Para-Hermitian Geometry, Dualities and Generalized Flux 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.

arXiv comment68 pages; v2: typos corrected; Final version to be published in Fortschritte der Physik

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• Noncommutative gauge theories on D-branes in non-geometric backgrounds
• Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory
• Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality
• Born sigma model for branes in exceptional geometry
• Global Aspects of Doubled Geometry and Pre-rackoid
• Poisson-Lie T-Duality of WZW Model via Current Algebra Deformation
• Global Double Field Theory is Higher Kaluza-Klein Theory
• Currents, charges and algebras in exceptional generalised geometry
• More on Doubled Aspects of Algebroids in Double Field Theory
• Hyperkähler, Bi-hypercomplex, Generalized Hyperkähler Structures and T-duality
• Complex Structures, T-duality and Worldsheet Instantons in Born Sigma Models
• Double Field Theory and Geometric Quantisation
• T-Dualities and Doubled Geometry of the Principal Chiral Model
• A note on Faddeev--Popov action for doubled-yet-gauged particle and graded Poisson geometry
• Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
• Extended Doubled Structures of Algebroids for Gauged Double Field Theory
• Born geometry via Künneth structures and recursion operators
• Born Lie algebras