Spherical T-Duality and the spherical Fourier-Mukai transform

TL;DR

This paper introduces a spherical Fourier-Mukai transform that establishes degree-shifting isomorphisms in K-theory for spherical T-dual pairs, extending previous results and clarifying conditions for duality existence.

Contribution

It defines a canonical Poincaré virtual line bundle and a spherical Fourier-Mukai transform, proving their role in inducing natural isomorphisms in 7-twisted K-theory for spherical T-dual pairs.

Findings

Spherical Fourier-Mukai transform implements degree shifts in K-theory.

All spherical T-dualities induce natural degree-shifting isomorphisms.

Results hold when the base manifold dimension is at most 4.

Abstract

In earlier papers, we introduced spherical T-duality, which relates pairs of the form $(P,H)$ consisting of an oriented $S^3$-bundle $P\rightarrow M$ and a 7-cocycle $H$ on $P$ called the 7-flux. Intuitively, the spherical T-dual is another such pair $(\hat P, \hat H)$ and spherical T-duality exchanges the 7-flux with the Euler class, upon fixing the Pontryagin class and the second Stiefel-Whitney class. Unless $\mathrm{dim}(M)\leq 4$, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. In this paper, we define a canonical Poincar\'e virtual line bundle $\mathcal{P}$ on $S^3 \times S^3$ (actually also for $S^n\times S^n$) and the spherical Fourier-Mukai transform, which implements a degree shifting isomorphism in K-theory on the trivial $S^3$-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 $(\hat P, \hat H)$ when $\mathrm{dim}(M)\leq 4$, improving our earlier results.