T-Duality: Topology Change from H-flux
P. Bouwknegt, J. Evslin, V. Mathai
TL;DR
The paper addresses topology change under T-duality for circle bundles with H-flux by proposing a concrete duality rule that exchanges $c_1(E)$ with the fiberwise integral of $H$ and proving isomorphisms in twisted cohomology and twisted K-theory via correspondence spaces. It develops a rigorous framework for T-duality in the twisted setting, deriving explicit maps (such as $T_*$ and $T_!$) and a twisted Grothendieck-Riemann-Roch relation, and it validates the construction through detailed 3D geometric examples and anomaly considerations. The work connects physical dualities from $E_8$ and S-duality perspectives with mathematical structures (Gysin sequences, pushforwards, and fusion products in decomposable twists), offering a unifying view of fluxes and topology under duality. It also highlights intriguing implications for M-theory flux quantization, gravitino anomalies, and the role of F-theory/F-duality pictures, suggesting extensions to higher tori and non-abelian settings with broad potential impact on string compactifications and duality dictionaries.
Abstract
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 RP^n, and the AdS^5 x S^5 to AdS^5 x CP^2 x S^1 with background H-flux of Duff, Lu 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 fails. 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.