We investigate quantisations of line bundles
on derived
Lagrangians
over
–shifted symplectic
derived Artin
–stacks
.
In our derived setting, a deformation quantisation consists of a curved
– deformation of the structure
sheaf
, equipped with a curved
–morphism to the ring of
differential operators on
;
for line bundles on smooth Lagrangian subvarieties of smooth
symplectic algebraic varieties, this simplifies to deforming
to a
DQ module over a DQ algebroid.
For each choice of formality isomorphism between the
– and
–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
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
–shifted
symplectic derived stacks.
