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Haagerup Symmetry in $(E_8)_1$?

Jan Albert, Yamato Honda, Justin Kaidi, Yunqin Zheng

TL;DR

The paper investigates whether Haagerup fusion categories ${ m H}_i$ can emerge as symmetries of well-known 2D conformal field theories, focusing on the chiral $(E_8)_1$ theory. It proposes that $(E_8)_1$ hosts a non-invertible ${ m H}_3 imes { m H}_3^{op}$ symmetry, realized by gauging the diagonal ${ m H}_3$ in the putative ${ m Z}({ m H}_3)$ theory, with the gauged theory matching ${ m Z}({ m H}_3)$. The work also uncovers a network of related gaugings, including ${ m Fib} imes { m Fib}^{op}$ in conformal embeddings such as $(G_2)_1 imes (F_4)_1 o (E_8)_1$ and near-group extensions for $(E_6)_1 imes (A_2)_1$, illustrating how Haagerup symmetry emerges across related theories. Complementing these constructions, modular bootstrap results provide supporting evidence for Haagerup-symmetric CFTs at central charges around $c o 2$ and $c o 6$, suggesting concrete targets for future non-invertible-gauging realizations and enriching the landscape of symmetry in RCFTs.

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.

Haagerup Symmetry in $(E_8)_1$?

TL;DR

The paper investigates whether Haagerup fusion categories can emerge as symmetries of well-known 2D conformal field theories, focusing on the chiral theory. It proposes that hosts a non-invertible symmetry, realized by gauging the diagonal in the putative theory, with the gauged theory matching . The work also uncovers a network of related gaugings, including in conformal embeddings such as and near-group extensions for , illustrating how Haagerup symmetry emerges across related theories. Complementing these constructions, modular bootstrap results provide supporting evidence for Haagerup-symmetric CFTs at central charges around and , suggesting concrete targets for future non-invertible-gauging realizations and enriching the landscape of symmetry in RCFTs.

Abstract

We suggest that the chiral theory -- in many senses the simplest VOA -- may have Haagerup symmetry for . Likewise, we suggest that the non-chiral WZW model may have symmetry, and that gauging the diagonal symmetry gives a theory with symmetry, which is the theory predicted in \cite{Evans:2010yr}. Along the way, we show that also has a symmetry, and that gauging the diagonal symmetry gives the WZW model, explaining the well-known conformal embedding . Finally, we suggest a relation to theories with symmetry at , complimenting the discussion with new modular bootstrap results.

Paper Structure

This paper contains 12 sections, 89 equations, 4 figures.

Table of Contents

  1. Warm-Up: Automorphisms of $E_8$
  2. Haagerup Symmetry
  3. Relation to $c=2,6$
  4. The category ${{\mathcal{H}}}_3$ and its Drinfeld center
  5. Lifting Verlinde lines to non-diagonal modular invariants
  6. Warm-up: from $(A_1)_4$ to $(A_2)_1$
  7. From $\mathcal{Z}({{\mathcal{H}}}_3)$ to $(E_8)_1$
  8. From $\mathcal{Z}({{\mathcal{H}}}_3)$ to $(E_6)_1 \times (A_2)_1$
  9. Additional examples: $\mathrm{Fib} \times \mathrm{Fib}^\mathrm{op}$
  10. Additional examples: from $(G_2)_3$ to $(E_6)_1$
  11. Modular bootstrap with non-invertible symmetries
  12. More on symmetries in the chiral $(\mathfrak{e}_8)_1$ WZW model

Figures (4)

  • Figure 1: Gaugings relating the $\mathcal{Z}({{\mathcal{H}}}_3)$, $(E_8)_1$, and $(E_6)_1 \times (A_2)_1$ theories. Lines in red denote symmetries which do not commute with the full chiral algebra.
  • Figure 2: Bounds on the dimension $\Delta$ of the lightest scalar in ${\mathcal{V}}_1$, as a function of $c$. The allowed region is shaded in blue. Note that both the $(A_2)_1$ and $(E_6)_1$ theories live at kinks on the boundary.
  • Figure 3: Bounds on the dimension $\Delta$ of the lightest ${\mathcal{V}}_{\pi_2} \cup {\mathcal{V}}_{\sigma_1}$ (i.e. ${\mathbb{Z}}_3$-charged) scalar, as a function of $c$. The $(A_2)_1$ theory again lives at a kink.
  • Figure 4: The web of gaugings relating the ${{\mathcal{H}}}_1, {{\mathcal{H}}}_2, {{\mathcal{H}}}_3$ Haagerup symmetries.