Two approaches to the holomorphic modular bootstrap

The holomorphic bootstrap attempts to classify rational conformal field theories. The straight ahead approach is hard to implement when the number of characters become large. We combine all characters of an RCFT to form a vector valued modular form with multiplier. Using known results from the theory of vector valued modular forms, given a known RCFT, we obtain new vector valued modular forms that share the same multiplier as the original RCFT. By taking particular linear combinations of the new solutions, we look for and find new admissible solutions. In the well-studied two character case, we reproduce all known admissible solutions with Wronskian indices $6$ and $8$. The method is illustrated with examples with up to six characters. The method using vector valued modular forms thus provides a new approach to the holomorphic modular bootstrap.