Five-branes, Seven-branes and Five-dimensional E_n field theories

TL;DR

This paper extends 5-brane web constructions by incorporating 7-branes to realize all E_n theories up to n=8, making their exceptional symmetries manifest and enabling detailed analysis of BPS states.

Contribution

It introduces a novel 7-brane extension to 5-brane webs, allowing the construction of all E_n theories and explicit determination of BPS state quantum numbers.

Findings

Successfully constructed E_n theories up to n=8 with 7-branes.

Made exceptional global symmetries manifest in the brane setup.

Determined BPS state quantum numbers using geometrical constraints.

Abstract

We generalize the (p,q) 5-brane web construction of five-dimensional field theories by introducing (p,q) 7-branes, and apply this construction to theories with a one-dimensional Coulomb branch. The 7-branes render the exceptional global symmetry of these theories manifest. Additionally, 7-branes allow the construction of all E_n theories up to n=8, previously not possible in 5-brane configurations. The exceptional global symmetry in the field theory is a subalgebra of an affine symmetry on the 7-branes, which is necessary for the existence of the system. We explicitly determine the quantum numbers of the BPS states of all E_n theories using two simple geometrical constraints.