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

TL;DR

This paper introduces a new BV operator generating higher Koszul brackets on differential forms, establishing its properties and placing it within a quantization framework related to $L_{}$-bialgebroids.

Contribution

The authors define a novel BV-type operator $ riangle_P$ on (pseudo)differential forms, analyze its properties, and embed it into a one-parameter family of operators representing a quantization of cotangent $L_{}$-bialgebroids.

Findings

Defined a new BV operator $ riangle_P$ generating higher Koszul brackets.

Established the operator's properties and its inclusion in a quantization family.

Developed a theory of formal $$-differential operators with grading structures.

Abstract

We introduce a formal $\hbar$-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 $\hbar$-differential operators, which can be understood as a quantization of the cotangent $L_{\infty}$-bialgebroid. We obtain symmetric description on both $\Pi TM$ and $\Pi T^*M$. For the purpose of the above, we develop in detail a theory of formal $\hbar$-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.