Classifying three-character RCFTs with Wronskian Index equalling $0$ or $2$

Abstract

In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers $\text{[n,l]}$. $\mathbf{n}$ is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and $\mathbf{l}$, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study $\text{[3,0]}$ and $\text{[3,2]}$ MLDEs in order to classify the corresponding CFTs. We reduce the problem to a "finite" problem: to classify CFTs with central charge $0<c\le 96$, we need to perform $6,720$ computations for the former and $20,160$ for the latter. Each computation involves (i) first finding a simultaneous solution to a pair of Diophantine equations and (ii) computing Fourier coefficients to a high order and checking for positivity. In the $\text{[3,0]}$ case, for $0<c\le 96$, we obtain many character-like solutions: two infinite classes and a discrete set of $303$. After accounting for various categories of known solutions, including Virasoro minimal models, WZW CFTs, Franc-Mason vertex operator algebras and Gaberdiel-Hampapura-Mukhi novel coset CFTs, we seem to have seven hitherto unknown character-like solutions which could potentially give new CFTs. We also classify $\text{[3,2]}$ CFTs for $0<c\le 96$: each CFT in this case is obtained by adjoining a constant character to a $\text{[2,0]}$ CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.

Added references and explanations and tables. an added section on possible GHM-dual pairs of CFTs. unchanged section three which contains the classification. 4 pages

References in corpus (6)

• Irreversibility of the renormalization group flow in non-unitary quantum field theory
• Holomorphic modular bootstrap revisited
• Bootstrapping Fermionic Rational CFTs with Three Characters
• Classification of RCFT from Holomorphic Modular Bootstrap: A Status Report
• On classification of extremal non-holomorphic conformal field theories
• Universal Correlators and Novel Cosets in 2d RCFT

Cited by in corpus (10)

• Bosonic Rational Conformal Field Theories in Small Genera, Chiral Fermionization, and Symmetry/Subalgebra Duality
• Bootstrapping Fermionic Rational CFTs with Three Characters
• Needles in a haystack: An algorithmic approach to the classification of 4d $\mathcal{N}=2$ SCFTs
• Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25
• Surface defects, flavored modular differential equations and modularity
• Hecke Relations, Cosets and the Classification of 2d RCFTs
• Meromorphic Cosets and the Classification of Three-Character CFT
• New meromorphic CFTs from cosets
• Two approaches to the holomorphic modular bootstrap
• Quasi-Characters in $\widehat{su(2)}$ Current Algebra at Fractional Levels