"Strings and Mathematics 2004"

This is a general introduction to how gerbes emerge in Quantum Field Theory, with links to the geometry of infinite-dimensional Grassmannians.

There are a number of constructions of a gerbe or bundle gerbe over a compact Lie group. I will give a simple construction for the case of SU(n).

T-duality, in its simplest form, is the R to 1/R symmetry of String Theory compactified on a circle of radius R. It can be generalized to manifolds which admit circle actions (e.g. circle bundles) or, more generally, torus actions. In the case of nontrivial torus bundles, and in the background of H-flux, T-duality relates manifolds of different topology and in particular provides isomorphisms between the twisted cohomologies and twisted K-theories of these manifolds. In this talk we will discuss these developments as well as provide some examples in the case of principal circle bundles. In a sequel to this talk Mathai will discuss the case of higher dimensional torii.

My talk will cover the general case of T-duality for principal torus bundles. The new phenomenon that occurs here is that not all background fluxes are T-dualizable, and some joint work with Bouwknegt and Hannabuss works out the precise class of T-dualizable background fluxes. The isomophisms in twisted K-theory and twisted cohomology also follow in this case. A big puzzle remained to explain these mysterious ``missing T-duals'' corresponding to non-T-dualizable background fluxes. In another joint work with Rosenberg, we give a complete characterization of T-duality on principal 2-torus-bundles with background flux. Here we show that this problem is resolved using noncommutative topology. It turns out that every principal 2-torus-bundle with background flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. This suggests an unexpected link between classical string theories and the ``noncommutative'' ones, obtained by ``compactifying'' along noncommutative tori.

In this talk, we shall discuss the idea of Pi stability for the B-type D- branes on Calabi-Yau (CY) manifolds, adopting a description of these branes as being objects in a certain category. The stability of the D-branes is exhibited as a function of the CY moduli and this approach then enables us to work over the entire moduli space and not just at the large radius limit. The monodromy transformations are then realised as autoequivalences of these categories which via Orlov's theorem are then implemented by Fourier-Mukai transformations.

M-Theory is an eleven-dimensional theory that unifies the five consistent ten-dimensional Superstring theories. The theory has proved to have rich physical and mathematical structures that are not yet fully understood. In this talk I will review the basics of M-Theory and the recent developments in understanding this theory.

The K-theory of a compact Lie group G with torsion free fundamental group was computed by Hodgkin to be an exterior algebra over the integers on m generators where m is the rank of G. We will give an elementary proof of this fact. If time permits we will comment briefly on the twisted case (this last part is still work in progress).