Para‐Hermitian Geometry, Dualities and Generalized Flux Backgrounds. Issue 3 (20th November 2018) - Marotta, Vincenzo E.; Szabo, Richard J.
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‐Kähler geometry which extends to a natural example of a Born geometry. The corresponding phase space geometry belongs to the family of natural almost para‐Kähler structures which we construct explicitly as deformations of the canonical para‐Kähler 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. Abstract : A survey is given on physical models which capture the main concepts of double field theory on para‐Hermitian manifolds. It is shown that the geometric theory of Lagrangian and Hamiltonian dynamical systems is an instance of para‐Kähler geometry which extends to a natural example of a Born geometry. The corresponding phase space geometry belongs to the family of natural almost para‐Kähler structures which are constructed explicitly as deformations of the canonical para‐Kähler structure by non‐linear connections. This framework will be extended to a class of non‐Lagrangian dynamical systems which naturally encodes the notion of fluxes in para‐Hermitian geometry. In this case the emergence of fluxes is described in terms of weak integrability defined by the D‐bracket. This construction is extended to arbitrary cotangent bundles allowing to reproduce the standard generalized fluxes of double field theory. The authors 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 are used to illustrate how changes of polarizations give rise to different fluxes and string backgrounds in para‐Hermitian geometry.