"Mathematics of String Theory 2006"

on Thursday 27th July 2006

9:00 am

Speaker: Peter Bouwknegt (ANU)

Title: T-duality and Generalized Geometry

Abstract: In this talk I will discuss how to incorporate T-duality for principal torus bundles into the framework of (a generalized) generalized geometry. This is joint work with Josh Garretson and Peggy Kao.

10:15 am

Speaker: Jacek Brodzki (University of Southampton, UK)

Title: D-brane charges and Poincare duality on noncommutative manifolds

Abstract: In classical differential geometry, one of the main properties of a compact diffierentiable manifold M of dimension n is the Poincare duality, which establishes an isomorphism between cohomology in degree k and homology in degree n-k, or equivalently, it provides a non degenerate complex valued pairing between cohomology groups in degree k and n-k. An analogue of this property, expressed in terms of Poincare duality in Kasparov's KK-theory, has found a place in Connes' axiomatic description of differentiable manifolds in noncommutative geometry. In this talk we shall give an introduction to the notion of Poincare duality in bivariant K-theory and provide applications of this formalism in the D-brane theory. We propose a general formula for D-brane charge.

11:30 am

Speaker: Gil Cavalcanti (Oxford University, UK)

Title: A surgery for generalized complex 4-manifolds

Abstract: I'll introduced a surgery for generalized complex 4-manifolds whose starting point is a symplectic 4-manifold with a symplectic 2-torus and whose output is a generalized complex manifold which is symplectic away from a 2-torus, where the structure becomes of complex type. Using this surgery in a specific symplectic manifold I'll give the first example of generalized complex manifold which does not admit either complex or symplectic structures.

2:00 pm

Speaker: Keith Hannabuss (Oxford University, UK)

Title: T-duality and tensor categories

Abstract: There are many examples of physical systems which can be described by apparently different models related by rather non-obvious symmetries. In string theory T-duality related spaces with group actions and H-fields. The simplest cases can be described in purely geometric terms, but others are harder to describe. One approach is to use the methods of non-commutative geometry, but in some cases even that is insufficient because the algebras are not associative. This talk will survey some of the ideas.

3:30 pm

Speaker: Katrin Wendland (University of Augsburg, Germany)

Title: Towards the boundary of spaces of conformal field theories

Abstract: Is there a notion of compactness, similar to the notion of Gromov compactness in geometry, governing moduli spaces of conformal field theories (CFTs)? While this has undisputedly been a difficult question of general importance for a while, which we certainly cannot answer at this stage, recent discussions in string theory have reignited interest in this problem. In joint work with Daniel Roggenkamp we give a possible first step to approach this topic. We establish an intrinsic notion of limiting processes in CFTs. The resulting limits can exhibit the structure of degenerate CFTs, resembling degeneration phenomena in geometry which are familiar from large volume limits of non-linear sigma models. In fact, by applying techniques from noncommutative geometry to such boundary points of moduli spaces of CFTs one can explicitly associate geometric interpretations to certain limits of CFTs.

4:45 pm

Speaker: Siye Wu (University of Colorado, USA. University of Hong Kong, HK)

Title: Fiber Integration of Deligne Cohomology Classes

Abstract: We begin with a review of how to integrate differential forms and cohomology classes along the fiber. Then we discuss the geometry of gerbes and Deligne cohomology classes. Finally we give an intrinsic definition of fiber integration of such objects and explain its significance, relating it to various types of fiber integration reviewed earlier. We complete the talk with an outlook of future development.