Hecke Relations, Cosets and the Classification of 2d RCFTs

TL;DR

This paper explores the relationships between Hecke operators, cosets, and 2d RCFTs, proposing a unifying framework for characterizing RCFTs via Hecke images and cosets, supported by extensive examples and new relations.

Contribution

It introduces a novel perspective linking Hecke relations and cosets to classify 2d RCFT characters, including many new examples and relations up to seven characters.

Findings

Characters of certain WZW models are realized as Hecke images of minimal models.

New Hecke relations involving Fisher group $Fi_{23}$ and Harada-Norton group $HN$.

Verification of the proposal for theories with up to five characters.

Abstract

We systemically study the Hecke relations and the $c=8k$ coset relations among 2d rational conformal field theories (RCFTs) with up to seven characters. We propose that the characters of any 2d RCFT -- unitary or non-unitary -- satisfying a holomorphic modular linear differential equation (MLDE) can be realized as either a Hecke image or the coset of a Hecke image with respect to a $c=8k$ theory. Benefited from the recent results on holomorphic modular bootstrap, we check this proposal for all admissible theories with up to five characters. We also find many new interesting Hecke relations. For example, the characters of WZW models $(E_{6})_2,(E_7)_2,(E_{7\frac12})_2$ can be realized as the Hecke images $\mathsf{T}_{13},\mathsf{T}_{19},\mathsf{T}_{19}$ of Virasoro minimal models $M_{\rm sub}(7,6),M(5,4),M_{\rm eff}(13,2)$ respectively. Besides, we find the characters associated to the second largest Fisher group $Fi_{23}$ and the Harada-Norton group $HN$ can be realized as the Hecke images $\mathsf{T}_{23},\mathsf{T}_{19}$ of the product theories $M_{\rm eff}(5,2)\otimes M_{\rm eff}(7,2)$ and $M_{\rm eff}(7,2)^{\otimes 2}$ respectively. Mathematically, our study provides a great many interesting examples of vector-valued modular functions up to rank seven.