Classification of RCFT from Holomorphic Modular Bootstrap: A Status Report

TL;DR

The note surveys the status of classifying Rational Conformal Field Theories in two dimensions via the holomorphic bootstrap, which builds admissible characters as holomorphic solutions to $n^{th}$-order Modular Linear Differential Equations with non-negative $q$-series coefficients and uses them to form modular-invariant partition functions. It organizes results by $(n,\ell)$, showing finite classifications for $(n,\ell)=(1,\ell<6)$ and for $(n,\ell)=(2,0)$, while revealing infinite candidate sets for $\ell\ge6$ generated by Hecke images and quasi-characters; three-character theories with $\ell=0$ are well-explored and constrained by irreducible monodromy, with a detailed $U$-series and notable coset structures, whereas the $\ell>0$ and $n>3$ cases remain largely open. The work highlights dual coset relations $c_1+c_2=24$, the role of Hecke operations and quasi-characters in expanding admissible spaces, and the irreducible monodromy criterion as a key filter for physically meaningful RCFTs, outlining clear progress and substantial open questions toward a comprehensive catalog. This status report clarifies where complete classifications have been achieved and where new techniques (e.g., monodromy constraints, coset constructions) are most needed to close the remaining gaps.

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.