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# Advances in Theoretical and Mathematical Physics

## Volume 7 (2003)

### Number 3

### Seiberg-Witten Curve for *E*-String Theory Revisited

Pages: 419 – 455

DOI: http://dx.doi.org/10.4310/ATMP.2003.v7.n3.a3

#### Authors

#### 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}R} 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} 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 *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.