Deformation quantisation of exact shifted symplectic structures, with an application to vanishing cycles
J. P. Pridham
TL;DR
This work extends the bridge between shifted symplectic and Poisson geometry by establishing an equivalence between exact $n$-shifted symplectic structures and non-degenerate $n$-shifted Poisson structures equipped with formal derivations, within a broad derived, analytic, and non-commutative framework. It develops a robust deformation-quantisation program, yielding essentially unique self-dual quantisations in many settings and connecting these to classical parametrisations (De Wilde–Lecomte, Fedosov) where appropriate. The main advances include a homotopical Legendre transformation, a precise Maurer–Cartan formalism for exact structures, and an operadic interpretation that unifies strict and curved/hd-derivation quantisations across algebras, Lagrangians, and co-isotropic structures. The vanishing cycles application is a central highlight: canonically quantising the canonical exact $(-1)$-shifted structure on a derived critical locus produces twisted $ ext{D}$-modules whose inverting $oldsymbol{ ext{h}}$ recovers the perverse sheaf of vanishing cycles with its monodromy, aligning with BBDJS and HHR constructions. Overall, the paper provides a principled, associator-independent pathway from shifted symplectic data to quantisation and vanishing cycles, with broad implications for derived geometry, noncommutative analogues, and the geometric Langlands program through perverse sheaves and D-modules.
Abstract
We extend the author's and CPTVV's correspondence between shifted symplectic and Poisson structures to establish a correspondence between exact shifted symplectic structures and non-degenerate shifted Poisson structures with formal derivation, a concept generalising constructions by De Wilde and Lecomte. Our formulation is sufficiently general to encompass derived algebraic, analytic and $\mathcal{C}^{\infty}$ stacks, as well as Lagrangians and non-commutative generalisations. We also show that non-degenerate shifted Poisson structures with formal derivation carry unique self-dual deformation quantisations in any setting where the latter can be formulated. One application is that for (not necessarily exact) $0$-shifted symplectic structures in analytic and $\mathcal{C}^{\infty}$ settings, it follows that the author's earlier parametrisations of quantisations are in fact independent of any choice of associator, and generalise Fedosov's parametrisation of quantisations for classical manifolds. Our main application is to complex $(-1)$-shifted symplectic structures, showing that our unique quantisation of the canonical exact structure, a sheaf of twisted $BD_0$-algebras with derivation, gives rise to BBDJS's perverse sheaf of vanishing cycles, equipped with its monodromy operator.