Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 012, 51 pages      arXiv:2006.10048      https://doi.org/10.3842/SIGMA.2021.012
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

Topological T-Duality for Twisted Tori

Paolo Aschieri abc and Richard J. Szabo abdef
a) Dipartimento di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale, Viale T. Michel 11, 15121 Alessandria, Italy
b) Arnold-Regge Centre, Via P. Giuria 1, 10125 Torino, Italy
c) Istituto Nazionale di Fisica Nucleare, Torino, Via P. Giuria 1, 10125 Torino, Italy
d) Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK
e) Maxwell Institute for Mathematical Sciences, Edinburgh, UK
f) Higgs Centre for Theoretical Physics, Edinburgh, UK

Received June 30, 2020, in final form January 22, 2021; Published online February 05, 2021

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.

Key words: noncommutative $C^*$-algebraic T-duality; nongeometric backgrounds; Mostow fibration of almost abelian solvmanifolds; $C^*$-algebra bundles of noncommutative tori.

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