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

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 $\mathrm{Bun}_{\mathbf{G}}(\mathcal{E})$ 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 $\mathfrak{B}$ 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 rôle in our story. We also comment on the higher-dimensional theories.