On classification of extremal non-holomorphic conformal field theories

TL;DR

This paper explores the classification of extremal non-holomorphic conformal field theories by analyzing their characters and representation theory, aiming to connect modular tensor categories with vertex operator algebras.

Contribution

It introduces extremal vertex operator algebras with minimal conformal dimensions and classifies their characters for central charges up to 48.

Findings

Finitely many characters for extremal VOAs with up to three modules

Complete list of possible characters for c ≤ 48

Supports conjecture linking modular tensor categories and conformal field theories

Abstract

Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $\mathcal{C}$ and a central charge $c$. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category $\mathcal{C}$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $\mathcal{C}_V$ is $\mathcal{C}$. In this paper, we initiate a mathematical program in and around this conjecture. We define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge, and non-trivial representation theory. We show that there are finitely many different characters of extremal vertex operator algebras V possessing at most three different irreducible modules. Moreover, we list all of the possible characters for such vertex operator algebras with $c$ at most 48.