Abstract / Description of output
We investigate quantisations of line bundles L on derived Lagrangians X over 0shifted symplectic derived Artin Nstacks Y. In our derived setting, a deformation quantisation consists of a curved A∞ deformation of the structure sheaf OY, equipped with a curved A∞ 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,OY) to a DQ module over a DQ algebroid.
For each choice of formality isomorphism between the E2 and P2 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 antiinvolutive 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 nshifted symplectic derived stacks.
For each choice of formality isomorphism between the E2 and P2 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 antiinvolutive 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 nshifted symplectic derived stacks.
Original language  English 

Number of pages  61 
Journal  Geometry & Topology 
Publication status  Accepted/In press  16 Jul 2021 
Keywords / Materials (for Nontextual outputs)
 math.AG
 math.QA
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Jon Pridham
 School of Mathematics  Personal Chair of Derived Algebraic Geometry
Person: Academic: Research Active