Quantization of (- 1)-Shifted Derived Poisson Manifolds

We investigate the quantization problem of (- 1) -shifted derived Poisson manifolds in terms of BV _∞ -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 sequence of Maurer–Cartan elements, each obtained from a short exact sequence of differential graded Lie algebras. At each step, 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 A , its corresponding linear (- 1) -shifted derived Poisson manifold A^∨[- 1] admits a canonical quantization. Finally, given a Lie algebroid A and a one-cocycle s∈ Γ (A^∨) , 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^∨ is shown to admit a canonical quantization in terms of Evens–Lu–Weinstein module.