Topological T-duality and T-folds

TL;DR

This paper constructs C*-algebras related to Topological T-duality, linking noncommutative geometries and T-folds in string theory, and analyzes D-brane charges within this framework.

Contribution

It explicitly constructs C*-algebras from string-theoretic data and identifies their role in modeling nongeometric backgrounds and T-folds in Topological T-duality.

Findings

Constructed continuous-trace algebras from gerbes on trivial bundles.

Identified noncommutative T-duals with nongeometric string backgrounds.

Described D-brane charge groups using K-theory bundles.

Abstract

We explicitly construct the C*-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 ${\mathbb R}^d$ unique up to exterior equivalence from the data of a smooth ${\mathbb T}^d$-equivariant gerbe on a trivial bundle $X = W \times {\mathbb T}^d$. We argue that the `noncommutative T-duals' of Mathai and Rosenberg, should be identified with the nongeometric backgrounds well-known in string theory. We also argue that the crossed-product C*-algebra ${\mathcal A} \rtimes_{\alpha|_{\KZ^d}} {\mathbb Z}^d$ should be identified with the T-folds of Hull 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 studied by Echterhoff, Nest and Oyono-Oyono give a natural description of these objects.