Supercharges, Quantum States and Angular Momentum for N=4 Supersymmetric Monopoles

TL;DR

This paper develops a quantum mechanical model for N=4 supersymmetric monopoles, extending previous N=2 results, and derives explicit expressions for angular momentum and supercharges on the moduli space.

Contribution

It introduces a detailed model for N=4 monopole quantum mechanics, highlighting the doubling of fermionic zero-modes and formulating the angular momentum operator with complex structures.

Findings

Quantum states are represented by differential forms on the moduli space.

Derived a general expression for the total angular momentum operator.

Expressed supercharges using twisted Dolbeault operators.

Abstract

We revisit the moduli space approximation to the quantum mechanics of monopoles in N=4 supersymmetric Yang-Mills-Higgs theory with maximal symmetry breaking. Starting with the observation that the set of fermionic zero-modes in N=4 supersymmetric Yang-Mills-Higgs theory can be viewed as two copies of the set of fermionic zero-modes in the N=2 version, we build a model to describe the quantum mechanics of N=4 supersymmetric monopoles, based on our previous paper [1] on the N=2 case, in which this doubling of fermionic zero-modes is manifest throughout. Our final picture extends the familiar result that quantum states are described by differential forms on the moduli space and that the Hamiltonian operator is the Laplacian acting on forms. In particular, we derive a general expression for the total angular momentum operator on the moduli space which differs from the naive candidate by the adjoint action of the complex structures. We also express all the supercharges in terms of (twisted) Dolbeault operators and illustrate our results by discussing, in some detail, the N=4 supersymmetric quantum dynamics of monopoles in a theory with gauge group SU(3) broken to U(1) x U(1).