Quantisation of derived Lagrangians

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∞ 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 non-degenerate 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.