Quantisation of derived Lagrangians

We investigate quantisations of line bundles $\mathcal{L}$ on derived Lagrangians $X$ over $0$-shifted symplectic Artin $N$-stacks $Y$. In our setting, a deformation quantisation consists curved $A_{\infty}$ the structure sheaf $\mathcal{O}_{Y}$, equipped with morphism to ring differential operators $\mathcal{L}$; for smooth Lagrangian subvarieties algebraic varieties, this simplifies deforming $(\mathcal{L}, \mathcal{O}_{Y})$ DQ module algebroid. For each choice formality isomorphism between $E_2$ and $P_2$ operads, we construct map from space non-degenerate power series coefficients in relative cohomology groups respective de Rham complexes. When is square root dualising bundle, leads an equivalence even certain anti-involutive quantisations, ensuring that always exist such bundles. This gives rise dg category Lagrangians, Fukaya form envisaged by Behrend Fantechi. also sketch generalisation these results higher $n$-shifted stacks.