Topological T-duality for torus bundles with monodromy

TL;DR

This paper introduces a simplified, general topological T-duality framework for torus bundles with monodromy, avoiding complex tools like Chern classes, and establishes conditions for T-dual existence and invariance of key structures.

Contribution

It provides a new, streamlined definition of topological T-duality applicable to all torus bundles with monodromy, including principal and non-principal cases, using gerbes and morphisms.

Findings

Necessary and sufficient conditions for T-dual existence are established.

Invariance of twisted cohomology, K-theory, and Courant algebroids is demonstrated.

T-duality can be iterated to compute twisted K-theory groups.

Abstract

We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficient conditions for existence of a T-dual in the case of affine torus bundles. This is general enough to include all principal torus bundles as well as torus bundles with arbitrary monodromy representations. We show that isomorphisms in twisted cohomology, twisted K-theory and of Courant algebroids persist in this general setting. We also give an example where twisted K-theory groups can be computed by iterating T-duality.