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Topological \(T\)-duality for twisted tori. (English) Zbl 1475.46055

Summary: We apply the \(C^*\)-algebraic formalism of topological \(T\)-duality due to V. Mathai and J. Rosenberg [Commun. Math. Phys. 253, No. 3, 705–721 (2005; Zbl 1078.58006)] 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.

MSC:

46L55 Noncommutative dynamical systems
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
16D90 Module categories in associative algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory

Citations:

Zbl 1078.58006
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References:

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