Supersymmetric Quantum Mechanics and Topology

Muhammad Abdul Wasay

doi:10.1155/2016/3906746

Abstract

Supersymmetric quantum mechanical models are computed by the path integral approach. In theβ→0limit, 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.

Highlights

Supersymmetry is a quantum mechanical space-time symmetry which induces transformations between bosons and fermions
Advances in High Energy Physics higher dimensional systems and systems with many particles to implement such ideas to problems in different branches of physics, for example, condensed matter physics, atomic physics, and statistical physics [24,25,26,27,28,29]. Another interesting application is [30], in which the low energy dynamics of kmonopoles in N = 2 supersymmetric Yang-Mills theory are determined by N = 4 supersymmetric quantum mechanics based on the moduli space of k static monopole solutions
For a manifold admitting spin structure, we study a more refined model which yields the index of Dirac operator. Both the Euler characteristic of a manifold M and the index of Dirac operator are the Witten indices of the appropriate supersymmetric quantum mechanical systems

Summary

Introduction

Supersymmetry is a quantum mechanical space-time symmetry which induces transformations between bosons and fermions. Advances in High Energy Physics higher dimensional systems and systems with many particles to implement such ideas to problems in different branches of physics, for example, condensed matter physics, atomic physics, and statistical physics [24,25,26,27,28,29] Another interesting application is [30], in which the low energy dynamics of kmonopoles in N = 2 supersymmetric Yang-Mills theory are determined by N = 4 supersymmetric quantum mechanics based on the moduli space of k static monopole solutions. For a manifold admitting spin structure, we study a more refined model which yields the index of Dirac operator Both the Euler characteristic of a manifold M and the index of Dirac operator are the Witten indices of the appropriate supersymmetric quantum mechanical systems.

Calculus of Grassmannian Variables

Gaussian Integrals

Supersymmetric Sigma Models

Summary and Conclusion