Bootstrapping Fermionic Rational CFTs with Three Characters

TL;DR

This paper extends the classification of fermionic rational conformal field theories (RCFTs) to three-character solutions of third-order modular linear differential equations, revealing connections to fermionized WZW models and sporadic group characters.

Contribution

It introduces a new analysis of three-character fermionic RCFTs from third-order MLDEs, identifying solutions related to fermionized WZW models and sporadic groups.

Findings

Most solutions map to fermionized WZW model characters.

Identified pairs of fermionic CFTs producing the $K( au)$ character.

Extended the classification to three-character fermionic RCFTs.

Abstract

Recently, the modular linear differential equation (MLDE) for level-two congruence subgroups $\Gamma_\theta, \Gamma^{0}(2)$ and $\Gamma_0(2)$ of $\text{SL}_2(\mathbb{Z})$ was developed and used to classify the fermionic rational conformal field theories (RCFT). Two character solutions of the second-order fermionic MLDE without poles were found and their corresponding CFTs are identified. Here we extend this analysis to explore the landscape of three character fermionic RCFTs obtained from the third-order fermionic MLDE without poles. Especially, we focus on a class of the fermionic RCFTs whose Neveu-Schwarz sector vacuum character has no free-fermion currents and Ramond sector saturates the bound $h^{\text{R}} \ge \frac{c}{24}$, which is the unitarity bound for the supersymmetric case. Most of the solutions can be mapped to characters of the fermionized WZW models. We find the pairs of fermionic CFTs whose characters can be combined to produce $K(\tau)$, the character of the $c=12$ fermionic CFT for $\text{Co}_0$ sporadic group.