Non-Abelian Berry Phases and BPS Monopoles

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.

15 pages. v2: footnotes added to point the reader towards later developments where conjectures made in this paper were subsequently proven. A shortened version of this paper was published in PRL under the title "Scheme for Building a 't Hooft-Polyakov Monopole"

References in corpus (9)

• Anomalous Hall Effect and Magnetic Monopoles in Momentum-Space
• The Moduli Space of Many BPS Monopoles for Arbitrary Gauge Groups
• Supersymmetry and the cohomology of (hyper)Kaehler manifolds
• The Geometric Phase in Supersymmetric Quantum Mechanics
• The Berry Phase of D0-Branes
• Berry Phase and Supersymmetry
• Non-abelian Berry's phase and Chern numbers in higher spin pairing condensates
• The Geometric Phase and Gravitational Precession of D-Branes
• Tunneling D0-branes

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• N=4, 3D Supersymmetric Quantum Mechanics in Non-Abelian Monopole Background
• Black Hole Berry Phase
• Non-Abelian Monopole in the Parameter Space of Point-like Interactions
• BPS Monopole in the Space of Boundary Conditions
• Models for the BPS Berry Connection
• Berry's connection, Kähler geometry and the Nahm construction of monopoles
• Rotor Spectra and Berry Phases in the Chiral Limit of QCD on a Torus
• Tunneling D0-branes
• Non-Abelian Berry Phase, Instantons and N=(0,4) Supersymmetry
• Supersymmetry and non-Abelian geometric phase for a free particle on a circle with point-like interactions
• Extended supersymmetry and its applications in quantum mechanical models associated with self-dual gauge fields
• Connection Constraints from Non-Abelian Supersymmetric Quantum Mechanics