BPS Monopole in the Space of Boundary Conditions

TL;DR

This paper explores the non-Abelian Berry connections in a quantum system with boundary conditions, revealing a connection to BPS monopoles and higher-derivative supersymmetry, thus linking quantum boundary conditions to gauge theory monopoles.

Contribution

It demonstrates that certain boundary conditions in quantum mechanics induce non-Abelian Berry connections equivalent to BPS monopoles, a novel link between boundary conditions and gauge theory monopoles.

Findings

Ground-state Berry connection is a BPS monopole.

Energy levels become doubly-degenerate under specific boundary conditions.

Excited states relate to non-BPS monopoles.

Abstract

The space of all possible boundary conditions that respect self-adjointness of Hamiltonian operator is known to be given by the group manifold $U(2)$ in one-dimensional quantum mechanics. In this paper we study non-Abelian Berry's connections in the space of boundary conditions in a simple quantum mechanical system. We consider a system for a free spinless particle on a circle with two point-like interactions described by the $U(2) \times U(2)$ family of boundary conditions. We show that, for a certain $SU(2) \subset U(2) \times U(2)$ subfamily of boundary conditions, all the energy levels become doubly-degenerate thanks to the so-called higher-derivative supersymmetry, and non-Abelian Berry's connection in the ground-state sector is given by the Bogomolny-Prasad-Sommerfield (BPS) monopole of $SU(2)$ Yang-Mills-Higgs theory. We also show that, in the ground-state sector of this quantum mechanical model, matrix elements of position operator give the adjoint Higgs field that satisfies the BPS equation. It is also discussed that Berry's connections in the excited-state sectors are given by non-BPS 't Hooft-Polyakov monopoles.