Integrable systems and supersymmetric gauge theory

TL;DR

This paper links integrable systems, specifically the periodic Toda lattice, to the geometry of N=2 supersymmetric Yang-Mills theory, revealing how spectral curves encode physical data like BPS states and pre-potentials.

Contribution

It demonstrates that the Seiberg-Witten curves for N=2 Yang-Mills theories are spectral curves of the Toda lattice associated with the dual group, connecting gauge theory and integrable systems.

Findings

Spectral curves correspond to Toda lattice for dual groups.

Seiberg-Witten differential expressed in Toda variables.

Pre-potential as free energy of a topological field theory.

Abstract

After the work of Seiberg and Witten, it has been seen that the dynamics of N=2 Yang-Mills theory is governed by a Riemann surface $\Sigma$. In particular, the integral of a special differential $\lambda_{SW}$ over (a subset of) the periods of $\Sigma$ gives the mass formula for BPS-saturated states. We show that, for each simple group $G$, the Riemann surface is a spectral curve of the periodic Toda lattice for the dual group, $G^\vee$, whose affine Dynkin diagram is the dual of that of $G$. This curve is not unique, rather it depends on the choice of a representation $\rho$ of $G^\vee$; however, different choices of $\rho$ lead to equivalent constructions. The Seiberg-Witten differential $\lambda_{SW}$ is naturally expressed in Toda variables, and the N=2 Yang-Mills pre-potential is the free energy of a topological field theory defined by the data $\Sigma_{\gg,\rho}$ and $\lambda_{SW}$.