On 2d Conformal Field Theories with two characters

Rational CFT's are classified by an integer ℓ, the number of zeroes of the Wronskian of their characters in moduli space. For ℓ=0 they satisfy non-singular modular-invariant differential equations, while for ℓ>0 the corresponding equations have singularities. We survey CFT's with two characters and ℓ=0,2,3,4 and verify the consistency, at the level of characters, of some candidate theories with ℓ≠0. For ℓ=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.