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.
Highlights
For n > 0, existence of quantisations of n-shifted Poisson structures is a formality, following from the equivalence En+1 Pn+1 of operads
Beyond the setting of smooth Deligne–Mumford stacks, unshifted symplectic structures only arise on objects incorporating both stacky and derived structures, as non-degeneracy of the symplectic form implies that the cotangent complex must have both positive and negative terms
Examples of such symplectic derived stacks include the derived moduli stack of perfect complexes on an algebraic K 3 surface, or the derived moduli stack of locally constant G-torsors on a compact oriented topological surface, for an algebraic group G equipped with a Killing form on its Lie algebra
Summary
For n > 0, existence of quantisations of n-shifted Poisson structures is a formality, following from the equivalence En+1 Pn+1 of operads. In order to adapt these constructions to 0-shifted symplectic structures, we replace polyvectors or differential operators with the Hochschild complex CCR(X ) of a derived Artin stack X , defined in terms of a resolution by stacky CDGAs (commutative bidifferential bigraded algebras). Since this has an E2-algebra structure, a choice w of Levi decomposition for the Grothendieck–Teichmüller group gives it a Deformation quantisation for unshifted symplectic. I would like to thank the anonymous referee for many helpful comments
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