Seiberg-Witten Curve for E-String Theory Revisited
Tohru Eguchi, Kazuhiro Sakai
TL;DR
The paper reexamines the Seiberg-Witten curve for the $E$-string theory, elucidating its affine $E_8$ structure as realized on a rational elliptic surface and clarifying the geometric role of the Wilson line parameters. It develops holomorphic sections for the $E_n$ and $\widehat{E}_8$ curves, interprets Wilson lines as intersection data on the elliptic fiber, and explores degenerations that preserve unbroken subgroups of $E_8$. A key result is a novel reduction to four dimensions that preserves $SL(2,\mathbb{Z})$, yielding the $\mathcal{N}=2$ $SU(2)$ theory with $N_f=4$ (and revealing $SO(8)$ triality) and a Donagi–Witten-type curve directly from the six-dimensional framework. Together, these findings illuminate the modular structure and cross-dimensional connections of $E$-string geometry, providing concrete tools for analyzing lower-dimensional theories within a unified geometric setting.
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 τand 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.