Spherical T-Duality and the spherical Fourier-Mukai transform

Abstract

In earlier papers, we introduced spherical T-duality, which relates pairs of the form $(P,H)$ consisting of an oriented $S^3$-bundle $P\to M$ 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 $\dim (M)\le 4$, 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 $\mathcal{P}$ 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 $\dim (M)\le 4$, improving our earlier results.

19 pages, clarifications added, typos corrected

References in corpus (4)

• On the Topology and Flux of T-Dual Manifolds
• Spherical T-Duality
• Spherical T-duality II: An infinity of spherical T-duals for non-principal SU(2)-bundles
• A noncommutative model for higher twisted K-Theory

Cited by in corpus (4)

• Topological T-duality via Lie algebroids and $Q$-flux in Poisson-generalized geometry
• On the Chern character in Higher Twisted K-theory and spherical T-duality
• Computations in higher twisted $K$-theory
• Bundles of strongly self-absorbing $C^{\ast }$-algebras with a Clifford grading