Recall that ( geometric ) T-duality is an operation acting on tuples roughly consisting of
The idea of topological T-duality is to disregard the Riemannian metric and the connection and study the remaining “topological” structure.
In the language of bi-branes a topological T-duality transformation is a bi-brane of a special kind between the two gerbes involved. The induced pull-push operation (in groupoidification and geometric function theory) on (sheaves of sections of, or K-classes of) (twisted) vector bundles is essentially the Fourier-Mukai transformation. More on the bi-brane interpretation of (topological and non-topological) T-duality is in SarkissianSchweigert08.
Two tuples consisting of a -bundle over a manifold and a line bundle gerbe over are topological T-duals if there exists an isomorphism of the two line bundle gerbes pulled back to the fiber product correspondence space :
The simplest version of topological T-duality, when is a principal circle bundle, was originally developed in
and the torus bundle case was introduced in
In these papers the gerbe does not appear, but an integral 3-form, representing the Dixmier-Douady class of a gerbe does. Note that if the integral cohomology group of has torsion in dimension three, not all gerbes will arise in this way. The formalization with the above data originates in
A refined description is in
There is a C*-algebraic version of toplogical T-duality, discussed for instance at
Jonathan Rosenberg has also written a little introductory book for mathematicians:
A discussion that instead of noncommutative spaces uses topological groupoids is in
The bi-brane perspective on T-duality is amplified in
A transcript a a talk by Varghese Mathai on topological T-duality is here: