### 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.

Original language | English |
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Article number | 1800093 |

Journal | Fortschritte der Physik |

Early online date | 20 Nov 2018 |

DOIs | |

Publication status | E-pub ahead of print - 20 Nov 2018 |

### ASJC Scopus subject areas

- Physics and Astronomy(all)

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## Cite this

*Fortschritte der Physik*, [1800093]. https://doi.org/10.1002/prop.201800093