String Junctions for Arbitrary Lie Algebra Representations

TL;DR

This paper develops a geometric framework using string junctions in IIB string theory to classify and analyze arbitrary Lie algebra representations, linking junctions to weight lattices and Dynkin labels.

Contribution

It introduces a method to characterize junction equivalence classes with invariant charges, defining a lattice and metric that relate to Lie algebra weight lattices, including arbitrary representations.

Findings

Junction equivalence classes are uniquely characterized by invariant charges.

A lattice with a metric derived from junction intersection pairing is constructed.

The framework relates junctions with asymptotic charges to Dynkin labels and fundamental weights.

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.