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

Seiberg–Witten geometry of mass deformed 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 spaceMof vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space BunG(E) of holomorphic GC-bundles on a (possibly degenerate) elliptic curve E defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group G. The integrable systems P underlying the special geometry of M are identified. The moduli spaces of framed Ginstantons on R2 × T2, of G-monopoles with singularities on R2 × S1, the Hitchin systems on curves with punctures, as well as various spin chains play an important rˆole in our story. We also comment on the higher-dimensional theories.