E_8 flavour multiplets

TL;DR

The work addresses how exceptional symmetries, notably $E_8$, emerge in string-theoretic brane setups by analyzing open strings on D7-branes arranged in $SO(8)\times SO(8)$ blocks and employing branch cuts and $SL(2,\mathbb{Z})$ dualities. It shows that $E_8$ can be viewed as the SL(2,Z) completion of perturbative $SO(16)$ and constructs the full adjoint, vector, and spinor content via explicit string configurations, including decoupling procedures that reduce spacetime dimensions from nine to eight to seven and relate to D8/D7 pictures. The $E_8$ flavour multiplet on seven D7-branes is built by combining perturbative quarks with $E_8$-adjoint configurations, interpreted as bound states of two dyons, with consistent dimensional reductions to five and four dimensions. Overall, the paper provides a concrete, brane-based mechanism for exceptional symmetry enhancement and a unified view linking perturbative open strings, non-perturbative dyons, and dual Type I' descriptions, while matching established results in the literature.

Abstract

We analyze gauge symmetry enhancements $SO(16)\to E_8$ on eight D7-branes and $SO(14)\times U(1)\to E_8$ on seven D7-branes from open strings. String configurations which we present in this paper are closely related to the ones given by Gaberdiel and Zwiebach. Our construction is based on $SO(8)\times SO(8)$ decomposition and its relation to the D8-brane case via T-duality is clearer. Then we study supersymmetric Yang-Mills theory on D3-brane near the D7-branes. This theory has flavour symmetry group which is equal to the gauge group on D7-branes. We suggest that when this symmetry is enhanced, two dyons make bound states which, together with elementary quarks, constitute an $E_8$ multiplet.

Figures (15)

• Figure 1: (a) One block contains four D7-branes (blobs) and an orientifold plane (small cross at the center). Branch cuts, which gives transformation $(p,q)\rightarrow (p-q,q)$, stretched between each D7-brane and the orientifold plane (solid lines). Furthermore, a branch cut reversing the string orientation goes from the orientifold plane to infinity (broken line). (b) Quantum effects split an orientifold plane into two 7-branes. We call them B- and C-brane, and the monodromies along paths $C_1$ and $C_2$ are $(p,q)\rightarrow(p,p+q)$ and $(p,q)\rightarrow(3p-4q,p-q)$, respectively.
• Figure 2: Open strings belonging to adjoint representation of $SO(8)$. The blobs and the small cross represent D7-branes and an orientifold plane, respectively. A and B should be distinguished. C is projected out by orientifold projection.
• Figure 3: String configurations belonging to vector representation of $SO(8)$
• Figure 4: From string configurations in ${\bf8}_v$ representation, configurations belonging to ${\bf8}_s$ (b) and ${\bf8}_c$ (c) representations can be constructed.
• Figure 5: Configurations which have self intersections which are not able to be removed by continuous deformation should be forbidden.