Uncovering the Symmetries on [p,q] 7-branes: Beyond the Kodaira Classification

TL;DR

The paper addresses how symmetry algebras arise on [p,q] 7-brane configurations in type IIB/F-theory, extending beyond Kodaira singularities. It develops a unified framework where the brane monodromy $K\in SL(2,\mathbb{Z})$ and the invariant $\ell$ classify the realized algebras via the asymptotic charge form $f(p,q)$, derived from $K$, and shows how adding branes transitions algebras according to $f(\mathbf z)$. By constructing two-brane kernels and mapping $K$ to binary quadratic forms, the authors reproduce the Kodaira series and generate finite, affine, and eventually indefinite algebras, organizing the landscape by $SL(2,\mathbb{Z})$ conjugacy classes. They provide explicit mechanisms for algebra enhancement and demonstrate equivalences among several $E_N$, $D_N$, and Argyres-Douglas series, while outlining a sequel addressing infinite-dimensional algebras. The work thus connects modular monodromy, brane topology, and Lie algebra representations to map the full spectrum of 7-brane backgrounds relevant for D3-brane probes and associated field theories.

Abstract

We begin a classification of the symmetry algebras arising on configurations of type IIB [p,q] 7-branes. These include not just the Kodaira symmetries that occur when branes coalesce into a singularity, but also algebras associated to other physically interesting brane configurations that cannot be collapsed. We demonstrate how the monodromy around the 7-branes essentially determines the algebra, and thus 7-brane gauge symmetries are classified by conjugacy classes of the modular group SL(2,Z). Through a classic map between the modular group and binary quadratic forms, the monodromy fixes the asymptotic charge form which determines the representations of the various (p,q) dyons in probe D3-brane theories. This quadratic form also controls the change in the algebra during transitions between different brane configurations. We give a unified description of the brane configurations extending the D_N, E_N and Argyres-Douglas H_N series beyond the Kodaira cases. We anticipate the appearance of affine and indefinite infinite-dimensional algebras, which we explore in a sequel paper.

Figures (4)

• Figure 1: A singlet junction as a loop with an asymptotic string of charge $\hbox{\boldmath $\bf z$}$.
• Figure 2: (a) Brane configuration of the ${\bf H}$, ${\bf D}$, ${\bf E}$ and the generalized ${\bf S_p}$ series. (b) Dynkin diagrams of the algebras $A_N$, $D_N$, $E_N$ and $T_{p+1,2,N-1-p}$.
• Figure 3: (a) Brane configuration of ${\cal G} = T_{r_1,r_2,r_3}$. (b) Dynkin diagram of ${\cal G} = T_{r_1,r_2,r_3}$.
• Figure 4: The enhanced configuration ${\bf G}^{enh}$ is obtained by adding a ${\bf Z_{[p,q]}}$-brane to the original configuration $\hbox{\boldmath $\bf G$}$. A generic junction has support on the new brane.