Seiberg-Witten Geometry of Four-Dimensional $\mathcal N=2$ Quiver Gauge Theories

TL;DR

This paper explores the geometric structure of four-dimensional $ =2$ ADE quiver gauge theories, linking their vacua to moduli spaces of holomorphic maps and integrable systems, revealing deep connections with instantons, monopoles, and Hitchin systems.

Contribution

It provides a detailed geometric and integrable systems description of the vacua of mass deformed $ =2$ superconformal ADE quiver gauge theories, connecting gauge theory to moduli spaces and integrable models.

Findings

Identified the vacuum moduli space with moduli of holomorphic maps to bundle moduli spaces.

Linked the special geometry of vacua to integrable systems such as Hitchin systems and spin chains.

Clarified the role of instantons, monopoles, and higher-dimensional theories in the geometric framework.

Abstract

Seiberg-Witten geometry of mass deformed $\mathcal N=2$ superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space $\mathfrak M$ of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space ${\rm Bun}_{\mathbf G} (\mathcal E)$ of holomorphic $G^{\mathbb C}$-bundles on a (possibly degenerate) elliptic curve $\mathcal E$ defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group $G$. The integrable systems $\mathfrak P$ underlying the special geometry of $\mathfrak M$ are identified. The moduli spaces of framed $G$-instantons on ${\mathbb R}^{2} \times {\mathbb T}^{2}$, of $G$-monopoles with singularities on ${\mathbb R}^{2} \times {\mathbb S}^{1}$, the Hitchin systems on curves with punctures, as well as various spin chains play an important r\^ole in our story. We also comment on the higher-dimensional theories.