The cotangent bundle $T^*X$ of a smooth space $X$ quantises (e.g. in the deformation quantisation sense) to the sheaf $D_X$ of differential operators on $X$.
What is the analogous quantisation of the shifted cotangent bundle $T^*[n]X$?
P.S. Deformation quantisation concerns quantising Poisson algebras to associative algebras. I'm not sure what the correct deformation problem is for $n$-shifted Poisson algebras, and this subquestion is implicitly part of the above.
