Spherical T-Duality and the spherical Fourier-Mukai transform

Abstract

In Bouwknegt et al. (2015) [3, 4], we introduced spherical T-duality, which relates pairs of the form (P, H) consisting of an oriented S3-bundle P → M and a 7-cocycle H on P called the 7-flux. Intuitively, the spherical T-dual is another such pair (Pˆ , Hˆ ) and spherical T-duality exchanges the 7-flux with the Euler class, upon fixing the Pontryagin class and the second Stiefel–Whitney class. Unless dim(M) ≤ 4, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. In this paper, we define a canonical Poincaré virtual line bundle P on S3 ×S3 (actually also for Sn ×Sn) and the spherical Fourier–Mukai transform, which implements a degree shifting isomorphism in K-theory on the trivial S3-bundle. This is then used to prove that all spherical T-dualities induce natural degreeshifting isomorphisms between the 7-twisted K-theories of the pairs (P, H) and(Pˆ , Hˆ )when dim(M) ≤ 4, improving our earlier results.