Berry Phase and Supersymmetry

TL;DR

This paper explores how supersymmetry constrains the Berry connection in quantum mechanics, linking it to well-known equations like tt* and monopole equations depending on the type of multiplets involved.

Contribution

It establishes a connection between supersymmetry constraints and specific geometric equations for the Berry connection, providing new insights into supersymmetric quantum systems.

Findings

Berry connection satisfies tt* equations for chiral multiplets

Berry connection satisfies monopole equations for vector multiplets

Results apply to supersymmetric quantum mechanics

Abstract

We study the constraints of supersymmetry on the non-Abelian holonomy given by U=P exp(i\int A), the path-ordered exponential of a connection A. For theories with four supercharges, we show that A satisfies the tt* equations if it is a function of chiral multiplets. In contrast, when A is a function of vector multiplets, it satisfies the Bogomolnyi monopole equations. We describe applications of these results to the Berry connection in supersymmetric quantum mechanics.