Discrete Nahm Equations for SU(N) Hyperbolic Monopoles

TL;DR

This paper extends the discrete Nahm equations framework from SU(2) to SU(N) hyperbolic monopoles, establishing a new correspondence between solutions of these equations and monopole configurations in hyperbolic space.

Contribution

It introduces the $(N-1)$-interval discrete Nahm equations and proves their solutions are equivalent to SU(N) hyperbolic monopoles, generalizing prior SU(2) results.

Findings

Established the $(N-1)$-interval discrete Nahm equations for SU(N) monopoles

Proved the correspondence between solutions and hyperbolic monopoles

Showed monopoles are determined by boundary U(1) fields

Abstract

In a paper of Braam and Austin, $\text{SU}(2)$ magnetic monopoles in hyperbolic space $H^{3}$ were shown to be the same as solutions to matrix-valued difference equations called the discrete Nahm equations. Here, I discover the $(N-1)$-interval discrete Nahm equations and show that their solutions are equivalent to $\text{SU}(N)$ hyperbolic monopoles. These discrete time evolution equations on an interval feature a jump in matrix dimensions at certain points in the evolution, which are given by the mass data of the corresponding monopole. I prove the correspondence with higher rank hyperbolic monopoles using localisation and Chern characters. I then prove that the monopole is determined up to gauge transformations by its "holographic image" of $\text{U}(1)$ fields at the asymptotic boundary of $H^{3}$.