Quantization of (-1)-Shifted Derived Poisson Manifolds

We investigate the quantization problem of $(-1)$-shifted derived Poisson manifolds in terms of ${\text{BV}}_{\mathrm{\infty }}$-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 $\text{L}$-algebroid $\text{Cc}\text{aV}$, its corresponding linear $(-1)$-shifted derived Poisson manifold $\text{Cc}{\text{aV}}^{\vee }[-1]$ admits a canonical quantization. Finally, given a Lie algebroid $A$ and a one-cocycle $s\in \text{sections}{A}^{\vee }$, 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.