Meromorphic Cosets and the Classification of Three-Character CFT

TL;DR

This paper classifies all unitary three-character conformal field theories with zero Wronskian index, identifying their modular forms and pairing relations, and provides constraints on higher central charge theories.

Contribution

It introduces a complete classification of three-character CFTs with vanishing Wronskian index, excluding certain central charges, and explores their modular and coset structures.

Findings

Classified all unitary three-character CFTs with zero Wronskian index (excluding c=8,16)

Identified two infinite affine series and 45 additional theories

Established constraints on higher central charge meromorphic theories

Abstract

We investigate the admissible vector-valued modular forms having three independent characters and vanishing Wronskian index and determine which ones correspond to genuine 2d conformal field theories. This is done by finding bilinear coset-type relations that pair them into meromorphic characters with central charges 8, 16, 24, 32 and 40. Such pairings allow us to identify some characters with definite CFTs and rule out others. As a key result we classify all unitary three-character CFT with vanishing Wronskian index, excluding $c=8,16$. The complete list has two infinite affine series $B_{r,1},D_{r,1}$ and 45 additional theories. As a by-product, at higher values of the total central charge we also find constraints on the existence or otherwise of meromorphic theories. We separately list several cases that potentially correspond to Intermediate Vertex Operator Algebras.