Instanton effects and Witten complex in supersymmetric quantum mechanics on SO(4)

TL;DR

This paper investigates instanton effects in supersymmetric quantum mechanics on SO(4), connecting vacuums and confirming the relation to de Rham cohomology, while providing criteria for instanton effect cancellations.

Contribution

It explicitly constructs instanton solutions on SO(4), analyzes their impact on vacuums, and relates findings to topological properties, advancing understanding of supersymmetric quantum mechanics.

Findings

Instanton solutions connect approximate vacuums on SO(4)

Hessian matrices determine true vacuums consistent with cohomology

Criteria established for cancellation of instanton effects

Abstract

We examine supersymmetric quantum mechanics on $SO(4)$ to realize Witten's idea. We find instanton solutions connecting approximate vacuums. We calculate Hessian matrices for these solutions to determine true vacuums. Our result is in agreement with de Rham cohomology of $SO(4)$. We also give a criterion for cancellation of instanton effects for a pair of instanton paths.