THE TOPOLOGY OF T-DUALITY FOR Tn-BUNDLES

Abstract

In string theory, the concept of T-duality between two principal Tn-bundles E and Ê over the same base space B, together with cohomology classes h ∈ H3(E,ℤ) and ĥ ∈ H3(Ê,ℤ), has been introduced. One of the main virtues of T-duality is that h-twisted K-theory of E is isomorphic to ĥ-twisted K-theory of Ê. 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 and 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.