Seiberg-Witten Curve for the E-String Theory

TL;DR

This work constructs a six-dimensional Seiberg--Witten curve for the $E$-string theory as an elliptic fibration over ${f P}^1$, with coefficient functions expressed in terms of affine $E_8$ characters up to level 6. Using a mirror-type transformation, it links the curve to the holomorphic (Jacobi) partition data $Z_n$ for torus-compactified strings, and imposes the holomorphic anomaly and gap conditions to determine the $Z_n$ recursively, obtaining results up to $Z_8$. The six-dimensional curve reduces to known five- and four-dimensional Seiberg--Witten curves in the appropriate limits, thereby providing a master framework that unifies the SW descriptions across dimensions via $E_8$ representation content. The construction also ties the BPS spectrum to holomorphic curve counts in the Calabi--Yau ${1\over 2}K_3$ and the toroidal ${\cal N}=4$ $U(n)$ Yang--Mills partition functions on ${1\over 2}K_3$, highlighting deep links between string/M-theory, algebraic geometry, and supersymmetric gauge dynamics.

Abstract

We construct the Seiberg-Witten curve for the E-string theory in six-dimensions. The curve is expressed in terms of affine E_8 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.

Figures (1)

• Figure 1: Dynkin diagram for $E_8$: numbers attached to nodes denote the levels of representations and the numbers in parentheses show their labels.