Seiberg–Witten Curve for E-String Theory Revisited

We discuss various properties of the Seiberg–Witten curve for the E-string theory which we have obtained recently in hep-th/0203025. Seiberg–Witten curve for the Estring describes the low-energy dynamics of a six-dimensional (1, 0) SUSY theory when compactified on R × T 2. It has a manifest affine E8 global symmetry with modulus τ and E8 Wilson line parameters {mi}, i = 1, 2, . . . , 8 which are associated with the geometry of the rational elliptic surface. When the radii R5, R6 of the torus T 2 degenerate R5, R6 → 0, E-string curve is reduced to the known Seiberg–Witten curves of fourand five-dimensional gauge theories. In this paper we first study the geometry of rational elliptic surface and identify the geometrical significance of the Wilson line parameters. By fine tuning these parameters we also study degenerations of our curve corresponding to various unbroken symmetry groups. We also find a new way of reduction to four-dimensional theories without taking a degenerate limit of T 2 so that the SL(2,Z) symmetry is left intact. By setting some of the Wilson line parameters to special values we obtain the four-dimensional SU(2) Seiberg–Witten theory with 4 flavors and also a curve by Donagi and Witten describing the dynamics of a perturbed N = 4 theory.