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arXiv · 1506.08736

Discrete Nahm Equations for SU(N) Hyperbolic Monopoles

Joseph Y C Chan
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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 $H^{3}$.

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Joseph Y C Chan. 2015-06-29. Discrete Nahm Equations for SU(N) Hyperbolic Monopoles. https://arxiv.org/abs/1506.08736

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