Supersymmetry and the Atiyah-Singer Index Theorem II: The Scalar Curvature Factor in the Schr{\" o}dinger Equation

TL;DR

This paper investigates the role of scalar curvature in the Schrödinger equation within the context of supersymmetry and the Atiyah-Singer index theorem, confirming the curvature term through path integral and loop expansion analysis.

Contribution

It demonstrates the emergence of the scalar curvature factor in the Hamiltonian from a supersymmetric quantization approach, providing a detailed loop expansion validation.

Findings

Scalar curvature term appears in the Hamiltonian as $rac{ mi ext{hbar}^2 R}{8}$

Path integral analysis confirms the scalar curvature contribution up to 2-loop order

Comparison with heat kernel expansion supports the existence of the curvature term

Abstract

The quantization of the superclassical system used in the proof of the index theorem results in a factor of $\hbar^{2}R/8 $ in the Hamiltonian. The path integral expression of the kernel is analyzed up to and including 2-loop order. The existence of the scalar curvature term is confirmed by comparing the linear term in the heat kernel expansion with the 2-loop order terms in the loop expansion.