Classification of RCFT from Holomorphic Modular Bootstrap: A Status Report

TL;DR

This paper reviews progress in classifying Rational Conformal Field Theories in 2D using the holomorphic bootstrap approach based on modular linear differential equations and admissible characters.

Contribution

It provides a status update on the ongoing project of classifying RCFTs through modular bootstrap methods since 1988.

Findings

Progress in classifying RCFTs using modular differential equations

Identification of admissible characters with non-negative integer coefficients

Construction of modular-invariant partition functions from characters

Abstract

Following the initial proposal in 1988, there has been much progress in classifying Rational Conformal Field Theories in 2 dimensions from the Holomorphic Bootstrap approach. This method starts by postulating a generic holomorphic Modular Linear Differential Equation of a given order and imposing the requirement of non-negative integrality of the coefficients in the series expansion of the solutions, which are then identified as admissible characters, from which a modular-invariant partition function is constructed. In this short note, the status of this project is summarised.