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Deformation quantisation for unshifted symplectic structures 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.

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Correspondence to J. P. Pridham.

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This work was supported by the Engineering and Physical Sciences Research Council (Grant Number EP/I004130/2).

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Pridham, J.P. Deformation quantisation for unshifted symplectic structures on derived Artin stacks. Sel. Math. New Ser. 24, 3027–3059 (2018). https://doi.org/10.1007/s00029-018-0414-2

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Mathematics Subject Classification

  • 14A22
  • 53D55
  • 14D23