Quasi-Characters for three-character Rational Conformal Field Theories

We revisit (3,0) and (3,3) admissible solutions obtained using the MLDE method. We show that all $(3,0)$ solutions can be written in terms of a universal formula involving the ${}_3F_2$ hypergeometric function that takes into account the monodromy at the elliptic points. We construct $(3,3)$ admissible solutions from (3,0) CFTs using a duality due to Bantay and Gannon. This enables us to compute their modular properties such as the S-matrix and the fusion rules. We find that only 7 of the 15 known (3,3) admissible solutions have proper fusion rules. Using the theory of matrix MLDE, starting with a known (3,0) and (3,3) solutions, we construct two other solutions, that are typically quasi-characters that share the same multiplier as the original solution. We then construct linear combinations that lead to new admissible solutions. We observe that admissible solutions arise as integer points that lie on a polytope. We construct all possible (3,6) and (3,9) admissible solutions that arise in this fashion. In some cases, we identify RCFT that arise from our (3,6) admissible solutions. In addition, we obtain a large family of admissible solutions with higher Wronskian index.