New Developments at the Interface Between

Geometry and Physics

University of Bath, 19-20 December 2016

This workshop is organised in conjunction with the LMS South Wales and South Wales Regional Meeting on 20 December.

Schedule (note late rearrangement of the talks by Pugh and Skinner)

Monday 19 December
from 12.30	Registration and coffee
13.20	Jeffrey Giansiracusa (Swansea) Tropical geometry and algebra over idempotent semirings
14.30	Bobby Acharya (ICTP & KCL) Particle physics and special holonomy
15.30	Coffee
16.00	David Skinner (Cambridge) Ambitwistor strings and the scattering equations
Tuesday 20 December
10.00	Katrin Wendland (Freiburg) On a special K3 theory
11.00	Coffee
11.30	Mathew Pugh (Cardiff) Braided subfactors associated to modular invariant partition functions
12.30	Lunch

All talks take place in the Wolfson Lecture Theatre, room 1.7 in building 4 West. Refreshments are served in the atrium on level 1 in 4 West. The programme continues with the LMS Meeting itself starting at 14.15.

Organiser: Johannes Nordström

Abstracts

• Bobby Acharya   Particle physics and special holonomy

  String theory and M theory predicts the existence of extra dimensions of space which are often modelled by manifolds with specific geometric structures. The geometry and topology of the extra dimensions can determine important properties of physics. In particular, G2-holonomy spaces provide a framework within which all of what is currently understood about particle physics (the Standard Model of Particles plus dark matter) could have a geometric description. Special kinds of singularity and submanifold play a crucial role in the picture, which I will try and sketch during this talk.

• Jeffrey Giansiracusa   Tropical geometry and algebra over idempotent semirings

  Tropical geometry is a tool that can reduce problems in algebraic geometry to piecewise polyhedral geometry and combinatorics, but it is also a new world of geometry in its own right. In this talk I will introduce the emerging picture of this kind of geometry as parallel to Grothendieck's vision of algebraic geometry. Here rings are replaced by idempotent semirings, and role of linear algebra in classical commutative algebra is replaced by the combinatorics of matroids.

• Mathew Pugh   Braided subfactors associated to modular invariant partition functions

  The modular invariant partition functions for SU(2) and SU(3) conformal field theories have been classified. The SU(2) theory is closely related to the preprojective algebras of Coxeter-Dynkin quivers. The corresponding finite dimensional superpotential algebras associated to the SU(3) invariants will be discussed. These algebras are finite dimensional analogues of Calabi-Yau algebras of dimension 3.

• David Skinner   Ambitwistor strings and the scattering equations

  Ambitwistor space is the space of complex null geodesics in a complex Riemannian manifold (M,g). It is naturally a holomorphic contact manifold, and a theorem of LeBrun states that (M, [g]) can be recovered from knowledge of this contact structure. However, the problem of how to encode the Einstein equations on ambitwistor space remains unsolved. I will explain a new perspective on this problem coming from the physics of perturbative scattering amplitudes and new, chiral string theories.

• Katrin Wendland   On a special K3 theory

  K3 theories comprise a type of conformal field theories that directly link geometry and physics. This manifests itself in the description of the moduli space of such theories, which makes immediate use of techniques borrowed from geometry. In the talk, we will recall the structure of this moduli space. We will then single out a special K3 theory, which realizes one of the largest possible symmetry groups among K3 theories. We will summarize a description of this theory by means of lattice techniques. This also allows us to apply "reflection", a procedure recently developed in joint work with Anne Taormina, which transforms certain superconformal field theories into holomorphic super vertex operator algebras and their admissible modules.