TY - JOUR

T1 - Quantization of (- 1)-Shifted Derived Poisson Manifolds

AU - Behrend, Kai

AU - Peddie, Matt

AU - Xu, Ping

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/9

Y1 - 2023/9

N2 - 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.

AB - 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.

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U2 - 10.1007/s00220-023-04762-1

DO - 10.1007/s00220-023-04762-1

M3 - Article

AN - SCOPUS:85165895903

SN - 0010-3616

VL - 402

SP - 2301

EP - 2338

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 3

ER -