The Ruijsenaars-Schneider Model in the Context of Seiberg-Witten Theory

TL;DR

This paper establishes a comprehensive Seiberg–Witten framework for five-dimensional $N=2$ SUSY YM compactified on a circle, showing that the resulting four-dimensional theory corresponds to the elliptic Ruijsenaars–Schneider integrable system with two key parameters $(R,ε)$. It derives the exact prepotential and spectral curves for $SU(N)$, including detailed SU(2) and SU(N) analyses, by exploiting Lax representations, determinant formulas, and free-fermion techniques, and demonstrates how the elliptic RS model degenerates to trigonometric RS and to rational curves in the perturbative limit. A central result is the explicit perturbative prepotential and its residue-form expression, together with special simplifications at $ε=π/2$ that connect to known results (e.g., BMMM1) and provide a bridge between different SEIBERG–Witten realizations. The work also outlines how these constructions extend to general $SU(N)$, highlights the role of Weierstrass and theta-function machinery, and discusses open questions about six-dimensional lift and spin-chain interpretations, with implications for understanding UV finiteness and dualities in higher-dimensional gauge theories.

Abstract

The compactification of five dimensional N=2 SUSY Yang-Mills (YM) theory onto a circle provides a four dimensional YM model with N=4 SUSY. This supersymmetry can be broken down to N=2 if non-trivial boundary conditions in the compact dimension, φ(x_5 +R) = e^{2πiε}φ(x_5), are imposed on half of the fields. This two-parameter (R,ε) family of compactifications includes as particular limits most of the previously studied four dimensional N=2 SUSY YM models with supermultiplets in the adjoint representation of the gauge group. The finite-dimensional integrable system associated to these theories via the Seiberg-Witten construction is the generic elliptic Ruijsenaars-Schneider model. In particular the perturbative (weak coupling) limit is described by the trigonometric Ruijsenaars-Schneider model.