Topological T-Duality for Twisted Tori

Abstract

We apply the $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-algebras rigorously describe the T-folds from non-geometric string theory.

Contribution to the SIGMA Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi for his 60th birthday

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Cited by in corpus (4)

• Looking for structure in the cobordism conjecture
• Braided Symmetries in Noncommutative Field Theory
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
• Towards a T-dual Emergent Gravity