Seiberg-Witten Curve for E-String Theory Revisited

Abstract

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 E-string describes the low-energy dynamics of a six-dimensional (1,0) SUSY theory when compactified on R⁴R} X T². It has a manifest affine E₈ global symmetry with modulus tau and E₈ Wilson line parameters m_i, i = 1,2, ... ,8 which are associated with the geometry of the rational elliptic surface. When the radii R₅, R₆ of the torus T² degenerate R₅, R₆ go to 0, E-string curve is reduced to the known Seiberg-Witten curves of four- and 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 T² 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.