Combinatorics

Discrete mathematics

Pure mathematics

Central charge

Mathematics

Wronskian

Algorithm

Mathematical analysis

Geometry

Character (mathematics)

Computation

Conformal field theory

Conformal map

Rational number

Coset

Journal of High Energy Physics

A bstract 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 [n,l] . n is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and l , the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study [3,0] and [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 &lt; c ≤ 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 [3,0] case, for 0 &lt; c ≤ 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 [3,2] CFTs for 0 &lt; c ≤ 96: each CFT in this case is obtained by adjoining a constant character to a [2,0] CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.

Classifying three-character RCFTs with Wronskian index equalling 0 or 2