Abstracts for Geometry and Quantum Field Theory

This talk reports on joint work with Mauro Spera. Given a principal bundle with connection one can define the induced "looped connection" on the looped principal bundle. We explain this procedure and apply it to the construction of Spin-c connections on the loop space of a finite-dimensional string manifold, and to show that the loop space of an arbitrary finite-dimensional manifold possesses a good cover in the sense of Leray. 

We report our joint work with Qingtao Chen and Fei Han, where we construct a mod 2 analogue of the Witten genus for $8k+2$ dimensional spin manifolds, as well as modular characteristic numbers for a class of spin^c manifolds which we call string^c manifolds. When these spin^c manifolds are actually spin, one recovers the original Witten genus on string manifolds. These genera vanish on string and string^c complete intersections respectively in complex projective spaces. 

We discuss C*-algebras associated with dynamical systems on a Cantor space. Of particular interest are systems arising from global fields.

Brylinski and McLaughlin have introduced a "transgression" functor that takes abelian gerbes with connection over a smooth manifold M to certain principal bundles over the free loop space LM of M. In my talk I will present a characterization of the image of this functor. Then I describe an inverse functor, called "regression", so that an equivalence of geometric categories over M and LM is obtained. It will become clear how the concept of a bundle gerbe fits nicely into this context.