Abstract
For the N=4 superconformal coset theory described by SU(N+2)/SU(N) (that contains a Wolf space) with N=3, the N=2 WZW affine current algebra with constraints is obtained. The 16 generators of the large N=4 linear superconformal algebra are described by those WZW affine currents explicitly. By factoring out four spin-1/2 currents and the spin-1 current from these 16 generators, the remaining 11 generators (spin-2 current, four spin-3/2 currents, and six spin-1 currents) corresponding to the large N=4 nonlinear superconformal algebra are obtained. Based on the recent work by Gaberdiel and Gopakumar on the large N=4 holography, the extra 16 currents, with spin contents (1, 3/2, 3/2, 2), (3/2, 2, 2, 5/2), (3/2, 2, 2, 5/2), and (2, 5/2, 5/2, 3) described in terms of N=2 multiplets, are obtained and realized by the WZW affine currents. As a first step towards N=4 W algebra (which is NOT known so far), the operator product expansions (OPEs) between the above 11 currents and these extra 16 higher spin currents are found explicitly. It turns out that the composite fields with definite U(1) charges, made of above (11+16) currents (which commute with the Wolf space subgroup SU(N=3) x SU(2) x U(1) currents), occur in the right hand sides of these OPEs.
Highlights
Gaberdiel and Gopakumar [1] have proposed that the large N = 4 higher spin theory onAdS3 based on the higher spin algebra is dual to the ’t Hooft limit of the two dimensional large N = 4 superconformal coset theory
The ’t Hooft coupling constant λ is identified with the free parameter of the large N = 4 superconformal algebra (this additional parameter is due to the fact that there exist two SU(2) affine algebras rather than one) and this ’t Hooft parameter λ is equal to the above μ parameter which is related to the mass of scalar field in the AdS3 bulk theory, according to this proposal [1]
Let us concentrate on the linear terms on the higher spin current appearing in the right hand side of operator product expansions (OPE) (There are linear or nonlinear terms containing 11 currents of large N = 4 nonlinear superconformal algebra in the full expressions)
Summary
We calculate the various OPEs between the 11 generators of the large N = 4 nonlinear superconformal algebra and the 16 higher spin currents.. Let us concentrate on the linear terms on the higher spin current appearing in the right hand side of OPEs (There are linear or nonlinear terms containing 11 currents of large N = 4 nonlinear superconformal algebra in the full expressions). For the Wolf × SU(2) × U(1) = SU(2) × U(1) WZW model with k+ = k + 1 and k− = 1, as described in the introduction, the 16 generators of the large N = 4 linear superconformal algebra are expressed in terms of N = 2 super stress energy tensor of super spin-1, two super and a super spin-0 current [22].
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