On 2d Conformal Field Theories with Two Characters

Rational CFT's are classified by an integer $\ell$, the number of zeroes of the Wronskian of their characters in moduli space. For $\ell=0$ they satisfy non-singular modular-invariant differential equations, while for $\ell>0$ the corresponding equations have singularities. We survey CFT's with two characters and $\ell=0,2,3,4$ and verify the consistency, at the level of characters, of some candidate theories with $\ell\ne 0$. For $\ell=2$ there are seven consistents sets of characters. We identify specific combinations of level-1 current algebras that are potential symmetries of the corresponding CFT's.