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

TL;DR

This paper systematically derives elliptic curves for 7-brane configurations related to affine Lie algebras and demonstrates their connection to five-dimensional $E_n$ theories compactified on a circle, supporting the D3 probe picture.

Contribution

It provides a systematic derivation of elliptic curves for affine 7-brane configurations and links them to 5D $E_n$ theories on $S^1$, confirming recent web constructions.

Findings

Elliptic curves for $\\wh E_n$ 7-branes are derived from cubic equations.

The $\\wh E_n$ 7-branes describe the discriminant locus of elliptic curves for 5D $E_n$ theories.

Supports the D3 probe picture for 5D $E_n$ theories on $\\bR^4 \\times S^1$.

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.