Non-Abelian Berry Phases and BPS Monopoles

TL;DR

This paper links non-Abelian Berry phases in a quantum mechanical model to SU(2) monopoles, showing how adjusting potentials reveals BPS monopoles with underlying supersymmetry and potential broader implications for supersymmetric sigma models.

Contribution

It demonstrates that the non-Abelian Berry phase corresponds to a 't Hooft-Polyakov monopole and introduces a BPS monopole with supersymmetry, connecting Berry phases to monopole solutions.

Findings

Berry phase equals the path-ordered exponential of a monopole.

Adjusting the potential yields a BPS monopole satisfying Bogomolnyi equations.

Hidden supersymmetry underlies the BPS monopole structure.

Abstract

We study a simple quantum mechanical model of a spinning particle moving on a sphere in the presence of a magnetic field. The system has two ground states. As the magnetic field is varied, the ground states mix through a non-Abelian Berry phase. We show that this Berry phase is the path ordered exponential of the smooth SU(2) 't Hooft-Polyakov monopole. We further show that, by adjusting a potential on the sphere, the monopole becomes BPS and obeys the Bogomolnyi equations. For this choice of potential, it turns out that there is a hidden supersymmetry underlying the system and the Bogomolnyi equations are analogous to the tt* equations of Cecotti and Vafa. We conjecture that the Bogomolnyi equations also govern the Berry phase of N=(2,2) supersymmetric sigma models with other target spaces.