedge product induces a non
degenerated symmetric bilinear form $g$ on $\Lambda^l V$:
$g(a,b):=\alpha(a\wedge b)$ which is invariant under $\sll(V)$. The
defining representation induces a map
$\sll(V)\to\so(\Lambda^lV,g)$. This is an isomorphism for $n=4$ when
$\dim\Lambda^lV=6$.
\end{remark}

\begin{definitions}
\summary{Automorphisms and derivations}
A linear isomorphism $\phi\from\g\to\g$ of a Lie algebra 
$\g$ such that $\phi(\Lieb{X,Y}) = \Lieb{\phi(X),\phi(Y)}$ for all $X,Y \in 
\g$ is called an \emph{automorphism}. The set of all automorphisms 
builds a group $\Aut(\g) \subset \GL(\g)$. A linear map $\delta 
\from \g \to \g$ is called a \emph{derivation}, if 
$\delta(\Lieb{X,Y})=\Lieb{\delta(X),Y}+\Lieb{X,\delta(Y)}$ for all $X,Y \in \g$ 
(this is the linearized automorphism condition). The set of all 
derivations builds a Lie algebra $\der(\g) \subset \gl(\g)$. 
The kernel of a derivation defines a subalgebra.
\end{definitions}

\begin{definition}
\summary{Ideals}
A sub space $I \subset \g$ of a Lie algebra $\g$ over the field 
$\F$ is called an \emph{ideal} of $\g$, if $X \in \g$ and $Y 
\in I$ implie $\Lieb{X,Y} \in I$. The kernel of a Lie algebra homomorphism is 
an ideal. The \emph{centre} of $\g$ is defined to be 
$\Centre(\g):=\setof{Z \in \g\property \Lieb{X,Z}=0, X \in \g}$ and 
defines a natural ideal. The \emph{derived algebra} $\Lieb{\g,\g} := 
\setof{\Lieb{X,Y} \property X,Y \in \g} \subset \g$ is another 
example of an ideal. If $I \subset \g$ is an ideal then the quotient 
space $\g/I$ inherits a natural Lie algebra structure.
\end{definition}

\begin{definition}
\summary{Simple Lie algebras and ideals} 
A non Abelian Lie algebra $\g$ is called \emph{simple}, if
$\setof{0}$ and $\g$ are the only ideals of $\g$. An ideal
$I \subset \g$ of a Lie algebra is called \emph{simple} if
$\setof{0}$ and $I$ are the only sub ideals of $\g$ in $I$.
\end{definition}

\begin{proposition}
Among the complex classical Lie algebras the following are simple:
$\sll(V)$ for $n\ge 2$, $\so(V)$ for $n=3$ and $n\ge 5$, $\sp(V)$ for
$n\ge 2$.
\end{proposition}

\begin{definition}
A finite dimensional $\g$-representation $\g\tens W \to W$ is called
\emph{semisimple}, if for any $\g$-invariant subspace $W'\subset W$
there is a $\g$-invariant complement in $W$. A Lie algebra $\g$ is
called \emph{semisimple}, if any finite dimensional representation is
semisimple.
\end{definition}

\begin{definition}
\summary{Trace form} If $\rep \from \g \to \gl(W)$ denotes a
finite dimensional $\g$-rep\-re\-senta\-tion, then $\rep$ induces a
bilinear symmetric form on $\g$, the so called \emph{trace form}
$Tr_{\rep} \in \Sym^2(\g^\dual)$, defined by $Tr_{\rep}(X,Y):=
\trace( \rep(X) \circ \rep(Y) )$.
\end{definition}

The trace form $Tr_{\rep}$ is an example of a $\g$-invariant vector
under the representation $\Sym^2(-\ad^\dual) \from \g \to
\Sym^2(\g^\dual)$. The \emph{radical} of $T_{\rep}$ is the space
$\setof{X \in \g \property Tr_{\rep}(X,Y)=0, Y\in\g}$ and defines an
ideal in $\g$. If the radical is zero, then the trace from is non
degenerated, i.e.~it defines an inner product on $W$ and the
representation takes values in $\rep \from \g \to \so(W,Tr_{\rep})$.

\begin{definition}
\summary{Killing form} The trace from $Tr_{\ad}$ corresponding to the
adjoint representation $\ad \from \g \to \gl(\g)$ is called
the \emph{Killing form}.
\end{definition}

\begin{examples}
Let $V$ be a $n$-dimensional vector space over $\F$. The Killing form
for $\gl(V)$ is given by $Tr_{\ad}(A,B)=(2n)\trace(A\circ
B)-2\trace(A)\trace(B)$. For the (complex) classical Lie algebras, the
Killing form is a multiple of the trace form of the defining
representation, the later is nondegenerate: for $\sll(n)$ we get
$Tr_{\ad}(A,B)=(2n)\trace(A\circ B)$, for $\so(n)$ we get
$Tr_{\ad}(A,B)=(n-2)\trace(A\circ B)$ and for $\sp(n)$ we get
$Tr_{\ad}(A,B)=(n+2)\trace(A\circ B)$. This shows, that the Killing
form for most of the classical Lie algebras is nondegenerated.
\end{examples}

\begin{theorem}
A Lie algebra is semisimple iff its Killing form $Tr_{\ad} \in 
\Sym^2(\g^\dual)$ is nondegenerated. Hence a simple Lie algebra is 
semisimple.
\end{theorem}

\begin{definition}
\summary{Reductive Lie algebra} Let $\g$ be a Lie algebra. $\g$ is called 
\emph{reductive}, if the derived algebra $\Lieb{\g,\g}$ is semisimple and 
we have a decomposition $\g=\Centre(\g) \dsum \Lieb{\g,\g}$.
\end{definition}
%
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\end{document}
\bye
