Doubled Geometry and T-Folds

TL;DR

Hull develops a duality-covariant framework for string theory by doubling torus fibers to form a target space with $T^{2d}$ fibers and an $O(d,d;\mathbb{Z})$ structure. A self-duality constraint is imposed, implemented via gauging of half the currents, to recover the conventional sigma-model on patches while preserving T-duality as a symmetry; a compensating topological term ensures equivalence at arbitrary genus and yields a duality-consistent dilaton. The formalism naturally handles T-folds, where transition functions include T-dualities, and extends to supersymmetry and a fully doubled base, connecting to generalised geometry and its limitations. Overall, the paper provides a rigorous quantum-mechanical basis for doubled geometry and its application to non-geometric backgrounds, with implications for modular invariance, dilaton dynamics, and broader generalisations of string geometry.

Abstract

The doubled formulation of string theory, which is T-duality covariant and enlarges spacetime with extra coordinates conjugate to winding number, is reformulated and its geometric and topological features examined. It is used to formulate string theory in T-fold backgrounds with T-duality transition functions and a quantum implementation of the constraints of the doubled formalism is presented. This establishes the quantum equivalence to the usual sigma-model formalism for world-sheets of arbitrary genus, provided a topological term is added to the action. The quantisation involves a local choice of polarisation, but the results are independent of this. The natural dilaton of the doubled formalism is duality-invariant and so T-duality is a perturbative symmetry for the perturbation theory in the corresponding coupling constant. It is shown how this dilaton is related to the dilaton of the conventional sigma-model which does transform under T-duality. The generalisation of the doubled formalism to the superstring is given and shown to be equivalent to the usual formulation. Finally, the formalism is generalised to one in which the whole spacetime is doubled.