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Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25

Sunil Mukhi, Brandon C. Rayhaun

TL;DR

This work advances the classification of two-dimensional unitary RCFTs with exactly two primaries and c<25 by leveraging holomorphic cosets, gluing, and coset techniques to connect p=2 theories to known p=1 theories (holomorphic VOAs).The authors develop and apply a framework based on genus, modular data, and MLDEs to systematically enumerate all admissible two-character theories in the specified central-charge window, proving finiteness and constructibility for c<25.They identify four Mathur–Mukhi–Sen MMS theories and show that every two-primary theory in this regime is either one of these MMS theories or a coset of a holomorphic VOA by an MMS theory, resulting in a comprehensive catalog of 123 theories, with many new; they also provide explicit characters and Kac–Moody data.Beyond delivering a bona fide RCFT classification, the paper outlines a general strategy potentially extensible to chiral algebras with low central charge beyond two primaries, and it opens pathways toward deeper connections with moonshine and Hecke structures.

Abstract

We classify all two-dimensional, unitary, rational conformal field theories with two primaries, central charge $c<25$, and arbitrary Wronskian index. In mathematical parlance, we classify all strongly regular vertex operator algebras (VOAs) with central charge $c<25$ and exactly two simple modules. We find that any such theory is either one of the Mathur-Mukhi-Sen (MMS) theories $\mathsf{A}_{1,1}$, $\mathsf{G}_{2,1}$, $\mathsf{F}_{4,1}$, or $\mathsf{E}_{7,1}$, or it is a coset of a chiral algebra with one primary operator (also known as a holomorphic VOA) by such an MMS theory. By leveraging existing results on the classification of holomorphic VOAs, we are able to explicitly enumerate all of the aforementioned cosets and compute their characters. This leads to 123 theories, most of which are new. We emphasize that our work is a bona fide classification of RCFTs, not just of characters. Our techniques are general, and we argue that they offer a promising strategy for classifying chiral algebras with low central charge beyond two primaries.

Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25

TL;DR

This work advances the classification of two-dimensional unitary RCFTs with exactly two primaries and c<25 by leveraging holomorphic cosets, gluing, and coset techniques to connect p=2 theories to known p=1 theories (holomorphic VOAs).The authors develop and apply a framework based on genus, modular data, and MLDEs to systematically enumerate all admissible two-character theories in the specified central-charge window, proving finiteness and constructibility for c<25.They identify four Mathur–Mukhi–Sen MMS theories and show that every two-primary theory in this regime is either one of these MMS theories or a coset of a holomorphic VOA by an MMS theory, resulting in a comprehensive catalog of 123 theories, with many new; they also provide explicit characters and Kac–Moody data.Beyond delivering a bona fide RCFT classification, the paper outlines a general strategy potentially extensible to chiral algebras with low central charge beyond two primaries, and it opens pathways toward deeper connections with moonshine and Hecke structures.

Abstract

We classify all two-dimensional, unitary, rational conformal field theories with two primaries, central charge , and arbitrary Wronskian index. In mathematical parlance, we classify all strongly regular vertex operator algebras (VOAs) with central charge and exactly two simple modules. We find that any such theory is either one of the Mathur-Mukhi-Sen (MMS) theories , , , or , or it is a coset of a chiral algebra with one primary operator (also known as a holomorphic VOA) by such an MMS theory. By leveraging existing results on the classification of holomorphic VOAs, we are able to explicitly enumerate all of the aforementioned cosets and compute their characters. This leads to 123 theories, most of which are new. We emphasize that our work is a bona fide classification of RCFTs, not just of characters. Our techniques are general, and we argue that they offer a promising strategy for classifying chiral algebras with low central charge beyond two primaries.
Paper Structure (20 sections, 9 theorems, 138 equations, 1 figure, 4 tables)

This paper contains 20 sections, 9 theorems, 138 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Organization
  3. Notation guide
  4. Preliminaries and Background
  5. RCFT basics
  6. Meromorphic theories
  7. Genus, cosets, and gluing
  8. Modular linear differential equations and quasi-characters
  9. Classification
  10. Modular data
  11. Admissible characters
  12. Enumeration of theories
  13. Theories with $0\leq c <8$
  14. Theories with $8\leq c <16$
  15. Theories with $16\leq c <25$
  16. ...and 5 more sections

Key Result

Proposition 3

Let $\mathcal{A}$ be any chiral algebra with $c=24$ and one simple module, except for $\mathbf{S}( \IfNoValueTF{7,4} {\mathsf{A}} {\IfNoValueTF{-NoValue-} {\mathsf{A}_{7,4}} {\mathsf{A}_{7,4}^{-NoValue-}} } \IfNoValueTF{1,1} {\mathsf{A}} {\IfNoValueTF{-NoValue-} {\mathsf{A}_{1,1}} {\mathsf{A}_{1,1}

Figures (1)

  • Figure 1: Different pairs of embeddings of $\mathcal{V}= \IfNoValueTF{1,1} {\mathsf{A}} {\IfNoValueTF{-NoValue-} {\mathsf{A}_{1,1}} {\mathsf{A}_{1,1}^{-NoValue-}} } , \IfNoValueTF{2,1} {\mathsf{G}} {\IfNoValueTF{-NoValue-} {\mathsf{G}_{2,1}} {\mathsf{G}_{2,1}^{-NoValue-}} } , \IfNoValueTF{4,1} {\mathsf{F}} {\IfNoValueTF{-NoValue-} {\mathsf{F}_{4,1}} {\mathsf{F}_{4,1}^{-NoValue-}} } ,$ or $\IfNoValueTF{7,1} {\mathsf{E}} {\IfNoValueTF{-NoValue-} {\mathsf{E}_{7,1}} {\mathsf{E}_{7,1}^{-NoValue-}} }$ into a $c=24$ chiral algebra $\mathcal{A}$ with one simple module.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Definition 4
  • Theorem 5: Theorem 3.7 of mason20082
  • Lemma 6
  • proof
  • Theorem 7
  • Proposition 8
  • Proposition 9
  • ...and 5 more