Towards a Classification of Two-Character Rational Conformal Field Theories

TL;DR

This work tackles the problem of classifying two-character RCFTs by leveraging modular linear differential equations (MLDEs) and introducing quasi-characters as a versatile, non-admissible-but-integer-coefficient foundation.The authors develop two complementary construction schemes—multiplicative (using powers of $j^{1/3}$) and additive (sums of same-$S$ quasi-characters)—to generate infinite families of admissible characters with arbitrarily large $\,\ell$, and prove the completeness of the additive method for all even $\,\ell$.A central contribution is showing that all admissible two-character spectra with $\,\ell eq 0$ can be built from a finite set of base quasi-characters corresponding to $\,\ell=0$ and $\,\ell=2$ (and $\,\ell=4$ via $j^{1/3}$), connecting to Kaneko–Zagier parametrisations and coset relations, and linking these constructions to Hecke images. The results provide a concrete, algorithmic path toward a full two-character RCFT classification, with implications for identifying physically meaningful theories among the large admissible set and clarifying the role of cosets and IVOA-like structures in this landscape.Overall, the paper bridges MLDE methods, number-theoretic constructions, and coset/Hecke techniques to advance a comprehensive, scalable program for two-character RCFT classification.

Abstract

We provide a simple and general construction of infinite families of consistent, modular-covariant pairs of characters satisfying the basic requirements to describe two-character RCFT. These correspond to solutions of generic second-order modular linear differential equations. To find these solutions, we first construct "quasi-characters" from the Kaneko-Zagier equation and subsequent works by Kaneko and collaborators, together with coset dual generalisations that we provide in this paper. We relate our construction to the Hecke images recently discussed by Harvey and Wu.