On a Batalin--Vilkovisky operator generating higher Koszul brackets on differential forms

Abstract

We introduce a formal $\hslash$-differential operator $\Delta$ that generates higher Koszul brackets on the algebra of (pseudo)differential forms on a $P_{\infty }$-manifold. Such an operator was first mentioned by Khudaverdian and Voronov in \texttt{arXiv:1808.10049}. (This operator is an analogue of the Koszul--Brylinski boundary operator ${\partial }_P$ which defines Poisson homology for an ordinary Poisson structure.) Here we introduce $\Delta ={\Delta }_P$ by a different method and establish its properties. We show that this BV type operator generating higher Koszul brackets can be included in a one-parameter family of BV type formal $\hslash$-differential operators, which can be understood as a quantization of the cotangent $L_{\infty }$-bialgebroid. We obtain symmetric description on both $\Pi TM$ and $\Pi T^{\ast }M$. For the purpose of the above, we develop in detail a theory of formal $\hslash$-differential operators and also of operators acting on densities on dual vector bundles. In particular, we have a statement about operators that can be seen as a quantization of the Mackenzie--Xu canonical diffeomorphism. Another interesting feature is that we are able to introduce a grading, not a filtration, on our algebras of operators. When operators act on objects on vector bundles, we obtain a bi-grading.

References in corpus (2)

• $Q$-Manifolds and Mackenzie Theory
• Q-manifolds and Mackenzie theory: an overview