Conformal Field Theories with Sporadic Group Symmetry

TL;DR

The paper develops a systematic program to realize conformal field theories with sporadic symmetry by decomposing the monster CFT’s stress tensor into commuting pieces and studying the resulting commutant VOAs. It introduces monstralizing commutant pairs and shows how known subVOAs (Ising, parafermions, lattice VOAs) can uplift to new theories with symmetry groups from the happy family, using McKay’s E8 correspondence as a guiding framework. The authors compute dual characters for these commutants via MLDEs, Hecke operators, and Rademacher sums, and demonstrate several explicit examples including the baby monster, Fischer groups Fi24', Fi23, Fi22, as well as Th and HN, often organizing them along McKay diagrams. This work provides a unifying VOA-centric perspective on moonshine-type correspondences and raises questions about the general existence and properties of further monstralizing commutant pairs and their potential genus-zero features.

Abstract

The monster sporadic group is the automorphism group of a central charge $c=24$ vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its $c=24$ stress tensor $T(z)$, this theory contains many other conformal vectors of smaller central charge; for example, it admits $48$ commuting $c=\frac12$ conformal vectors whose sum is $T(z)$. Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups ${}^2{E}_6(2)$ and ${F}_4(2)$ of Lie type. Many of these examples are naturally associated to McKay's $\widehat{E_8}$ correspondence, and we use the structure of Norton's monstralizer pairs more generally to organize our presentation.

Figures (5)

• Figure 1: A diagram of the simple sporadic groups, based on data taken from the Atlas of Finite Groups atlas. An arrow from $H$ to $G$ indicates that $H$ is a subquotient of $G$. The groups $G$ which are encircled by solid lines as opposed to dashed lines are those for which a chiral algebra ${\textsl{VG}}^\natural$ with inner automorphism group $G$ is known. These chiral algebras embed into one another in the same way their associated groups do as subquotients.
• Figure 2: Dynkin diagram of $\widehat{E_8}$, decorated by conjugacy classes of the monster sporadic group $\mathbb{M}$. We propose to further decorate each node by the inner automorphism group of the commutant of $\mathcal{W}_{D_{n\mathrm{X}}}$ in $V^\natural$.
• Figure 3: The subset of monstralizer pairs $(G,\widetilde{G})$ whose associated $\mathbb{M}$-com pairs $(\mathcal{W}_G,\mathcal{W}_{\widetilde{G}})$ are treated in this paper. A red line from a pair $(H,\widetilde{H})$ up to $(G,\widetilde{G})$ indicates that $\widetilde{H}$ is a subgroup of $\widetilde{G}$ and $G$ is a subgroup of $H$. The associated chiral algebras mirror these inclusions, $\mathcal{W}_{\widetilde{H}}\hookrightarrow \mathcal{W}_{\widetilde{G}}$ and $\mathcal{W}_G\hookrightarrow \mathcal{W}_H$.
• Figure 4: Dynkin diagram of $\widehat{E_7}$, decorated by conjugacy classes of the baby monster sporadic group $\mathbb{B}$. We propose to further decorate each node by the inner automorphism group of the commutant of $\mathcal{W}_{B(n\mathrm{X})}$ in ${\textsl{V}}\mathbb{B}^\natural$ (c.f. §\ref{['subsec:mckaye7']}). The cases 2B and 4B do not appear to coincide with any of the constructions considered in §\ref{['subsec:mckaye8']}, and we leave their study to future work.
• Figure 5: Dynkin diagram of $\widehat{E_6}$, decorated by conjugacy classes of the Fischer group ${\textsl{Fi}}_{24}$. We propose to further decorate each node by the inner automorphism group of the commutant of $\mathcal{W}_{F(n\mathrm{X})}$ in ${\textsl{VF}}_{24}^\natural$ (c.f. §\ref{['subsec:mckaye7']}). The case 3A does not appear to coincide with any of the constructions considered in §\ref{['subsec:mckaye8']}, and we leave its study to future work.

Theorems & Definitions (2)

• Example 2.1: Bosonization
• Example 2.2: Toroidal CFT