Spherical T-duality and the spherical Fourier–Mukai transform

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 degree-shifting isomorphisms between the 7-twisted K-theories of the pairs (P,H) and(Pˆ,Hˆ) when dim(M)≤4, improving our earlier results.