Conformal characters and the modular representation

TL;DR

This work addresses the problem of reconstructing Rational Conformal Field Theory (RCFT) characters from a given modular representation $\rho$ of $SL_{2}(\mathbb{Z})$. It introduces admissible representations and a canonical basis of vector-valued modular forms, showing that the character vector $\mathbb{X}$ is determined by $\rho$ together with the singular parts of its components, and that $\mathcal{M}(\rho)$ is a finitely generated $\mathbb{C}[J]$-module via the Hauptmodul $J$. Differential relations built from Eisenstein series and the discriminant lead to first-order systems governing the canonical basis, enabling recursive computation of $q$-expansions; invariants and covariants provide practical tools to generate new solutions from a given $\mathbb{X}$. The method is demonstrated in detail for the Yang-Lee model and the Ising model, obtaining explicit canonical basis elements and illustrating how positivity and integrality constraints select physically meaningful RCFT character vectors. The results offer a principled route to classify and construct RCFT characters from modular data, with broader implications for modular forms, invariants, and covariant techniques in conformal field theory.

Abstract

A general procedure is presented to determine, given any suitable representation of the modular group, the characters of all possible Rational Conformal Field Theories whose associated modular representation is the given one. The relevant ideas and methods are illustrated on two non-trivial examples: the Yang-Lee and the Ising models.