  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.
