English

Deformation quantisation for unshifted symplectic structures on derived Artin stacks

Algebraic Geometry 2018-04-13 v4 Quantum Algebra

Abstract

We prove that every 00-shifted symplectic structure on a derived Artin nn-stack admits a curved AA_{\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 nn-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 AA_{\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.

Keywords

Cite

@article{arxiv.1604.04458,
  title  = {Deformation quantisation for unshifted symplectic structures on derived Artin stacks},
  author = {J. P. Pridham},
  journal= {arXiv preprint arXiv:1604.04458},
  year   = {2018}
}

Comments

27pp.; v2 Propositions 1.23 and 3.10 added; v3 several small additions; v4 several changes following referee's comments, to appear in Selecta

R2 v1 2026-06-22T13:33:14.115Z