Supersymmetric Quantum Mechanics of Magnetic Monopoles: A Case Study

TL;DR

This paper analyzes the supersymmetric quantum mechanics of charge-(1,1) monopoles in N=2 supersymmetric Yang-Mills-Higgs theory, demonstrating the equivalence of formalisms, deriving supercharges, and showing the absence of bound states with scattering states and cross sections.

Contribution

It provides a detailed comparison of two formalisms for monopole quantum mechanics, derives explicit supercharges, and investigates the bound state spectrum.

Findings

No quantum bound states of charge-(1,1) monopoles in the moduli space approximation.

Explicit expressions for supercharges as differential operators.

Computed scattering states and differential cross sections.

Abstract

We study, in detail, the supersymmetric quantum mechanics of charge-(1,1) monopoles in N=2 supersymmetric Yang-Mills-Higgs theory with gauge group SU(3) spontaneously broken to U(1) x U(1). We use the moduli space approximation of the quantised dynamics, which can be expressed in two equivalent formalisms: either one describes quantum states by Dirac spinors on the moduli space, in which case the Hamiltonian is the square of the Dirac operator, or one works with anti-holomorphic forms on the moduli space, in which case the Hamiltonian is the Laplacian acting on forms. We review the derivation of both formalisms, explicitly exhibit their equivalence and derive general expressions for the supercharges as differential operators in both formalisms. We propose a general expression for the total angular momentum operator as a differential operator, and check its commutation relations with the supercharges. Using the known metric structure of the moduli space of charge-(1,1) monopoles we show that there are no quantum bound states of such monopoles in the moduli space approximation. We exhibit scattering states and compute corresponding differential cross sections.