Saddle point equations in Seiberg-Witten theory

TL;DR

This work develops a unified framework to extract Seiberg-Witten curves and meromorphic differentials for $\mathcal{N}=2$ Yang-Mills theories with classical gauge groups and diverse matter, by analyzing the instanton partition function through localization and the ADHM/universal-bundle formalism. The authors reduce the problem to a thermodynamic saddle-point problem for a density $\\rho(x)$ or profile $f(x)$, whose minimization yields the Seiberg-Witten curve via a resolvent $F(z)$ and the differential $\\lambda = (1/2\\pi i) z \, dy/y$. They demonstrate that the resulting 1-instanton corrections, computed from the curves, agree with known direct calculations across models, and they provide explicit hyperelliptic approximations and mapping relations between different theories. The work connects instanton counting, SW geometry, and hyperelliptic truncations into a coherent method, enabling systematic determination of low-energy effective actions for a broad class of $\\mathcal{N}=2$ gauge theories.

Abstract

N=2 supersymmetric Yang-Mills theories for all classical gauge groups, that is, for SU(N), SO(N), and Sp(N) is considered. The equations which define the Seiberg-Witten curve are proposed. In some cases they are solved. It is shown that for (almost) all models allowed by the asymptotic freedom the 1-instanton corrections which follows from these equations agree with the direct computations and with known results.

Figures (4)

• Figure 1: Conformal map for $SU(3)$, $N_f=2$
• Figure 2: Basic cycles
• Figure 3: Conformal map for $Sp(1)$, $N_f=1$
• Figure 4: Conformal map for $SU(2)$, and matter in symmetric and antisymmetric representation