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\(T\)-duality: topology change from \(H\)-flux. (English) Zbl 1062.81119
Summary: T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the \(H\)-flux, as we motivate using \(E_8\) and also using \(S\)-duality. We present known and new examples including NS5-branes, nilmanifolds, lens spaces, both circle bundles over \(\mathbb{R} P^n\), and the AdS\(^5 \times S^5\) to AdS\(^5\times \mathbb{C} P^2\times S^1\) with background \(H\)-flux of Duff, Lü and Pope. When T-duality leads to M-theory on a non-spin manifold the gravitino partition function continues to exist due to the background flux, however the known quantization condition for \(G_4\) receives a correction. In a more general context, we use correspondence spaces to implement isomorphisms on the twisted \(K\)-theories and twisted cohomology theories and to study the corresponding Grothendieck-Riemann-Roch theorem. Interestingly, in the case of decomposable twists, both twisted theories admit fusion products and so are naturally rings.

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)
58J90 Applications of PDEs on manifolds
19L10 Riemann-Roch theorems, Chern characters
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