Holomorphic modular bootstrap revisited

TL;DR

This paper advances the classification of rational conformal field theories by analyzing modular differential equations and representation theory, enabling systematic determination of allowed parameters for theories with up to five characters.

Contribution

It introduces a new method leveraging PSL(2,Z_n) representation theory to classify allowed central charges and weights, overcoming previous limitations.

Findings

Classified consistent characters up to d=5

Identified constraints on Wronskian index for d≤5

Fixed modular differential equations in terms of (c,h_i)

Abstract

In this work we revisit the "holomorphic modular bootstrap", i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the representation theory of ${\rm PSL}(2,\mathbb{Z}_n)$, we describe a method to classify allowed central charges and weights $(c,h_i)$ for theories with any number of characters $d$. This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to $d=5$ whose modular differential equations are uniquely fixed in terms of $(c,h_i)$. In the process, we identify the full set of constraints on the allowed values of the Wronskian index for fixed $d\leq 5$.