Derivation of Index theorem by Supersymmetry

TL;DR

This paper demonstrates specific cases of the Atiyah-Singer index theorem using supersymmetric sigma-models, employing technical and geometric methods to handle complex determinant calculations.

Contribution

It provides a detailed derivation of index theorem cases via supersymmetry, introducing novel calculation techniques and geometric insights.

Findings

Successful derivation of special Atiyah-Singer index theorem cases

Development of new methods for determinant calculations in supersymmetric models

Use of geometric arguments to simplify complex spin field computations

Abstract

The present paper gives calculations in detail to prove several special cases of Atiyah-Singer theorem through supersymmetric $\sigma$-models. Some technical tricks are employed to calculate the determinants of fluctuation operators of the path integrals. An intuitive and geometric argument is applied to overcome the complicated calculation on spin fields twisted by gauge fields.