Conformal Field Theories with Sporadic Group Symmetry

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 = 1 2 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 E 8 correspondence, and we use the structure of Norton's monstralizer pairs more generally to organize our presentation.

Bootstrapping Fermionic Rational CFTs with Three Characters

“…For instance, the characters of the Ising model and babymonster CFT are combined to produce J(τ ) [9]. The bilinear relation has been utilized to analyze the monstralizing commutant pair in [24]. It turns out that the solutions of the fermionic MLDE also exhibit bilinear relations for some cases.…”

“…Here we present the Fourier coefficients of the vector-valued modular form by exploiting Rademacher's idea. An application of the Rademacher expansion to the vector-valued modular form has been discussed in the literature, e.g., [24,33].…”

“…It has been known that the process of the above decomposition can be formulated by using the mathematical notion of the commutant subalgebra of Monster VOA. The explicit examples of Monster deconstruction have been discussed in [24,39] with choosing t 1 (z) as the stress-energy tensor of the minimal models or Z k parafermion model. As a consequence, the Monster deconstruction leads us to specific RCFTs which exhibit sporadic groups in their modular invariant partition function.…”

Fun with $F_{24}$

“…BKM algebras were originally introduced by Borcherds [7] in his proof of the monstrous moonshine conjectures of Conway and Norton [14], and Thompson [49,50]. 3 The monster BKM arises from BRST quantization of a chiral bosonic string theory and elucidates connections between modular functions, the monster sporadic simple group, and the physics of 2d CFT.…”

“…2 A self-dual SVOA W is one that is rational and the unique irreducible W-module (up to isomorphism.) 3 For reviews of moonshine, see, e.g., [21,1].…”