Seiberg-Witten Curve forE-String Theory

Abstract
We construct the Seiberg-Witten curve for the E-string theory in six-dimensions. The curve is expressed in terms of affine E8 characters up to level 6 and is determined by using the mirror-type transformation so that it reproduces the number of holomorphic curves in the Calabi-Yau manifold and the amplitudes of N = 4 U(n) Yang-Mills theory on (1/2)K3. We also show that our curve flows to known five- and four-dimensional Seiberg-Witten curves in suitable limits. We further find new type of reduction to some particular four-dimensional theories such as the SU(2) Seiberg-Witten theory with 4 flavors, without taking a degenerate limit of T2 so that the SL(2,Z) symmetry is left intact.