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

TL;DR

The paper generalizes the 5-brane web construction by incorporating 7-branes, enabling an explicit realization of the affine symmetry and allowing all E_n theories up to E8 to be realized in this framework. It develops a 7-brane/string-junction approach that preserves supersymmetry, converts certain 5-5 states into 5-7 junctions, and connects the global symmetry to affine extensions while linking BPS spectra to del Pezzo geometry. By deriving a mutual-intersection constraint and establishing a dictionary between del Pezzo curves and brane junctions, it provides explicit BPS spectra for the entire E_n series and clarifies how symmetry enhances at the fixed point. The work unifies brane dynamics with geometric methods, offering a robust toolkit for analyzing 5D superconformal field theories with exceptional symmetry and their BPS content.

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.

Figures (8)

• Figure 1: a. The 5-brane web corresponding to the $E_1$ theory. b. The 5-brane web corresponding to the $E_2$ theory, with horizontal external leg (D5-brane) generating a flavor.
• Figure 2: A D7-brane added to the $E_1$ configuration, adding a flavor and producing $E_2$.
• Figure 3: a. The external 5-branes of the $E_1$ web can end on 7-branes. b. The 7-branes move within the 5-brane face, removing the external 5-branes entirely.
• Figure 4: The W-boson (top) and instanton (bottom) can be continuously deformed from being 5-5 stings to 5-7 string junctions.
• Figure 5: a. The toric skeleton for $\,{\hbox{P}\!\!\!\!\!\hbox{I}\,\,\,}^2$, which gives the $E_0$ theory; the slopes are the degenerating cycles. This is identical to the corresponding $(p,q)$ 5-brane web. b. The toric skeleton for ${\cal B}_1$, $\,{\hbox{P}\!\!\!\!\!\hbox{I}\,\,\,}^2$ blown up at a generic point, here chosen to be on the boundary.