Free Field Representations and Screening Operators for the N = 4 Doubly Extended Superconformal Algebras

We present explicit free field representations for the N = 4 doubly extended superconformal algebra, Ãγ. This algebra generalizes and contains all previous N = 4 superconformal algebras. We have found Ãγ to be obtained by hamiltonian reduction of the Lie superalgebra D(2|1;α). In addition, screening operators are explicitly given and the associated singular vectors identified. We use this to present a natural conjecture for the Kac determinant generalizing a previous conjecture by Kent and Riggs for the singly extended case. The results support and illuminate several aspects of the characters of this algebra previously obtained by Taormina and one of us. In recent works, we [1] and two of us [2], have outlined how essentially all known superconformal algebras – including (super-W -type) ones (for N ≥ 4) for which the algebra only closes on composites of generators – may be classified and indeed be given a unified treatment, using the techniques of hamiltonian reductions of WZNW models on Lie (super)groups [3]. In ref.[2] this program was described at the level of the classical hamiltonian reduction, whereas in ref. [1] the quantum case was considered as well using explicit free field realizations. In the case of superconformal algebras with an affine ŜO(N) algebra such a realization has been known for some time [4, 5]. In [1] it was further discussed and explicit realizations were described for the non-exceptional type Lie superalgebras, sl(N |2), osp(N |2) and osp(4|2N). For the N = 4 singly extended algebra containing an affine ŜU(2) [6] it has been recently provided by Matsuda, [7]. In the present letter we describe the case of the N = 4 “doubly extended” superconformal algebra of Sevrin, Troost and Van Proeyen [8] (see also [9]-[11]), commonly denoted Ãγ (see [12]-[14]) and further analyzed by several authors ([15]-[17]). The algebra, Ãγ, was originally given in a form with auxiliary free fermionic generators and an extra U(1) current. That algebra is denoted Aγ and has the superalgebra D(2|1;α) as its finite dimensional sub-algebra. We have found that Ãγ is obtained as a result of carrying out the hamiltonian reduction of the WZNW model on the Lie supergroup of the Lie superalgebra D(2|1;α). We first provide a brief account of how to translate between the notations of [1] using the Lie superalgebra language, and the original notations for Ãγ (see for example [8, 12]). The exceptional Lie superalgebra, D(2|1;α) (cf. ref. [18]) depends on the parameter α or γ ≡ α 1−α , and has rank three. Define the simple roots, α1, α2, α3 with metric given by α 1 = 0, α 2 2 = −2γ, α 3 = −2(1− γ), α1 · α2 = γ, α1 · α3 = 1− γ, α2 · α3 = 0, (1)