# Connections in Geometry and Physics 2010

Conference Date:
Friday, May 7, 2010 (All day) to Sunday, May 9, 2010 (All day)

GAP 2010 will focus on the following three main research areas: mathematical relativity, gauge theory, and mirror symmetry. Although these research areas all originated in physics, they have now also become important fields of study in mathematics. The aim of the workshop is to present some of the exciting new developments in these areas, as well as to expose local area graduate students and postdocs to these ideas and to introduce them to geometers and physicists in the region.

 Spyros Alexakis, University of Toronto Vincent Bouchard, University of Alberta Jim Bryan, University of British Columbia Freddy Cachazo, Perimeter Institute Emre Coskun, University of Western Ontario Magdalena Czubak, University of Toronto Norman Do, McGill University Charles Doran, University of Alberta Joel Kamnitzer, University of Toronto Melissa Liu, Columbia University David Morrison, University of California at Santa Barbara Nikita Nekrasov, Institut des Hautes Etudes Scientifiques Andrew Neitzke, University of Texas at Austin Harvey Reall, University of Cambridge Abhijnan Rej, Fields Institute Natalia Saulina, Perimeter Institute David Skinner, Perimeter Institute Mu-Tao Wang, Columbia University Shing-Tung Yau, Harvard University

Spyros Alexakis, University of Toronto

A black hole uniqueness theorem

I will discuss recent joint work with A. Ionescu and S. Klainerman on the black hole uniqueness problem. A classical result of Hawking (building on earlier work of Carter and Robinson) asserts that any vacuum, stationary black hole exterior region must be isometric to the Kerr exterior, under the restrictive assumption that the space-time metric shouldbe analytic in the entire exterior region. We prove that Hawking's theorem remains valid without the assumption of analyticity, for black hole exteriors which are apriori assumed to be "close" to the Kerr exterior solution in a very precise sense. Our method of proof relies on certain geometric Carleman-type estimates for the wave operator. Time permitting, some more recent developments will also be surveyed.

Vincent Bouchard, University of Alberta

Topological recursion and mirror symmetry

The topological recursion of Eynard and Orantin has found many applications in various areas of mathematics. In this talk I will discuss the recursion from the point of view of Hurwitz numbers and local mirror symmetry. I will explain the mathematics underlying the recursion, its relation with the cut-and-join equation, and explore first steps towards proving (and understanding geometrically) the appearance of the recursion in local mirror symmetry.

Jim Bryan, University of British Columbia

Motivic degree zero Donaldson-Thomas invariants

The Hilbert scheme X[n] of n points on variety X parameterizes length n, zero dimensional subschemes of X. When X is a smooth surface, X[n] is also smooth and a beautiful formula for its motive was determined by Gottsche. When X is a threefold, X[n] is in general singular, of the wrong dimension, and reducible. However if X is a smooth Calabi-Yau threefold, X[n] has a canonical virtual motive --- a motification of the degree zero Donaldson-Thomas invariants. We give a formula analogous to Gottsche's for the virtual motive of X[n]. The key computation gives a q-refinement of the classical formula of MacMahon which counts 3D partitions.

Freddy Cachazo, Perimeter Institute

What Do Grassmannians And Particle Colliders Have In Common?

In the past year, motivated by physics, a rich structure has emerged from studying certain contour integrals in Grassmannians. Physical considerations single out a natural meromorphic form in G(k,n) with a cyclic structure. The residues obtained from these contour integrals have been shown to be invariants of a Yangian algebra. These residues also control what happens deep inside collisions of protons taking place at colliders like the Large Hadron Collider or LHC at CERN. Applications of the Global Residue Theorem give rise to relations among residues which ensure important physical properties.

Emre Coskun, University of Western Ontario

Representations of Generalized Clifford Algebras

Clifford algebras arose in Dirac's work on the relativistic wave equation in quantum mechanics. Using the Clifford algebra associated to a quadratic form on a finite dimensional vector space, one can reduce the relativistic wave equation, a PDE of order two, to a system of linear PDEs. Similarly, one can use matrix representations of generalized (i.e. higher degree) Clifford algebras to reduce a PDE of higher degree. These generalized Clifford algebras have been the subject of ongoing research since late 1980s. In this talk, we will discuss generalized Clifford algebras, known results about their representations, and results of ongoing work in this direction.

Magdalena Czubak, University of Toronto

Topological defects in gauge theories.

We discuss recent progress on the rigorous description of the dynamics of the energy concentration sets in the abelian Higgs model. This is joint work with R. Jerrard.

Norman Do, McGill University

Lattice points in moduli spaces of curves

There appear to be only two essentially distinct ways to understand intersection numbers on moduli spaces of curves --- via Hurwitz numbers or symplectic volumes. In this talk, we will consider polynomials defined by Norbury which bridge the gap between these two pictures. They appear in the enumeration of lattice points in moduli spaces of curves and it appears that their coefficients store interesting information. We will also describe a connection between these polynomials and the topological recursion defined by Eynard and Orantin.

Charles Doran, University of Alberta

Normal Forms for Lattice Polarized K3 Surfaces and the Kuga-Satake Hodge Conjecture

We introduce a projective hypersurface "normal form" for a class of K3 surfaces which generalizes the classical Weierstrass normal form for complex elliptic curves.  A geometric two-isogeny relates these K3 surfaces to the Kummer K3 surfaces of principally polarized abelian surfaces, with the normal form coefficients naturally identifying with the Igusa basis of Siegel modular forms of degree two.  These results are reinterpreted through the lens of the Kuga-Satake Hodge Conjecture, and seen as a prediction coming from mirror symmetry.

Joel Kamnitzer, University of Toronto

Categorical Lie algebra actions and braid group actions

We will discuss the notion of categorical Lie algebra actions, as introduced by Rouquier and Khovanov-Lauda. In particular, we will give examples of categorical Lie algebra actions on derived categories of coherent sheaves. We will show that such categorical Lie algebra actions lead to actions of braid groups.

Melissa Liu, Columbia University

The coherent-constructible correspondence and homological mirror symmetry for toric varieties

I will discuss (i) a categorification of Morelli's theorem relating equivariant coherent sheaves on the toric variety to constructible sheaves on R^n, (ii) microlocalization functor relating the Fukaya category of a cotangent bundle to constructible sheaves on the base (due to Nadler-Zaslow, Nadler), and (iii) SYZ transformation relating equivariant (nonequivariant) coherent sheaves on the toric variety to Lagrangians in the cotangent of R^n (T^n). If time permits, I will also describe similar results for toric orbifolds. This talk is based on joint work with Bohan Fang, David Treumann, and Eric Zaslow.

David Morrison, University of California at Santa Barbara

Noncommutative algebras and (commutative) algebraic geometry

The study of D-branes at singular points of Calabi-Yau threefolds has revealed interesting connections between certain noncommutative algebras and singular algebraic varieties.  In many respects, the choice of an appropriate noncommutative algebra is analogous to finding a resolution of singularities of the variety.  We will explain this connection in detail, and outline a program for studying such "noncommutative resolutions'' globally, for compact algebraic (Calabi--Yau) threefolds.

Andrew Neitzke, University of Texas at Austin

Wall-crossing and hyperkahler geometry

I will give an overview of recent work with Davide Gaiotto and Greg Moore.  This work relates the phenomenon of "wall-crossing" for BPS states in four-dimensional N=2 theories to a new construction of hyperkahler metrics. These metrics include in particular the metrics on moduli spaces of solutions to Hitchin equations.  I will also briefly describe some extensions of this work to incorporate line and surface operators in the N=2 theory (in progress).

Nikita Nekrasov, Institut des Hautes Etudes Scientifiques

Thermodynamic Bethe ansatz from instantons in super-Yang-Mills theory

We show that the generating function of the equivariant (generalized) Donaldson invariants of ${\bf R}^2 X {\Sigma}$ is captured by the solution of a thermodynamic Bethe ansatz equation. Based on a joint work with S. Shatashvili.

Harvey Reall, University of Cambridge

Algebraically special solutions in higher dimensions

The Petrov classification of the Weyl tensor is an important tool in the study of exact solutions of the Einstein equation in 4d. For example, the Kerr solution was discovered in a study of spacetimes with algebraically special Weyl tensors. Algebraic classification of the Weyl tensor has been extended to higher dimensions. I shall review this classification and describe known families of algebraically special solutions. Recent progress towards obtaining a higher dimensional generalization of the Goldberg-Sachs theorem will be described.

Abhijnan Rej, Fields Institute

Periods, Geometry and Quantum Fields

I give a brief introduction to periods in the sense of Kontsevich and Zagier and to multiple zeta values which are, at least conjecturally, examples of periods. I discuss recent work on the appearance of multiple zeta values in quantum field theory and explain the general strategy in explaining the same. I conclude with some possible relationships with mirror symmetry.

Natalia Saulina, Perimeter Institute

Chern-Simons-Rozansky-Witten topological field theory

I will discuss a hybrid between Chern-Simons and Rozansky-Witten models. In particular, Wilson loops in this topological field theory are objects of a quantum deformation of the equivariant derived category of coherent sheaves.

David Skinner, Perimeter Institute

Twistor-String Theory

I'll give an introduction to twistor-string theory, which is an attempt to reformulate supersymmetric gauge theory in four-dimensional space-time in terms of a certain generalisation of Gromov-Witten theory in twistor space. The resulting theory is closely related to the multi-dimensional residue calculus in G(k,n) (introduced in Cachazo's talk).

Mu-Tao Wang, Columbia University

On the notion of quasilocal mass in general relativity

There have been many attempts to define quasilocal mass for a spacelike 2-surface in a spacetime by the Hamilton-Jacobi method. The essential difficulty in this approach is the choice of the background configuration to be subtracted from the physical Hamiltonian. The quasilocal mass should be positive for general surfaces, but on the other hand should be zero for surfaces in the flat spacetime. In this talk, I shall describe how to use isometric embeddings into the Minkowski space to overcome this difficulty and propose a new definition of gauge-independent quasi-local mass that has the desired properties, in addition to other natural requirements for a mass. This talk is based on a joint work with Shing-Tung Yau at Harvard.

Shing-Tung Yau, Harvard Universit

Heterotic string and complex Monge-Ampère equation

I will talk about the work that I did with Jixiang Fu and Jun Li on the Strominger system and their role in string theory.

Marco Gualtieri, University of Toronto

Funding provided in part by: