Genera from an algebraic index theorem for supermanifolds

We prove a super-version of Nest-Tsygan’s algebraic index theorem. This work is inspired by the appearance of the same cobordism invariants in three related stories: index theory, trace methods in the deformation theory of algebras, and partition functions in quantum field theory. We show that one can recover the cobordism invariant appearing in supersymmetric quantum mechanics using trace methods for the associated deformation quantization problem.