Nahm's Equations and Rational Maps from $\mathbb{C}\mathrm{P}^1$ to $\mathbb{C}\mathrm{P}^n$

Author:

Max Schult

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), High Energy Physics - Theory (hep-th)

journal:

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date:

2023-10-26 16:00:00

Abstract

We consider Nahm's equations on a bounded open interval with first order poles at the ends. By imposing further boundary conditions, extended moduli spaces are identified with spaces of rational maps from $\mathbb{C}\mathrm{P}^1$ to $\mathbb{C}\mathrm{P}^n$. Having the residues at one end define sums of irreducible representations of $\mathfrak{su}(2)$, the dimensions of those summands correspond to holomorphic charge on the rational map side. Technical difficulties arise due to half powers in the boundary conditions. Further, a symplectomorphic identification with moduli spaces corresponding to spaces of rational maps into complete flag varieties is given.