Quantisation of derived Lagrangians

Pridham, Jonathan P¹

¹ School of Mathematics and Maxwell Institute, The University of Edinburgh, Edinburgh, United Kingdom

Geometry & topology, Tome 26 (2022) no. 6, pp. 2405-2489.

Résumé

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 nondegenerate 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.

Détail

Classification : 14A22, 14D23, 53D55
Keywords: deformation quantisation, derived algebraic geometry, Lagrangians

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