Metric algebroids, para-Hermitian structures and T-duality

Abstract

The geometry of para-Hermitian vector bundles is introduced and generalised metrics are defined on such vector bundles. In particular, the properties of Born metrics are demonstrated. Their application to classical Lagrangian dynamics is highlighted. Metric algebroids are presented and their existence problem is addressed together with their compatibility with para-Hermitian structures. The example of pre-Courant algebroids is thoroughly discussed and applied to the special case of Courant algebroids. In this setting the notion of Dirac structure is recalled in order to introduce Dirac-Riemannian foliations. Para-Hermitian manifolds endowed with a Born metric and their compatible metric algebroid structure are used to define sigma-models in a duality-symmetric formulation. Their Lie algebroid gauging is studied and the geometric interpretation of the gauging conditions as Dirac-Riemannian structures is presented. In particular, a detailed analysis of gauged sigma-models for regularly foliated manifolds is given. This construction is applied to describing a geometric picture of generalised Tduality, where the para-Hermitian manifold is supposed to admit different maximally isotropic foliations so that T-dual sigma models are recovered on their leaf spaces, which represent the physical space-times. The main examples presented in this work are given by Lie groups endowed with invariant para-Hermitian structures and the doubled twisted torus. In particular, for the latter its full T-duality chain is recovered by using these techniques.