Quantized higher derived bracket construction for Courant algebroids

Abstract: The present talk is a logical continuation of the talk I gave two weeks ago, however I will keep it self-contained (as long as you can believe in usefulness of Lie algebroids and Lie bialgebroids, aka Courant algebroids).

For a Lagrangian subbundle L in the Courant algebroid $T+T^*$, the exterior algebroid $\bigwedge L$ acts naturally on the de Rham forms $\Omega$. Whenever a transverse Lagrangian $M$ is chosen, we can embed the (graded) algebra of $L$-forms $\Omega_L$ into the (graded) algebra of (graded) differential operators on $\Omega$. There is also a distinguished differential operator on $\Omega$ – the de Rham differential! This setting allow us to apply (a modification of) the Voronov’s higher derived bracket method, by which we construct a natural family of brackets on $\Omega_L$. The bracket have arity 0,1,2 and 3 respectively, and are compatible in appropriate sense (that is, they form an $L_\infty$ algebra on $\Omega_L$). In the case, when both $L$ and $M$ are Dirac structures, the $L_\infty$ structure specializes to the differential graded Lie algebra controlling the deformation theory of $L$. Finally, we will discuss how does the constructed family of brackets on $\Omega_L$ depend on the choice of complementary Lagrangian $M$.