Uncovering Infinite Symmetries on [p,q] 7-branes: Kac-Moody Algebras and Beyond

TL;DR

The paper investigates infinite-dimensional algebras realized on type IIB [p,q] 7-branes by classifying configurations through SL(2,Z) conjugacy classes of brane monodromies. It develops a junction-based framework in which affine algebras arise at $f_K(p,q)=1$, with the imaginary root $\boldsymbol{\delta}$ encoding loop structures, and extends to hyperbolic and indefinite algebras when $f_K(p,q)>1$, including $\hat{E}_9$ and $E_{10}$. The authors connect these algebras to F-theory compactifications on $K3$ and the ${\cal B}_9$ surface via junction lattices that mirror the $E_8$-based homology lattices, revealing rich correspondences between BPS spectra and brane geometry. They outline a systematic classification by SL(2,Z) conjugacy, discuss the appearance of affine semisimple and double-loop structures, and propose a framework for exploring the spectrum of D3-brane probes, while leaving open questions about full classification and physical realization.

Abstract

In a previous paper we explored how conjugacy classes of the modular group classify the symmetry algebras that arise on type IIB [p,q] 7-branes. The Kodaira list of finite Lie algebras completely fills the elliptic classes as well as some parabolic classes. Loop algebras of E_N fill additional parabolic classes, and exotic finite algebras, hyperbolic extensions of E_N and more general indefinite Lie algebras fill the hyperbolic classes. Since they correspond to brane configurations that cannot be made into strict singularities, these non-Kodaira algebras are spectrum generating and organize towers of massive BPS states into representations. The smallest brane configuration with unit monodromy gives rise to the loop algebra \hat{E}_9 which plays a central role in the theory. We elucidate the patterns of enhancement relating E_8, E_9, \hat{E}_9 and E_10. We examine configurations of 24 7-branes relevant to type IIB compactifications on a two-sphere, or F-theory on K3. A particularly symmetric configuration separates the 7-branes into two groups of twelve branes and the massive BPS spectrum is organized by E_10 + E_10.

Figures (9)

• Figure 1: The imaginary root junction as a loop surrounding the enhanced configuration.
• Figure 2: A junction on an affine configuration.
• Figure 3: The enhancement of the Dynkin diagram of $E_3$ to that of $\widehat{E}_3$.
• Figure 4: ${\bf \widehat{E}_9}$ viewed as an enhancement of ${\bf E_8}$ (grey oval) by branes with charges $[-1,0]$ and $[3,1]$. Both imaginary roots are presented as loops with certain charges.
• Figure 5: Calculating the intersection form for $\widehat{E}_9$.