Modular invariant partition functions for the doubly extended N = 4 superconformal algebras

Abstract

Non-trivial modular properties of characters of the doubly extended N = 4 superconformal algebras A γ, A ̃ γ are derived from two different points of view. First, we use realizations on Wolf spaces, in particular when one of the levels of the two commuting affine SU(2) subalgebras takes the value 2. We emphasize how these realizations involve rational torus theories, and how some specific combinations of massless characters transform under the modular group as affine SU(2) characters. Second, we show how these combinations, and generalizations thereof, emerge from a study of the explicit form of the characters when angular variables are partly restricted, but the levels are not. The two results are then combined to give stringent constraints on the modular invariant A ̃ γ partition functions and they give rise to a partial classification of the latter, closely related to that of affine SU(2).