Two-dimensional RCFT's without Kac-Moody symmetry

TL;DR

This paper tackles the classification of two- and three-character Rational Conformal Field Theories that lack Kac-Moody (spin-1) symmetry by leveraging modular-invariant differential equations and the Wronskian framework indexed by the integer $\\ell$. It shows that for small $\\ell$ one can systematically identify candidate RCFTs, recovering both unitary and non-unitary minimal models and revealing intriguing dual theories whose characters combine to reproduce the Moonshine module via precise bilinear relations; in the unitary three-character case, the dual is tied to the Baby Monster module, providing a concrete link between Monster and Baby Monster representations. The results suggest a coset-like structure without affine currents and illuminate Moonshine-type pairings across multiple RCFTs, offering a roadmap to discover further Moonshine-related pairs among minimal models and their duals. Overall, the work deepens the connection between algebraic symmetries, modular constraints, and sporadic-group representations in two-dimensional CFTs, with potential generalizations to higher character counts and larger $\\\ell$.

Abstract

Using the method of modular-invariant differential equations, we classify a family of Rational Conformal Field Theories with two and three characters having no Kac-Moody algebra. In addition to unitary and non-unitary minimal models, we find "dual" theories whose characters obey bilinear relations with those of the minimal models to give the Moonshine Module. In some ways this relation is analogous to cosets of meromorphic CFT's. The theory dual in this sense to the Ising model has central charge 47/2 and is related to the Baby Monster Module.