Para-Hermitian Geometry, Dualities and Generalized Flux Backgrounds

TL;DR

This paper explores the geometric structures underlying double field theory, focusing on para-Hermitian manifolds, fluxes, and dualities, and introduces new frameworks connecting flux backgrounds with para-Kahler and Born geometries.

Contribution

It provides a comprehensive geometric framework for understanding fluxes and dualities in double field theory using para-Hermitian and Born geometries, including explicit constructions and examples.

Findings

Describes para-Hermitian geometry of Drinfel'd doubles.

Reproduces standard generalized fluxes within this geometric setting.

Illustrates how polarization changes lead to different string backgrounds.

Abstract

We survey physical models which capture the main concepts of double field theory on para-Hermitian manifolds. We show that the geometric theory of Lagrangian and Hamiltonian dynamical systems is an instance of para-Kahler geometry which extends to a natural example of a Born geometry. The corresponding phase space geometry belongs to the family of natural almost para-Kahler structures which we construct explicitly as deformations of the canonical para-Kahler structure by non-linear connections. We extend this framework to a class of non-Lagrangian dynamical systems which naturally encodes the notion of fluxes in para-Hermitian geometry. In this case we describe the emergence of fluxes in terms of weak integrability defined by the D-bracket, and we extend the construction to arbitrary cotangent bundles where we reproduce the standard generalized fluxes of double field theory. We also describe the para-Hermitian geometry of Drinfel'd doubles, which gives an explicit illustration of the interplay between fluxes, D-brackets and different polarizations. The left-invariant para-Hermitian structure on a Drinfel'd double in a Manin triple polarization descends to a doubled twisted torus, which we use to illustrate how changes of polarizations give rise to different fluxes and string backgrounds in para-Hermitian geometry.