Supersymmetry, Path Integration, and the Atiyah-Singer Index Theorem

TL;DR

This paper presents a novel supersymmetric proof of the Atiyah-Singer index theorem using path integral methods, explicitly computing the measure and propagator, and confirming the scalar curvature factor in the quantum Hamiltonian.

Contribution

It introduces a new supersymmetric proof employing path integrals, exact measure computation, and detailed analysis of the quantum system related to the index theorem.

Findings

Exact computation of path integral measure and Feynman propagator

Confirmation of scalar curvature factor in the Schrödinger equation

Agreement between loop and heat kernel expansions

Abstract

A new supersymmetric proof of the Atiyah-Singer index theorem is presented. The Peierls bracket quantization scheme is used to quantize the supersymmetric classical system corresponding to the index problem for the twisted Dirac operator. The problem of factor ordering is addressed and the unique quantum system that is relevant to the index theorem is analyzed in detail. The Hamiltonian operator is shown to include a scalar curvature factor, $\hbar^2R/8$. The path integral formulation of quantum mechanics is then used to obtain a formula for the index. For the first time, the path integral "measure" and the Feynman propagator of the system are exactly computed. The derivation of the index formula relies solely on the definition of a Gaussian superdeterminant. The two-loop analysis of the path integral is also carried out. The results of the loop and heat kernel expansions of the path interal are in complete agreement. This confirms the existence of the scalar curvature factor in the Schr\"odinger equation and validates the supersymmetric proof of the index theorem. Many other related issues are addressed. Finally, reviews of the index theorem and the supersymmetric quantum mechanics are presented.