Free Field Realization of N = 2 Super W 3 Algebra

We study the quantum N = 2 super-W3 algebra using the free field realization, which is obtained from the supersymmetric Miura transformation associated with the Lie superalgebra A(2|1). We compute the full operator product expansions of the algebra explicitly. It is found that the results agree with those obtained by the OPE method. ∗Address after March 1, 1993: Institute of Physics, University of Tsukuba, Ibaraki 305, Japan E-mail: [email protected] W -algebraic extensions of the N = 2 superconformal algebra [1] have attracted much interest in the context of conformal field theories[2, 3], topological W -gravities[4] and more recently in non-critical W -strings[5]. There are at least two approaches to study quantumW -algebras[6]. One is based on the OPE method[7], in whichW -algebras can be constructed by requiring the associativity and closure of the algebra. The other approach is the free field realization[8], in which the generators are expressed by free fields through the quantum Miura transformations. In this approach the closure of the algebra is not obvious at quantum level, in particular for the W -algebras associated with non simply laced Lie algebra[9]. In the case of N = 2 super-W algebras, it has been understood that the free field realization can be obtained from the quantum hamiltonian reduction [10] of affine Lie superalgebras A(n|n− 1) [2, 3]. The classical Poisson brackets structure in the case of A(2|1) has been studied by using techniques such as the super Gel’fand-Dickii bracket [11] and Polyakov’s soldering procedure [12, 13]. In the quantum case, the full algebra has been presented by using the OPE method [14, 15]. In the free field approach, however, only some of the operator product expansions have been calculated [3]. In this letter, we study the free field realization of N = 2 super-W3 algebra and compute the full operator product expansions. We find that the present free field approach gives the same algebra as the one obtained by the OPE method. We begin with the supersymmetric Miura transformation based on a Lie superalgebra A(n|n − 1) = sl(n + 1|n). We take the purely odd simple root system {α1, . . . , α2n}, with the Cartan matrix αi · αj = (−1)δi+1,j . Let {λ1, . . . , λ2n} be the fundamental weights of A(n|n−1) satisfying αi ·λj = δij . Denote N = 1 super-holomorphic coordinate by Z = (z, θ) and a super-derivative by D = ∂ ∂θ + θ∂ (∂ ≡ ∂ ∂z ). We introduce 2n free bosons φ(z) = (φ1(z), . . . , φ2n(z)) and real fermions ψ(z) = (ψ1(z), . . . , ψ2n(z)) satisfying