Discrete Nahm equations for SU(N) hyperbolic monopoles

In a paper of Braam and Austin, SU(2) magnetic monopoles in hyperbolic space H3 with half-integer mass and maximal symmetry breaking, 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 SU(N) hyperbolic monopoles of integer or half-integer mass, and maximal symmetry breaking. 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 U(1) fields at the asymptotic boundary of H3.