Topological T-duality for general circle bundles

TL;DR

This paper generalizes topological T-duality to all circle bundles, establishing existence, uniqueness, and isomorphisms of twisted cohomology and K-theory, including novel twists involving real line bundles.

Contribution

It extends T-duality to general circle bundles, introduces new types of twists, and demonstrates isomorphisms of twisted invariants in this broader setting.

Findings

Existence and uniqueness of T-duals for general circle bundles.

Isomorphism of twisted cohomology and K-theory under T-duality.

Examples of T-dual non-oriented circle bundles with computed twisted K-theory.

Abstract

We extend topological T-duality to the case of general circle bundles. In this setting we prove existence and uniqueness of T-duals. We then show that T-dual spaces have isomorphic twisted cohomology, twisted $K$-theory and Courant algebroids. A novel feature is that we must consider two kinds of twists in de Rham cohomology and $K$-theory, namely by degree 3 integral classes and a less familiar kind of twist using real line bundles. We give some examples of T-dual non-oriented circle bundles and calculate their twisted $K$-theory.