Quantization of (-1)-Shifted Derived Poisson Manifolds

TL;DR

This paper studies the quantization of (-1)-shifted derived Poisson manifolds, linking it to Maurer-Cartan elements and Poisson cohomology, and provides conditions and canonical methods for quantization.

Contribution

It establishes a criterion for quantizability based on second Poisson cohomology and constructs canonical quantizations for certain derived Poisson structures.

Findings

Quantization is equivalent to lifting Maurer-Cartan elements in dg Lie algebras.

A (-1)-shifted derived Poisson manifold is quantizable if its second Poisson cohomology vanishes.

Canonical quantizations are constructed for linear (-1)-shifted derived Poisson manifolds and derived intersections of coisotropic submanifolds.

Abstract

We investigate the quantization problem of $(-1)$-shifted derived Poisson manifolds in terms of $\BV_\infty$-operators on the space of Berezinian half-densities. We prove that quantizing such a $(-1)$-shifted derived Poisson manifold is equivalent to the lifting of a consecutive sequences of Maurer-Cartan elements of short exact sequences of differential graded Lie algebras, where the obstruction is a certain class in the second Poisson cohomology. Consequently, a $(-1)$-shifted derived Poisson manifold is quantizable if the second Poisson cohomology group vanishes. We also prove that for any $\L$-algebroid $\Cc{\aV}$, its corresponding linear $(-1)$-shifted derived Poisson manifold $\Cc{\aV}^\vee[-1]$ admits a canonical quantization. Finally, given a Lie algebroid $A$ and a one-cocycle $s\in \sections{A^\vee}$, the $(-1)$-shifted derived Poisson manifold corresponding to the derived intersection of coisotropic submanifolds determined by the graph of $s$ and the zero section of the Lie Poisson $A^\vee$ is shown to admit a canonical quantization in terms of Evens-Lu-Weinstein module.