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Spherical T-duality and the spherical Fourier-Mukai transform. (English) Zbl 1398.81187

Summary: In [Commun. Math. Phys. 249, No. 2, 383–415 (2004; Zbl 1062.81119); Phys. Rev. Lett. 92, No. 18, Article ID181601, 3 p. (2004; Zbl 1267.81264)], 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 \(\dim(M) \leq 4\), not all pairs admit spherical T-duals and the spherical T-duals are not always unique. In this paper, we define a canonical 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 \(\dim(M) \leq 4\), improving our earlier results.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
55R25 Sphere bundles and vector bundles in algebraic topology
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References:

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