BPS Monopole in the Space of Boundary Conditions

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.

13 pages, 4 eepic figures; minor corrections

References in corpus (3)

• Magnetic Monopole Dynamics, Supersymmetry, and Duality
• Non-Abelian Berry Phases and BPS Monopoles
• Non-Abelian Monopole in the Parameter Space of Point-like Interactions

Cited by in corpus (6)

• Extended supersymmetry with central charges in Dirac action with curved extra dimensions
• 5d Dirac fermion on quantum graph
• Instantons and Berry's connections on quantum graph
• Models for the BPS Berry Connection
• Extended supersymmetry with central charges in higher dimensional Dirac action
• Correspondence of topological classification between quantum graph extra dimension and topological matter