T-duality with H-flux: non-commutativity, T-folds and G x G structure

TL;DR

The paper unifies three strands of T-duality with NSNS $H$-flux—non-commutative torus fibrations, T-folds, and $SU(3)\times SU(3)$ structure mirrors—through generalized complex geometry and BV formalism. It shows that open-string non-commutativity is governed by a holomorphic Poisson bivector or β-transform, which may vary over the base and even be non-decomposable in non-geometric cases, while T-duality along two directions yields T-folds that cannot be globally polarized. By analyzing D-branes as generalized complex submanifolds and deriving the β-deformation to the Poisson sigma model, the authors connect brane monodromies, star-products, and non-commutative deformations in a consistent framework. In SU(3) × SU(3) structure compactifications with non-parallel spinors, a non-trivial bivector governs non-commutativity, linking mirror symmetry and non-geometric backgrounds and suggesting a generalized SYZ picture for NSNS flux backgrounds.

Abstract

Various approaches to T-duality with NSNS three-form flux are reconciled. Non-commutative torus fibrations are shown to be the open-string version of T-folds. The non-geometric T-dual of a three-torus with uniform flux is embedded into a generalized complex six-torus, and the non-geometry is probed by D0-branes regarded as generalized complex submanifolds. The non-commutativity scale, which is present in these compactifications, is given by a holomorphic Poisson bivector that also encodes the variation of the dimension of the world-volume of D-branes under monodromy. This bivector is shown to exist in SU(3) x SU(3) structure compactifications, which have been proposed as mirrors to NSNS-flux backgrounds. The two SU(3)-invariant spinors are generically not parallel, thereby giving rise to a non-trivial Poisson bivector. Furthermore we show that for non-geometric T-duals, the Poisson bivector may not be decomposable into the tensor product of vectors.