Global Aspects of T-Duality, Gauged Sigma Models and T-Folds

TL;DR

Hull develops a global formalism for T-duality via gauged sigma-models, showing that obstructions to gauging are weaker than obstructions to T-duality and that T-duality can act fibrewise on torus fibrations even when global isometries are absent. By introducing extra scalars and the doubled torus $\hat{M}$, the work overcomes gauging obstructions and derives globally defined transformations under $O(d,d;\mathbb{Z})$, including the exchange of first Chern classes and $H$-classes. The analysis extends to torus fibrations with local Killing vectors, clarifying when T-duality yields geometric duals and when it produces non-geometric T-folds, with a careful treatment of large gauge transformations and global consistency. The results provide a robust, globally-defined approach to T-duality and its generalisations in non-geometric string backgrounds, highlighting the role of T-folds as legitimate string backgrounds and connecting them to the doubled geometry framework.

Abstract

The gauged sigma-model argument that string backgrounds related by T-dual give equivalent quantum theories is revisited, taking careful account of global considerations. The topological obstructions to gauging sigma-models give rise to obstructions to T-duality, but these are milder than those for gauging: it is possible to T-dualise a large class of sigma-models that cannot be gauged. For backgrounds that are torus fibrations, it is expected that T-duality can be applied fibrewise in the general case in which there are no globally-defined Killing vector fields, so that there is no isometry symmetry that can be gauged; the derivation of T-duality is extended to this case. The T-duality transformations are presented in terms of globally-defined quantities. The generalisation to non-geometric string backgrounds is discussed, the conditions for the T-dual background to be geometric found and the topology of T-folds analysed.