Quantisation of derived Poisson structures

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.

48 pp; v5 overhauled, with many additional details and several simplified arguments; v6 further details added, final version to appear in G&T