Quantisation of derived Poisson structures

TL;DR

This paper proves that all 0-shifted Poisson structures on derived Artin n-stacks with perfect cotangent complexes can be quantized using curved A-infinity deformations, extending to smooth and analytic contexts.

Contribution

It introduces a new approach to quantization that leverages anti-involution properties of the Hochschild complex, differing from traditional invariance-based methods.

Findings

Quantization exists for 0-shifted Poisson structures on derived Artin n-stacks.

The method applies to LCI schemes, providing DQ algebroid quantizations.

Analogous quantization results are established in smooth and analytic settings.

Abstract

We prove that every $0$-shifted Poisson structure on a derived Artin $n$-stack admits a curved $A_{\infty}$ deformation quantisation whenever the stack has perfect cotangent complex; in particular, this applies to LCI schemes, where it gives a DQ algebroid quantisation. Whereas the Kontsevich--Tamarkin approach to quantisation for smooth varieties hinges on invariance of the Hochschild complex under affine transformations, we instead exploit the observation that the Hochschild complex carries an anti-involution, and that such anti-involutive deformations of the complex of polyvectors are essentially unique. We also establish analogous statements for deformation quantisations in $\mathcal{C}^{\infty}$ and analytic settings.