Seiberg-Witten Curve for E-String Theory Revisited

TL;DR

This paper revisits the Seiberg-Witten curve for E-string theory, exploring its geometric properties, symmetry degenerations, and novel reduction methods to connect with lower-dimensional gauge theories.

Contribution

It provides a detailed geometric analysis of the E-string Seiberg-Witten curve, including new reduction techniques that preserve SL(2,Z) symmetry and links to known four-dimensional theories.

Findings

Identified the geometric significance of Wilson line parameters.

Studied degenerations corresponding to unbroken symmetry groups.

Developed a new reduction method preserving SL(2,Z) symmetry.

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 R^4 x T^2. It has a manifest affine E_8 global symmetry with modulus \tau and E_8 Wilson line parameters {m_i},i=1,2,...,8 which are associated with the geometry of the rational elliptic surface. When the radii R_5,R_6 of the torus T^2 degenerate R_5,R_6 --> 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 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.