Batalin--Vilkovisky quantization and supersymmetric twists

We show that a family of topological twists of a supersymmetric mechanics with a K"ahler target exhibits a Batalin–Vilkovisky quantization. Using this observation we make a general proposal for the Hilbert space of states after a topological twist in terms of the cohomology of a certain perverse sheaf. We give several examples of the resulting Hilbert spaces including the categorified Donaldson–Thomas invariants, Haydys–Witten theory and the 3-dimensional A-model.