Ulrich Bunke

In string theory, the concept of T-duality between two principal T^n-bundles E_1 and E_2 over the same base space B, together with cohomology classes h_1\in H^3(E_1) and h_2\in H^3(E_2), has been introduced. One of the main virtues of T-duality is that h_1-twisted K-theory of E_1 is isomorphic to h_2-twisted K-theory of E_2. In this paper, a new, very topological concept of T-duality is introduced. We construct a classifying space for pairs as above with additional "dualizing data", with a forgetful map to the classifying space for pairs (also constructed in the paper). On the first classifying space, we have an involution which corresponds to passage to the dual pair, i.e. to each pair with dualizing data exists a well defined dual pair (with dualizing data). We show that a pair (E,h) can be lifted to a pair with dualizing data if an only if h belongs to the second step of the Leray-Serre filtration of E (i.e. not always), and that in general many different lifts exist, with topologically different dual bundles. We establish several properties of the T-dual pairs. In particular, we prove a T-duality isomorphism of degree -n for twisted K-theory.

The Chern classes of a K-theory class which is represented by a vector bundle with connection admit refinements to Cheeger-Simons classes in Deligne cohomology. In the present paper we consider similar refinements in the case where the classes in K-theory are represented by geometric families of Dirac operators. In low dimensions these refinements correspond to the exponentiated eta-invariant, the determinant line bundle with Quillen metric and Bismut-Freed connection, and Lott's index gerbe with connection and curving. We give a unified treatement of these cases as well as their higher generalizations. Our main technical tool is a variant of local index theory for Dirac operators of families of manifolds with corners.

We show that the inertia stack of a topological stack is again a topological stack. We further observe that the inertia stack of an orbispace is again an orbispace. We show how a U(1)-banded gerbe over an orbispace gives rise to a flat line bundle over its inertia stack. Via sheaf theory over topological stacks it gives rise to the twisted delocalized cohomology of the orbispace. With these results and constructions we generalize concepts, which are well-known in the smooth framework, to the topological case. In the smooth case we show, that our sheaf-theoretic definition of twisted delocalized cohomology of orbispaces coincides with former definitions using a twisted de Rham complex.

We study the topology of T-duality for pairs of U(1)-bundles and three-dimensional integral cohomology classes over orbispaces. In particular, our results apply to U(1)-spaces with finite isotropy. We generalize the theory developed in our previous paper math.GT/0405132 from spaces to orbispaces.

The goal of the present paper is the calculation of the equivariant twisted K-theory of a compact Lie group which acts on itself by conjugations, and elements of a TQFT-structure on the twisted K-groups. These results are originally due to D.S.Freed, M.J.Hopkins and C.Teleman. In this paper we redo their calculations in the framework of topological and differentiable stacks. We also show how moduli spaces of flat connections on surfaces give rise to trivializations of twists.

In the present paper we develop a framework in which questions of quantum ergodicity for operators acting on sections of hermitian vector bundles over Riemannian manifolds can be studied. We are particularly interested in the case of locally symmetric spaces. For locally symmetric spaces, we extend the recent construction of Silberman and Venkatesh (math.RT/0407413) of representation theoretic lifts to vector bundles.

For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev's 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific class of rank one homotopy quantum field theories.