Haagerup Symmetry in $(E_8)_1$?

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.

💡 Research Summary

The paper proposes that the level‑one E₈ Wess–Zumino–Witten (WZW) model, both in its chiral (e₈)₁ vertex operator algebra (VOA) form and in the non‑chiral (E₈)₁ theory, carries non‑invertible symmetries described by the Haagerup fusion categories H₁, H₂ and H₃ and their opposite categories. By examining well‑known conformal embeddings—first (E₇)₁ × (A₁)₁ ⊂ (E₈)₁ and then (E₆)₁ × (A₂)₁ ⊂ (E₈)₁— the authors illustrate how ordinary Verlinde lines (which commute with the full chiral algebra) can be “lifted’’ to non‑invertible topological lines that do not commute with the chiral algebra. In the (E₇)₁ × (A₁)₁ case the lifted symmetry is the Z₂A subgroup of Aut(E₈); in the (E₆)₁ × (A₂)₁ case it is the Z₃B conjugacy class whose centralizer is E₆ × A₂.

The central observation is that the vacuum character of (e₈)₁ can be written as a linear combination of four characters of the Drinfeld center Z(H₃): χ_{E₈,0}=χ₀+χ_{π₁}+2χ_{π₂}=χ₀+χ_{π₁}+2χ_{σ₁}, showing that the modular data of Z(H₃) is embedded in the E₈ theory. Using this, the authors define a diagonal Frobenius algebra A_diag = 1 ⊕ α ⊕ α² ⊕ ρ ⊕ αρ ⊕ α²ρ inside the product category H₃ × H₃^{op}. Gauging this algebra (i.e. performing a non‑invertible gauging) yields a theory whose partition function exactly matches that of the putative c = 8 Z(H₃) theory described in Evans–Gannon (2010). The computation involves solving the tube algebra with the known F‑symbols of H₃, evaluating twisted and twined partition functions (denoted Z|α, Z|ρ, etc.), and confirming that the resulting modular invariant coincides with the one built from the twelve characters of Z(H₃).

Beyond H₃, the paper shows that (E₈)₁ also admits a Fib × Fib^{op} symmetry. This follows from the conformal embedding (G₂)₁ × (F₄)₁ ⊂ (E₈)₁, where each factor possesses a Fibonacci fusion category symmetry. Gauging the diagonal Fibonacci symmetry reproduces (E₈)₁, while lifting the Fibonacci Verlinde lines gives the claimed Fib × Fib^{op} symmetry of (E₈)₁. The authors further demonstrate that gauging the Z₃B symmetry of (E₈)₁ takes one to the (E₆)₁ × (A₂)₁ model, which in turn contains a quotient (H₃ × H₃^{op})/Z₃B equivalent to a double near‑group extension of Z₂³. By gauging a specific simple object in this extension, one recovers the Z(H₃) theory again, establishing a web of dualities among these models.

Finally, the authors discuss possible extensions to central charges c ≈ 2 and c ≈ 6. Numerical studies and recent modular bootstrap analyses have hinted at a c ≈ 2 RCFT with Haagerup symmetry; the present work supplies additional evidence by locating the (A₂)₁ and (E₆)₁ theories at kinks of bootstrap bounds and by showing how their symmetry structures descend from the E₈ construction. An analogous c ≈ 6 theory is conjectured, with similar Haagerup‑type non‑invertible symmetries.

In summary, the paper provides the first concrete proposal that the simplest VOA, (e₈)₁, realizes Haagerup fusion categories as non‑invertible symmetries, explains how gauging these symmetries yields the Drinfeld center Z(H₃) theory, and embeds these ideas into a broader network of conformal embeddings, Fibonacci symmetries, and modular bootstrap constraints. This opens a new avenue for realizing exotic non‑group‑like symmetries in well‑understood CFTs and suggests many directions for future exploration, including explicit construction of the Z(H₃) theory, extension to other central charges, and applications to the Monster CFT.