Inozemtsev System as Seiberg-Witten Integrable system

TL;DR

This paper demonstrates that the Inozemtsev integrable system precisely encodes the Coulomb branch physics of certain 4d $ ext{USp}(2N)$ gauge theories, establishing a detailed correspondence between spectral data and field theory parameters.

Contribution

It explicitly maps the spectral curves and differentials of the Inozemtsev system to Seiberg-Witten curves for specific gauge theories, revealing a new link between integrable systems and supersymmetric gauge theories.

Findings

Explicit transformation for N=1 and N=2 cases

Matching of elliptic curve moduli to gauge couplings

Identification of system couplings with mass parameters

Abstract

In this work we establish that the Inozemtsev system is the Seiberg-Witten integrable system encoding the Coulomb branch physics of 4d $\mathcal{N}=2$ USp(2N) gauge theory with four fundamental and (for $N \geq 2$) one antisymmetric tensor hypermultiplets. We describe the transformation from the spectral curves and canonical one-form of the Inozemtsev system in the $N=1$ and $N=2$ cases to the Seiberg-Witten curves and differentials explicitly, along with the explicit matching of the modulus of the elliptic curve of spectral parameters to the gauge coupling of the field theory, and of the couplings of the Inozemtsev system to the field theory mass parameters. This result is a particular instance of a more general correspondence between crystallographic elliptic Calogero-Moser systems with Seiberg-Witten integrable systems, which will be explored in future work.