Seiberg-Witten theory, monopole spectral curves and affine Toda solitons

TL;DR

This paper explores the connection between Seiberg-Witten theory, monopole spectral curves, and affine Toda solitons, revealing how monopole solutions relate to integrable systems through geometric and algebraic structures.

Contribution

It explicitly links monopole spectral curves to affine Toda solitons, providing a geometric derivation and explicit construction of monopole solutions via Toda lattice theory.

Findings

Monopole spectral curves can be obtained from symmetric n-monopole curves via quotienting.

The relation between monopoles and Toda lattice is made explicit through the ADHMN construction.

Two SU(2) monopole solutions are generated by affine Toda soliton solutions.

Abstract

Using Seiberg-Witten theory it is known that the dynamics of N=2 supersymmetric SU(n) Yang-Mills theory is determined by a Riemann surface. In particular the mass formula for BPS states is given by the periods of a special differential on this surface. In this note we point out that the surface can be obtained from the quotient of a symmetric n-monopole spectral curve by its symmetry group. Known results about the Seiberg-Witten curves then implies that these monopoles are related to the Toda lattice. We make this relation explicit via the ADHMN construction. Furthermore, in the simplest case, that of two SU(2) monopoles, we find that the general two monopole solution is generated by an affine Toda soliton solution of the imaginary coupled theory.