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 non-degenerate 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.
Categories: math.AG math.QA