Supersymmetric quantum mechanics and topology

TL;DR

This paper explores how supersymmetric quantum mechanics models, computed via path integrals, relate to the topology of the target space, enabling exact index calculations through localization techniques.

Contribution

It demonstrates the connection between supersymmetric quantum mechanics and topological invariants of the target space, providing explicit computations for Euler characteristics and Dirac operator indices.

Findings

Exact index computations via localization

Relationship between supersymmetry and topological invariants

Explicit calculations for Euler characteristics and Dirac indices

Abstract

Supersymmetric quantum mechanical models are computed by the Path integral approach. In the $\beta\rightarrow0$ limit, the integrals localize to the zero modes. This allows us to perform the index computations exactly because of supersymmetric localization, and we will show how the geometry of target space enters the physics of sigma models resulting in the relationship between the supersymmetric model and the geometry of the target space in the form of topological invariants. Explicit computation details are given for the Euler characteristics of the target manifold, and the index of Dirac operator for the model on a spin manifold.