Exceptional groups from open strings

TL;DR

This paper addresses the absence of exceptional gauge groups in perturbative open-string constructions by introducing multi-pronged open strings (string junctions) that can end on multiple 7-branes in type IIB/F-theory on S^2. By incorporating junctions and brane-crossing dynamics, the authors construct explicit open-string representations for E6, E7, and E8 and show how their products reproduce the correct adjoint content and generate the additional generators needed for these algebras. The framework leverages SL(2,ℤ) monodromies and specific nonlocal 7-brane configurations (A,B,C branes) to realize the exceptional Lie algebras, with detailed geodesic and junction pictures confirming charge assignments and multiplication rules. The results suggest a nonperturbative open-string picture in which string junctions provide a natural language for BPS states and gauge structures beyond perturbative limits, while leaving open questions about selection rules and the full moduli-space dependence of these states.

Abstract

We consider type IIB theory compactified on a two-sphere in the presence of mutually nonlocal 7-branes. The BPS states associated with the gauge vectors of exceptional groups are seen to arise from open strings connecting the 7-branes, and multi-pronged open strings capable of ending on more than two 7-branes. These multi-pronged strings are built from open string junctions that arise naturally when strings cross 7-branes. The different string configurations can be multiplied as traditional open strings, and are shown to generate the structure of exceptional groups.

Figures (9)

• Figure 1: An $({r\atop s})$ string is shown crossing in the anti-clockwise direction the branch cut associated to a $[p,q]$ 7-brane. The outgoing string is the $M_{p,q} \cdot ({r\atop s})$ string.
• Figure 2: A 1-brane is created as the string crosses the 7-brane.
• Figure 3: Five versions of the $AA$ geodesic which transforms as $({\vcenter{}}{\vcenter{}}{\vcenter{}}{\vcenter{}}^2_a, {\vcenter{}}{\vcenter{}}{\vcenter{}}{\vcenter{}} )$ of $su(n_A) \times su(n_C)$.
• Figure 4: Different versions of the indirect $CC$ geodesic which transforms as $({\vcenter{}}{\vcenter{}}{\vcenter{}}{\vcenter{}}^4_a, \cdot )$ of $su(n_A) \times su(n_C)$.
• Figure 5: The left figure is a modified version of Fig. 4 (c). A couple of moves relates it to the right figure which is recognized as equivalent to Fig. 4 (a).