String Junctions for Arbitrary Lie Algebra Representations

TL;DR

This paper develops a Lie-algebraic framework for string junctions on IIB 7-brane backgrounds realizing ADE algebras. It defines invariant charges for junctions, constructs a canonical junction lattice via basis strings, and introduces a a bilinear intersection form that identifies ADE roots with self-intersection $-2$ junctions, including Jordan open-string realizations. By extending the basis with asymptotic $(p,q)$ charges, the authors build an extended weight lattice and derive explicit maps between junctions and Dynkin labels for $A_n$, $D_n$, and $E_n$; they also classify junctions corresponding to fundamental representations through self-intersection $-1$. The results connect the junction lattice to the Lie-algebra weight lattice, clarifying how conjugacy classes map to subsets of asymptotic charges and providing explicit constructions for exceptional algebras $E_6,E_7,E_8$. This provides a Lie-theoretic underpinning of brane-based gauge enhancements and a systematic way to realize arbitrary representations via string junctions, with potential links to BPS spectra and geometric engineering.

Abstract

We consider string junctions with endpoints on a set of branes of IIB string theory defining an ADE-type gauge Lie algebra. We show how to characterize uniquely equivalence classes of junctions related by string/brane crossing through invariant charges that count the effective number of prongs ending on each brane. Each equivalence class defines a point on a lattice of junctions. We define a metric on this lattice arising from the intersection pairing of junctions, and use self-intersection to identify junctions in the adjoint and fundamental representations of all ADE algebras. This information suffices to determine the relation between junction lattices and the Lie-algebra weight lattices. Arbitrary representations are built by allowing junctions with asymptotic (p,q) charges, on which the group of conjugacy classes of representations is represented additively. One can view the (p,q) asymptotic charges as Dynkin labels associated to two new fundamental weight vectors.

Figures (17)

• Figure 1: The rule indicating how the charge of a string changes upon crossing counterclockwise the cut of a brane it cannot end on.
• Figure 2: (a) A generic junction with an arbitrary presentation. (b) Removing the crossing of branch cuts at the expense of prongs and a possibly complicated graph shown shaded. (c) Shrinking the graph. (d) Canonical presentation.
• Figure 3: (a) A junction obtained by adding an $\hbox{\boldmath $\bf a$}$ string to a $\hbox{\boldmath $\bf b$}$ string. (b) The junction $(\hbox{\boldmath $\bf a$} - \hbox{\boldmath $\bf a$}')$ has no asymptotic charge and can be thought as a string starting on the $A$ brane and ending on the $A'$ brane.
• Figure 4: Deriving the intersection number associated to crossing strings by using the known self-intersection of a three-string junction.
• Figure 5: Dynkin diagram for $su(n)$ and associated brane configuration with junctions representing the simple roots.