Genera from an algebraic index theorem for supermanifolds

TL;DR

This paper proves a super-version of the algebraic index theorem, connecting cobordism invariants across index theory, deformation theory, and quantum field theory, with applications to supersymmetric quantum mechanics.

Contribution

It introduces a super-analogue of Nest-Tsygan's algebraic index theorem, linking cobordism invariants in multiple mathematical and physical contexts.

Findings

Recovered cobordism invariants in supersymmetric quantum mechanics

Established a super-version of the algebraic index theorem

Connected index theory, deformation theory, and quantum field theory

Abstract

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.