Equivariant Topological T-Duality

TL;DR

This paper extends topological T-duality to an equivariant setting, establishing an isomorphism in equivariant K-theory for compact Lie group actions, with a focus on the mathematical structure and properties of the duality.

Contribution

It formulates equivariant topological T-duality, defines the T-duality transformation in equivariant K-theory, and proves it is an isomorphism with natural properties.

Findings

T-duality transformation is an isomorphism in equivariant K-theory.

The transformation is its own inverse.

It is uniquely characterized by naturality and normalization.

Abstract

Topological T-duality is a relationship between pairs (E, P ) over a fixed space X, where E over X is a principal torus bundle and P over E is a twist, such as a gerbe of principal PU(H)-bundle. This is of interest to topologists because of the T-duality transformation: a T-duality relation between pairs (E, P ) and (F, Q ) comes with an isomorphism (with degree shift) between the twisted K-theory of E and the twisted K-theory of F. We formulate topological T-duality in the equivariant setting, following the definition of Bunke, Rumpf, and Schick. We define the T-duality transformation in equivariant K-theory and show that it is an isomorphism for actions of compact Lie groups, equal to its own inverse and uniquely characterized by naturality and a normalization for trivial situations.