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Quantisation of derived Lagrangians

Jonathan P Pridham

Geometry & Topology 26 (2022) 2405–2489
DOI: 10.2140/gt.2022.26.2405
Abstract

We investigate quantisations of line bundles $\mathsc{ℒ}$ on derived Lagrangians $X$ over $0$–shifted symplectic derived Artin $N$–stacks $Y\phantom{\rule{-0.17em}{0ex}}$. In our derived setting, a deformation quantisation consists of a curved ${A}_{\infty }$– deformation of the structure sheaf ${\mathsc{𝒪}}_{Y}$, equipped with a curved ${A}_{\infty }$–morphism to the ring of differential operators on $\mathsc{ℒ}$; for line bundles on smooth Lagrangian subvarieties of smooth symplectic algebraic varieties, this simplifies to deforming $\left(\mathsc{ℒ},{\mathsc{𝒪}}_{Y}\right)$ 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 nondegenerate quantisations to power series with coefficients in relative cohomology groups of the respective de Rham complexes. When $\mathsc{ℒ}$ 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.

Keywords
deformation quantisation, derived algebraic geometry, Lagrangians
Mathematical Subject Classification 2010
Primary: 14A22, 14D23, 53D55
Publication
Received: 11 September 2018
Revised: 15 June 2021
Accepted: 16 July 2021
Published: 13 December 2022
Proposed: Richard P Thomas
Seconded: Mark Gross, Dan Abramovich
Authors
 Jonathan P Pridham School of Mathematics and Maxwell Institute The University of Edinburgh Edinburgh United Kingdom