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 modularinvariant 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.