Affine 7-brane Backgrounds and Five-Dimensional $E_N$ Theories on $S^1$

TL;DR

This work constructs explicit elliptic curves for affine 7-brane backgrounds realizing the affine algebras $\\widehat{E}_N$ and $\\widehat{\\widetilde{E}}_N$ by starting from the rational elliptic surface for $\\widehat{E}_9$ and removing $A$-branes via the Tate algorithm. These curves encode the Coulomb branches of 5D $\\mathcal{N}=1$ $E_N$ theories on $\\mathbb{R}^4\\times S^1$ and reproduce known 5D brane-web results, supporting a D3-brane probe interpretation with BPS states arising from $(p,q)$ strings. The authors also analyze massless (degenerate) curves realizing $E_N$ singularities and study the compactification to 4D, obtaining Seiberg-Witten curves for 4D $\\mathcal{N}=2$ theories with appropriate flavor content. The results illuminate the affine and degeneration structures of 7-brane backgrounds and provide a concrete bridge between 7-brane configurations, elliptic curves, and low-dimensional gauge theories, with potential extensions to junction lattices and dualities.

Abstract

Elliptic curves for the 7-brane configurations realizing the affine Lie algebras $\wh E_n$ $(1 \leq n \leq 8)$ and $\wh{\wt E}_n$ $(n=0,1)$ are systematically derived from the cubic equation for a rational elliptic surface. It is then shown that the $\wh E_n$ 7-branes describe the discriminant locus of the elliptic curves for five-dimensional (5D) N=1 $E_n$ theories compactified on a circle. This is in accordance with a recent construction of 5D N=1 $E_n$ theories on the IIB 5-brane web with 7-branes, and indicates the validity of the D3 probe picture for 5D $E_n$ theories on $\bR^4 \times S^1$. Using the $\wh E_n$ curves we also study the compactification of 5D $E_n$ theories to four dimensions.