Classifying three-character RCFTs with Wronskian index equalling 3 or 4

A bstract In the Mathur-Mukhi-Sen (MMS) classification scheme for rational conformal field theories (RCFTs), a RCFT is identified by a pair of non-negative integers [ n , ℓ ], with n being the number of characters and ℓ the Wronskian index. The modular linear differential equation (MLDE) that the characters of a RCFT solve are labelled similarly. All RCFTs with a given [ n , ℓ ] solve the modular linear differential equation (MLDE) labelled by the same [ n , ℓ ]. With the goal of classifying [ 3, 3 ] and [ 3, 4 ] CFTs, we set-up and solve those MLDEs, each of which is a three-parameter non-rigid MLDE, for character-like solutions. In the former case, we obtain four infinite families and a discrete set of 15 solutions, all in the range 0 < c ≤ 32. Amongst these [ 3, 3 ] character-like solutions, we find pairs of them that form coset-bilinear relations with meromorphic CFTs/characters of central charges 16 , 24 , 32 , 40 , 48 , 56 , 64. There are six families of coset-bilinear relations where both the RCFTs of the pair are drawn from the infinite families of solutions. There are an additional 23 coset-bilinear relations between character-like solutions of the discrete set. The coset-bilinear relations should help in identifying the [ 3, 3 ] CFTs. In the [ 3, 4 ] case, we obtain nine character-like solutions each of which is a [ 2, 2 ] character-like solution adjoined with a constant character.