Generalised Kinematics for Double Field Theory

Understand

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.