Quantisation of derived Lagrangians

Pridham, Jonathan P

doi:10.2140/gt.2022.26.2405

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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 $ℒ$ 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 $𝒪_{Y}$ , equipped with a curved $A_{\infty}$ –morphism to the ring of differential operators on $ℒ$ ; for line bundles on smooth Lagrangian subvarieties of smooth symplectic algebraic varieties, this simplifies to deforming $(ℒ, 𝒪_{Y})$ 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 nondegenerate quantisations to power series with coefficients in relative cohomology groups of the respective de Rham complexes. When $ℒ$ 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.

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Keywords

deformation quantisation, derived algebraic geometry, Lagrangians

Mathematical Subject Classification 2010

Primary: 14A22, 14D23, 53D55

References

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

Volume 26, issue 6 (2022)