# Generalised Kinematics for Double Field Theory

## TL;DR

This paper extends the kinematical framework of Double Field Theory by incorporating a para-Hermitian structure, unifying it with Generalised Geometry and allowing for more general, non-flat backgrounds.

## Contribution

It introduces a new formalism with a canonical connection and generalized Lie derivative on a para-Hermitian manifold, broadening the scope of Double Field Theory.

## Key findings

- Constructed a canonical connection and generalized Lie derivative.
- Derived integrability conditions for symmetry algebra closure.
- Unified Double Field Theory with Generalised Geometry.

## Abstract

We formulate a kinematical extension of Double Field Theory on a $2d$-dimensional para-Hermitian manifold $(\mathcal{P},\eta,\omega)$ where the $O(d,d)$ metric $\eta$ is supplemented by an almost symplectic two-form $\omega$. Together $\eta$ and $\omega$ define an almost bi-Lagrangian structure $K$ which provides a splitting of the tangent bundle $T\mathcal{P}=L\oplus\tilde{L}$ into two Lagrangian subspaces. In this paper a canonical connection and a corresponding generalised Lie derivative for the Leibniz algebroid on $T\mathcal{P}$ are constructed. We find integrability conditions under which the symmetry algebra closes for general $\eta$ and $\omega$, even if they are not flat and constant. This formalism thus provides a generalisation of the kinematical structure of Double Field Theory. We also show that this formalism allows one to reconcile and unify Double Field Theory with Generalised Geometry which is thoroughly discussed.