Topological T-Duality for Twisted Tori

TL;DR

This paper extends topological T-duality to a broad class of spaces including nilmanifolds and solvmanifolds, providing explicit constructions of T-duals as noncommutative tori and linking them to non-geometric string theory.

Contribution

It develops a general method for constructing T-duals using $C^*$-algebras, especially for almost abelian solvmanifolds, and characterizes when classical T-duals exist.

Findings

Explicit T-dual constructions for nilmanifolds and solvmanifolds.

Necessary and sufficient criteria for classical T-duals in group-theoretic terms.

Identification of T-duals as noncommutative tori with non-trivial Dixmier-Douady classes.

Abstract

We apply the $C^*$-algebraic formalism of topological T-duality due to Mathai and Rosenberg to a broad class of topological spaces that include the torus bundles appearing in string theory compactifications with duality twists, such as nilmanifolds, as well as many other examples. We develop a simple procedure in this setting for constructing the T-duals starting from a commutative $C^*$-algebra with an action of ${\mathbb R}^n$. We treat the general class of almost abelian solvmanifolds in arbitrary dimension in detail, where we provide necessary and sufficient criteria for the existence of classical T-duals in terms of purely group theoretic data, and compute them explicitly as continuous-trace algebras with non-trivial Dixmier-Douady classes. We prove that any such solvmanifold has a topological T-dual given by a $C^*$-algebra bundle of noncommutative tori, which we also compute explicitly. The monodromy of the original torus bundle becomes a Morita equivalence among the fiber algebras, so that these $C^*$-algebras rigorously describe the T-folds from non-geometric string theory.