Haagerup Symmetry in $(E_8)_1$?

TL;DR

This paper explores the potential Haagerup symmetries in the $(E_8)_1$ vertex operator algebra, proposing new symmetry structures, gauging procedures, and connections to known models and modular bootstrap results.

Contribution

It introduces the possibility of Haagerup symmetry in $(E_8)_1$, analyzes symmetry gauging, and links these to known conformal embeddings and modular bootstrap findings.

Findings

$(E_8)_1$ may have Haagerup symmetry $ ext{Haagerup}_i$ for $i=1,2,3$.

Gauging diagonal symmetries yields models with $ ext{Fib} imes ext{Fib}^ ext{op}$ and $(G_2)_1 imes (F_4)_1$ symmetries.

Connections to theories with $ ext{Haagerup}_3$ symmetry at $c=2,6$ are suggested.

Abstract

We suggest that the chiral $(\mathfrak{e}_8)_1$ theory -- in many senses the simplest VOA -- may have Haagerup symmetry $\mathcal{H}_i$ for $i=1,2,3$. Likewise, we suggest that the non-chiral $(E_8)_1$ WZW model may have $\mathcal{H}_i \times \mathcal{H}_i^\textrm{op}$ symmetry, and that gauging the diagonal symmetry gives a $c=8$ theory with $\mathcal{Z}(\mathcal{H}_3)$ symmetry, which is the theory predicted in \cite{Evans:2010yr}. Along the way, we show that $(E_8)_1$ also has a $\mathrm{Fib} \times \mathrm{Fib}^\text{op}$ symmetry, and that gauging the diagonal symmetry gives the $(G_2)_1 \times (F_4)_1$ WZW model, explaining the well-known conformal embedding $(G_2)_1 \times (F_4)_1 \subset (E_8)_1$. Finally, we suggest a relation to theories with $\mathcal{H}_3$ symmetry at $c=2,6$, complimenting the discussion with new modular bootstrap results.