Quantisation of derived Poisson structures

Pridham, Jonathan P

doi:10.2140/gt.2025.29.3717

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Quantisation of derived Poisson structures

Jonathan P Pridham

Geometry & Topology 29 (2025) 3717–3771

DOI: 10.2140/gt.2025.29.3717

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 $𝒞^{\infty}$ and analytic settings.

Keywords

deformation quantisation, derived algebraic geometry

Mathematical Subject Classification

Primary: 14A30, 53D55

References

Publication

Received: 5 September 2023

Revised: 7 December 2024

Accepted: 4 January 2025

Published: 10 October 2025

Proposed: Richard P Thomas

Seconded: Jim Bryan, Dan Abramovich

Authors

Jonathan P Pridham
School of Mathematics and Maxwell Institute The University of Edinburgh Edinburgh United Kingdom

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Volume 29, issue 7 (2025)