By analyzing SU(3)×U(1) invariant stationary point, studied earlier by Nicolai and Warner, of gauged N = 8 supergravity, we find that the deformation of S 7 gives rise to nontrivial renormalization group flow in a three-dimensional boundary super conformal field theory from N = 8, SO(8) invariant UV fixed point to N = 2, SU(3) × U(1) invariant IR fixed point. By explicitly constructing 28-beins u, v fields, that are an element of fundamental 56-dimensional representation of E 7 , in terms of scalar and pseudo-scalar fields of gauged N = 8 supergravity, we get A 1 , A 2 tensors. Then we identify one of the eigenvalues of A 1 tensor with "superpotential" of de Wit-Nicolai scalar potential and discuss four-dimensional supergravity description of renormalization group flow, i.e. the BPS domain wall solutions which are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N = 8 supergravity. A numerical analysis of the steepest descent equations interpolating two critical points is given.

For the N = 4 superconformal coset theory by× U(1) (that contains an orthogonal Wolf space) with N = 4, the N = 2 WZW affine current algebra is obtained. The 16 generators (or 11 generators) of the large N = 4 linear (or nonlinear) superconformal algebra are described by these WZW affine currents explicitly. Along the line of large N = 4 holography, the extra 16 currents with spins (2, , 4) are obtained in terms of the WZW affine currents. The lowest spin of this N = 4 multiplet is two rather than one which is for an unitary Wolf space. The operator product expansions (OPEs) between the above 11 currents and these extra 16 higher spin currents are found explicitly.

We construct the lowest higher spin-2 current in terms of the spin-1 and the spin- Wolf space coset theory for general N. The remaining fifteen higher spin currents are determined. We obtain the three-point functions of bosonic (higher) spin currents with two scalars for finite N and k (the level of the spin-1 current). By multiplying SU(2) × U(1) into the above Wolf space coset theory, the other fifteen higher spin currents together with the above lowest higher spin-2 current are realized in the extension of the large N = 4 linear superconformal algebra. Similarly, the three-point functions of bosonic (higher) spin currents with two scalars for finite N and k are obtained. Under the large N 't Hooft limit, the two types of three-point functions in the nonlinear and linear versions coincide as in the unitary coset theory found previously.

In the N = 1 supersymmetric coset minimal model based on (B (1) N ) at level k, the higher spin-2 ′ Casimir current was obtained previously. By acting the above spin-3 2 current on the higher spin-2 ′ Casimir current, its superpartner, the higher spin-5 2 current, can be generated and combined as the first higher spin supercurrent with spins (2 ′ , 5 2 ). By calculating the operator product expansions (OPE) between the higher spin supercurrent and itself, the next higher spin supercurrent can be generated with spins ( 7 2 , 4). Moreover, the other higher spin supercurrent with spins (4 ′ , 9 2 ) can be generated by calculating the OPE between the first higher spin supercurrent with spins (2 ′ , 5 2 ) and the second higher spin supercurrent with spins ( 7 2 , 4). Finally, the higher spin supercurrent, ( 11 2 , 6), can be extracted from the right hand side of OPE between the higher spin supercurrents, (2 ′ , 5 2 ) and (4 ′ , 9 2 ).

We compute the operator product expansion (OPE) between the spin-4 current and itself in the W D 4 coset minimal model with SO(8) current algebra. The right hand side of this OPE contains the spin-6 Casimir current which is also a generator of W D 4 coset minimal model. Based on this N = 8 result, we generalize the above OPE for the general N(in the W D N 2 coset minimal model) by using two N-generalized coupling constants initiated by Hornfeck sometime ago: the simplest OPE for the lowest higher spin currents. We also analyze the similar OPE in the W B 3 (and W B N−1 2 ) coset minimal model with SO(7) current algebra. The large N 't Hooft limits are discussed. Our results in two dimensional conformal field theory provide the asymptotic symmetry, at the quantum level, of the higher spin AdS 3 gravity found by Chen et al.