Topological T-duality and T-folds

We explicitly construct the C*8-algebras arising in the formalism of Topological T-duality due to Mathai and Rosenberg from string-theoretic data in several key examples. We construct a continuous-trace algebra with an action of R{double-struck}^d unique up to exterior equivalence from the data of a smooth T{double-struck}^d-equivariant gerbe on a trivial bundle X = W × T{double-struck}^d. We argue that the "non-commutative T-duals" of Mathai and Rosenberg [7] should be identified with the non-geometric backgrounds well known in string theory. We also argue that the C*-algebra should be identified with the T-folds of Hull [1] and Belov et al. [2] which geometrize these backgrounds. We identify the charge group of D-branes on T-fold backgrounds in the C*-algebraic formalism of Topological T-duality. We also study D-branes on T-fold backgrounds. We show that the K-theory bundles of [13] give a natural description of these objects.