Deformation quantisation for unshifted symplectic structures on derived Artin stacks

• J. P. Pridham ^[1]
  1. [1] University of Edinburgh

     University of Edinburgh

     Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 4, 2018, págs. 3027-3059
• Idioma: inglés
• DOI: 10.1007/s00029-018-0414-2
• Enlaces
  • Texto completo (pdf)
• Resumen

  • We prove that every 0-shifted symplectic structure on a derived Artin n-stack admits a curved A∞ 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∞ 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.