Deformation quantisation for (-1)-shifted symplectic structures and vanishing cycles

Abstract

We formulate a notion of $E_0$ quantisation of $(-1)$-Poisson structures on derived Artin $N$-stacks, and construct a map from $E_0$ quantisations of $(-1)$-shifted symplectic structures to power series in de Rham cohomology. For a square root of the dualising line bundle, this gives an equivalence between even power series and self-dual quantisations. In particular, there is a canonical quantisation of any such square root, which localises to recover the perverse sheaf of vanishing cycles on derived DM stacks, thus giving a form of derived categorification of Donaldson--Thomas invariants.

34 pp; v2 Artin details added, some material moved to arXiv:1504.01940; v3 several additions and corrections, notably in 1.2.1; v4 some details added and small corrections; v5 minor changes, to appear in Algebraic Geometry

References in corpus (2)

• Symmetries and stabilization for sheaves of vanishing cycles
• Derived Algebraic Geometry and Deformation Quantization

Cited by in corpus (5)

• Derived coisotropic structures II: stacks and quantization
• Calabi-Yau structures on topological Fukaya categories
• Derived stacks in symplectic geometry
• Gaiotto's Lagrangian subvarieties via derived symplectic geometry
• A differential graded model for derived analytic geometry