Supersymmetry and the Atiyah-Singer Index Theorem I: Peierls Brackets, Green's Functions, and a Supersymmetric Proof of the Index Theorem

TL;DR

This paper applies the Peierls bracket quantization scheme to a supersymmetric system related to the twisted spin index theorem, providing a new SUSY proof through Green's functions and exact Feynman propagator calculations.

Contribution

It introduces a novel supersymmetric proof of the Atiyah-Singer index theorem using Green's functions and Peierls brackets.

Findings

Exact Feynman propagator computed for the supersymmetric system

Green's function method yields a direct derivation of the index formula

Provides a new SUSY-based proof of the index theorem

Abstract

The Peierls bracket quantization scheme is applied to the supersymmetric system corresponding to the twisted spin index theorem. A detailed study of the quantum system is presented, and the Feynman propagator is exactly computed. The Green's function methods provide a direct derivation of the index formula. Note: This is essentially a new SUSY proof of the index theorem.