Topological T-duality is a relationship between pairs (E, P) over a fixed space X, where \(E \rightarrow X\) is a principal torus bundle and \(P \rightarrow E\) is a twist, such as a gerbe for a principal \(PU({\mathcal {H}})\) -bundle. This is of interest to topologists because of the T-duality transformation: a T-duality relation between pairs (E, P) and \(({\hat{E}}, {\hat{P}})\) comes with an isomorphism (with degree shift) between the twisted K-theory of E and the twisted K-theory of \({\hat{E}}\) . We formulate topological T-duality for circle bundles 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 all compact Lie groups, equal to its own inverse and uniquely characterized by naturality and a normalization for trivial situations.
