Deformation quantisation for unshifted symplectic structures on derived Artin stacks

TL;DR

This paper develops a new method to quantize unshifted symplectic structures on derived Artin stacks, establishing a correspondence with de Rham cohomology and providing canonical quantizations that generalize known results for smooth varieties.

Contribution

It introduces a novel approach to deformation quantization for derived Artin stacks, connecting DQ algebroid quantizations with de Rham cohomology and extending Kontsevich--Tamarkin quantization.

Findings

Constructed a map from DQ algebroid quantizations to de Rham power series.

Established an equivalence between even power series and involutive quantizations.

Provided a canonical quantization for every symplectic structure on derived Artin stacks.

Abstract

We prove that every $0$-shifted symplectic structure on a derived Artin $n$-stack admits a curved $A_{\infty}$ deformation quantisation. The classical method of quantising smooth varieties via quantisations of affine space does not apply in this setting, so we develop a new approach. We construct a map from DQ algebroid quantisations of unshifted symplectic structures on a derived Artin $n$-stack to power series in de Rham cohomology, depending only on a choice of Drinfeld associator. This gives an equivalence between even power series and certain involutive quantisations, which yield anti-involutive curved $A_{\infty}$ deformations of the dg category of perfect complexes. In particular, there is a canonical quantisation associated to every symplectic structure on such a stack, which agrees for smooth varieties with the Kontsevich--Tamarkin quantisation for even associators.