Quantisation of derived Lagrangians

Abstract

We investigate quantisations of line bundles $\mathcal{L}$ on derived Lagrangians $X$ over $0$-shifted symplectic derived Artin $N$-stacks $Y$. In our derived setting, a deformation quantisation consists of a curved $A_{\infty }$ deformation of the structure sheaf ${\mathcal{O}}_Y$, equipped with a curved $A_{\infty }$ morphism to the ring of differential operators on $\mathcal{L}$; for line bundles on smooth Lagrangian subvarieties of smooth symplectic algebraic varieties, this simplifies to deforming $(\mathcal{L},{\mathcal{O}}_Y)$ to a DQ module over a DQ algebroid. For each choice of formality isomorphism between the $E_2$ and $P_2$ operads, we construct a map from the space of non-degenerate quantisations to power series with coefficients in relative cohomology groups of the respective de Rham complexes. When $\mathcal{L}$ is a square root of the dualising line bundle, this leads to an equivalence between even power series and certain anti-involutive quantisations, ensuring that the deformation quantisations always exist for such line bundles. This gives rise to a dg category of algebraic Lagrangians, an algebraic Fukaya category of the form envisaged by Behrend and Fantechi. We also sketch a generalisation of these quantisation results to Lagrangians on higher $n$-shifted symplectic derived stacks.

61 pp; v2 minor additions and refs updated; v3 expanded generally, with some new material in final section; v4 further expanded, final version to appear in G&T

References in corpus (8)

• Deformation theory via differential graded Lie algebras
• Derived coisotropic structures II: stacks and quantization
• Derived coisotropic structures I: affine case
• Braces and Poisson additivity
• Quantisation of derived Poisson structures
• Quantization of line bundles on Lagrangian subvarieties
• A note on étale atlases for Artin stacks and Lie groupoids, Poisson structures and quantisation
• Microlocal analysis and beyond

Cited by in corpus (6)

• Derived coisotropic structures II: stacks and quantization
• Derived coisotropic structures I: affine case
• Shifted Coisotropic Correspondences
• Quantisation of derived Poisson structures
• A note on étale atlases for Artin stacks and Lie groupoids, Poisson structures and quantisation
• A differential graded model for derived analytic geometry