GAP 2012 (Geometry And Physics)

Saturday, May 5, to Monday, May 7, 2012

To be held jointly at both the University of Waterloo
and at the Perimeter Institute for Theoretical Physics

Organizing Committee:   Marco Gualtieri (Toronto), Spiro Karigiannis (Waterloo), Ruxandra Moraru (Waterloo), Rob Myers (Perimeter), Pedro Vieira (Perimeter), McKenzie Wang (McMaster).

MORE SPEAKERS TO BE ANNOUNCED SOON.

***** REGISTRATION INFORMATION IS COMING SOON *****

The history of GAP (now in its fourth year!)

The first "Connections in Geometry And Physics" conference (GAP 2009) was held in May 2009 at the Perimeter Institute for Theoretical Physics in Waterloo, Canada. The primary goal was to bring together geometers, from Canada and around the world, who work at the interface between mathematics and physics. An important secondary objective was to further increase Canada's presence and visibility in geometry within the international mathematical community. GAP 2010 was held in May 2010, also at the Perimeter Institute. GAP 2011 was held in May 2011, at the Fields Institute for Mathematical Sciences in Toronto, Canada.

Each year, the format of the conference combines three separate but related themes in geometry and physics. This approach results in a very interesting mixture of talks. We expect that GAP will be an annual event. The aim of our conference is to present some of the exciting new developments in the areas of our chosen themes, as well as to expose local area graduate students and postdocs to these ideas. In fact, the previous conferences involved heavy participation by graduate students and postdoctoral fellows (over one third of all participants) from the universities of Toronto, Queen's, Waterloo, McMaster, and Western Ontario, the Perimeter Institute, as well as other institutions in Canada and around the world.

This year's themes are: special holonomy; generalized geometry; and integrable systems.

Previous GAP conferences

Overview of this year's topics:

1. Special holonomy. Manifolds with special holonomy include SU(m) (Calabi-Yau), Sp(m) (hyperKähler), G₂, and Spin(7) manifolds. Compact examples result from work of Yau, Joyce, and many others. They are also characterized by the fact that they admit parallel or Killing spinors, which are important ingredients in theories of physics that incorporate supersymmetry. All such manifolds come equipped with one or more parallel differential forms that determine special classes of calibrated submanifolds, part of the data of "supersymmetric cycles" in physics. It is interesting that calibrated submanifolds seem to belong to two different families (sometimes called instantons and branes), exhibiting strikingly different behaviour. Explicit examples of calibrated submanifolds in Euclidean spaces are also often connected with the theory of integrable systems.

On manifolds with special holonomy, one can also consider "calibrated" connections defined by an algebraic condition on their curvature forms. Examples include Hermitian-Yang-Mills connections on bundles over Calabi-Yau manifolds, and Donaldson-Thomas connections on G₂ and Spin(7) manifolds. These can be viewed as higher-dimensional analogues of anti-self-dual connections over Riemannian 4-manifolds. Calibrated connections in the G₂ and Spin(7) settings are still not as well understood as they are in the Calabi-Yau setting.

Manifolds with special holonomy are also believed to exhibit the phenomenon of mirror symmetry, currently best understood in the hyperKähler and Calabi-Yau cases. Understanding the geometric aspects of mirror symmetry will involve studying their moduli spaces of calibrated submanifolds and of calibrated connections. More precisely, under certain conditions, a manifold M with special holonomy is expected to fibre over a base space B, with the generic fibre being a calibrated torus in M. We knows such a fibration must have singular fibres, that are very difficult to deal with analytically and geometrically. The idea is that the "mirror manifold" should be obtained by "dualizing" the smooth part of this fibration, and somehow compactifying to deal with the singular fibres. Although some results have been obtained, by many mathematicians, toward understanding this picture in the case of Calabi-Yau manifolds, there has been much less progress in the cases of the exceptional holonomy groups.

2. Generalized geometry. In studying geometric structures determined by differential forms, such as symplectic, Calabi-Yau, and G₂ structures, Hitchin discovered a new class of geometries by relaxing the condition that the differential form should be of a fixed degree. These new structures, which involve generalizations of Riemannian metrics, complex structures, and other geometries of special holonomy, comprise the subject called generalized geometry. The hallmark of the generalized context is that the symmetries of a geometric structure involve an extension of the diffeomorphisms by the group of B-field gauge transformations.

Generalized geometry was rapidly adopted by physicists studying supersymmetric sigma models and supergravity theories, because it provided a conceptual explanation for the sometimes arcane transformation rules and dualities governing the constituent fields of the models. Indeed, T-duality and the Buscher rules which govern the simplest form of mirror symmetry are completely explained as symmetries in the generalized sense. Another success of the generalized framework was the explanation of the geometry of (2,2)-supersymmetric sigma models, which could be interpreted simply as generalized Kähler or Calabi-Yau geometry. This allowed not only the construction of many new examples, but also the solution of several outstanding problems which were raised in the 80s when the models were first proposed.

In addition to its usefulness in quantum field theory, generalized geometry has opened new avenues of study in differential geometry. For example, a generalized complex structure interpolates between complex and symplectic structures, and exotic examples on manifolds not admitting complex or symplectic structures have recently been discovered. The properties of these intermediate structures are currently being investigated, using a combination of methods from algebraic geometry, symplectic geometry, the Riemannian geometry of skew torsion, and Poisson geometry.

3. Integrable systems. Classical integrable systems are mechanical systems whose solutions are quasi-periodic. In the language of Hamiltonian mechanics, they correspond to systems that have a maximal number of integrals of motion; and in finite dimensions, integrability can be formulated as the existence of a Lagrangian fibration on the phase space. Classical examples include spinning tops, harmonic oscillators, Calogero-Moser systems, Toda lattices, and the Korteweg-de Vries (KdV) equation, to name a few.

More recent occurences of integrable systems include the Hitchin systems, which were obtained by Hitchin in the late 80s as reductions of the Yang-Mills equations, and are examples of complex integrable systems. Moreover, real integrable systems appear in mirror symmetry, where one looks for a special kind of Lagrangian fibration, by special Lagrangian submanifolds. Remarkably, integrable systems also arise in the Seiberg-Witten solution of N = 2 supersymmetric Yang-Mills theory for SU(2) gauge group, where Seiberg-Witten curves are identified with the spectral curves of known integrable systems such as Calogero-Moser systems and Hitchin systems. Furthermore, the Virasoro conjecture, a generalisation of Witten's KdV conjecture (proved by Kontsevich in the early 90s), relates the Gromov-Witten invariants of compact symplectic manifolds to integrable systems.

In addition to their vast presence in physics, integrable systems are important mathematical objects in their own right. For example, in algebraic geometry, moduli spaces of sheaves on holomorphic Poisson (symplectic) varieties often admit structures of integrable systems, which can be used to study the geometry and topology of these moduli. Recently, Gaiotto-Moore-Neitzke used complex integrable systems to construct explicit examples of hyperKähler metrics. Moreover, integrable systems play an essential role in the geometric Langlands Program, where automorphic forms of the Langlands Program are replaced by appropriate Hitchin systems.

Financial support is provided by	Perimeter Institute for Theoretical Physics University of Toronto Department of Mathematics University of Waterloo Faculty of Mathematics Centre de Recherches Mathématiques McMaster University Faculty of Science Fields Institute for Mathematical Sciences

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