Non-Abelian target space duals of Thurston geometries

TL;DR

This paper explores the Poisson-Lie T-duality of Thurston geometries, identifying their isometry groups and deriving non-Abelian duals, contributing to understanding duality in string theory on these geometries.

Contribution

It systematically finds non-Abelian T-duals of Thurston geometries using Poisson-Lie duality, focusing on their isometry subalgebras and duality conditions.

Findings

Identified all isometry subgroups as Bianchi type algebras.

Derived non-Abelian T-duals without B-field for Thurston geometries.

Commented on conformal invariance of the dual models.

Abstract

In this study, we proceed to investigate the Thurston geometries from the point of view of their Poisson-Lie (PL) T-dualizability. First of all, we find all subalgebras of Killing vectors that generate group of isometries acting freely and transitively on the three-dimensional target manifolds, where the Thurston metrics are defined. It is shown that three-dimensional Lie subalgebras are isomorphic to the Bianchi type algebras. We take the isometry subgroup of the metric as the first subgroup of Drinfeld double. In order to investigate the non-Abelian T-duality, the second subgroup must be chosen to be Abelian. Accordingly, the non-Abelian target space duals of these geometries are found via PL T-duality approach in the absence of $B$-field. We also comment on the conformal invariance conditions of the T-dual $\sigma$-models under consideration.