target space, background gauge field twisted smooth cohomology in string theory landscape of string theory vacua Special and general types Special notions Variants Extra structure Operations Theorems Recall that geometric T-duality is an operation acting on tuples roughly consisting of a smooth manifold with the structure of a torus-principal bundle – modelling spacetime equipped with a circle 2-bundle with connection – modelling the Kalb-Ramond field and in twisted K-theory refined to elements in differential twisted K-theory – modelling the RR-field and notably equipped with a (pseudo)Riemannian metric – modelling the field of gravity. The idea of topological T-duality (due to Bouwknegt-Evslin-Mathai 04, Bouwknegt-Hannabus-Mathai 04) is to disregard the Riemannian metric and the connection and study the remaining “topological” structure. While the idea of T-duality originates in string theory, topological T-duality has become a field of study in pure mathematics in its own right. 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 integral transform (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 (Sarkissian-Schweigert 08). 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 bundle gerbes pulled back to the fiber product correspondence space : of a certain prescribed integral transform-form (Bunke-Rumpf-Schick 08, p. 9)). T-duality topological T-duality differential T-duality T-duality 2-group spherical T-duality The concept of topological T-duality was introduced on the level of differential form-data in Peter Bouwknegt, Jarah Evslin, Varghese Mathai: T-Duality: Topology Change from H-flux, Commun. Math. Phys. 249 (2004)...