Haagerup Symmetry in $(E_8{)}_1$?

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 ${\mathcal{H}}_i\times {\mathcal{H}}_i^{\text{op}}$ 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.

13 pages, 4 figures; v2: references added