## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 7, Number 3 (2003), 419-455.

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

Tohru Eguchi and Kazuhiro Sakai

#### 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.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 7, Number 3 (2003), 419-455.

**Dates**

First available in Project Euclid: 4 April 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1112627374

**Mathematical Reviews number (MathSciNet)**

MR2030056

**Zentralblatt MATH identifier**

1065.81105

#### Citation

Eguchi, Tohru; Sakai, Kazuhiro. Seiberg-Witten Curve for E -String Theory Revisited. Adv. Theor. Math. Phys. 7 (2003), no. 3, 419--455. https://projecteuclid.org/euclid.atmp/1112627374