Integrability and Seiberg-Witten Theory: Curves and Periods

TL;DR

The article surveys how exact low-energy actions in 4d $N=2$ SUSY YM theories can be recast in terms of 1d integrable systems, with Seiberg–Witten data (spectral curves and period integrals) mapping directly to spectral curves, Lax representations, and action variables of integrable models. It highlights the elliptic Calogero system as a natural UV-finite framework encoding the flow from $N=4$ to $N=2$ and shows how, in various degeneration limits, this system reduces to the Toda chain, reproducing the SW prepotential through Whitham dynamics and the quasiclassical $\tau$-function. The work also discusses how Donagi–Witten polynomials and rational Calogero limits fit into this dictionary, and argues that the observed correspondences are robust evidence for a deep integrable structure underlying 4d SUSY gauge theories. Overall, the paper argues for a unified picture in which Seiberg–Witten theory and integrable systems illuminate each other, with potential extensions to affine algebras and Calabi–Yau geometries.

Abstract

Interpretation of exact results on the low-energy limit of $4d$ $N=2$ SUSY YM in the language of $1d$ integrability theory is reviewed. The case of elliptic Calogero system, associated with the flow between $N=4$ and $N=2$ SUSY in $4d$, is considered in some detail.