Affine Lie Algebras, String Junctions and 7-Branes

TL;DR

The paper demonstrates a brane-based realization of affine ADE Lie algebras in Type IIB/F-Theory by encoding affine roots as string junctions on configurations of 7-branes. The imaginary root $\boldsymbol{\delta}$ is realized as a loop around the branes, and the representation level $k$ is fixed by asymptotic charges $(p,q)$, yielding an affine inner product through the junction intersection form. This framework reproduces the full affine structure, shows how decoupling a single brane reduces to a finite $ADE$ algebra, and provides explicit constructions for $\widehat{su(2)}$ and $\widehat{E_8}$, including spectrum constraints and basis choices for junctions. The results bridge geometric brane configurations with the representation theory of affine algebras and have implications for the spectra of 4D $\mathcal{N}=2$ theories arising on D3-branes near 7-branes and for dual heterotic/non-critical string constructions.

Abstract

We consider the realization of affine ADE Lie algebras as string junctions on mutually non-local 7-branes in Type IIB string theory. The existence of the affine algebra is signaled by the presence of the imaginary root junction ``delta'', which is realized as a string encircling the 7-brane configuration. The level k of an affine representation partially constrains the asymptotic (p,q) charges of string junctions departing the configuration. The junction intersection form reproduces the full affine inner product, plus terms in the asymptotic charges.