Path Integral Methods in Index Theorems

TL;DR

This paper introduces path integral methods from quantum mechanics to prove key index theorems in geometry, making complex concepts accessible to advanced students with basic physics and geometry background.

Contribution

It provides a pedagogical approach to applying quantum path integrals to index theorems, bridging physics and differential geometry for learners.

Findings

Demonstrates the use of path integrals in proving the Gauss-Bonnet-Chern theorem

Explains the application of stationary phase and localization in index theorems

Connects supersymmetry concepts with geometric index results

Abstract

This paper provides a pedagogical introduction to the quantum mechanical path integral and its use in proving index theorems in geometry, specifically the Gauss-Bonnet-Chern theorem and Lefschetz fixed point theorem. It also touches on some other important concepts in mathematical physics, such as that of stationary phase, supersymmetry and localization. It is aimed at advanced undergraduates and beginning graduates, with no previous knowledge beyond undergraduate quantum mechanics assumed. The necessary mathematical background in differential geometry is reviewed, though a familiarity with this material is undoubtedly helpful.