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) × SU(2) × U(1) currents), occur in the right hand sides of these OPEs.

It was found that deformation of S 7 gives rise to renormalization group(RG) flow from N = 8, SO(8)-invariant UV fixed point to N = 1, G 2 -invariant IR fixed point in four-dimensional gauged N = 8 supergravity. Also BPS supersymmetric domain wall configuration interpolated between these two critical points. In this paper, we use the G 2 -invariant RG flow equations for both scalar fields and domain wall amplitude and apply them to the nonlinear metric ansatz developed by de Wit, Nicolai and Warner some time ago. We carry out the M-theory lift of the G 2 -invariant RG flow through a combinatoric use of the four-dimensional RG flow equations and eleven-dimensional Einstein-Maxwell equations. The non-trivial r(that is the coordinate transverse to the domain wall)-dependence of vacuum expectation values makes the Einstein-Maxwell equations consistent not only at the critical points but also along the supersymmetric RG flow connecting two critical points. By applying an ansatz for an eleven-dimensional threeform gauge field with varying scalars, we discover an exact solution to the eleven-dimensional Einstein-Maxwell equations corresponding to the M-theory lift of the G 2 -invariant RG flow.------E-mail addresses: ahn@knu.ac.kr (C. Ahn), taichi@hepth.hanyang.ac.kr (T. Itoh)

By studying various, known extrema of 1) SU(3) sectors, 2) SO(5) sectors and 3) SO(3) × SO(3) sectors of gauged N = 8 supergravity in four-dimensions, one finds that the deformation of seven sphere S 7 gives rise to non-trivial renormalization group(RG) flow in three-dimensional boundary conformal field theory from UV fixed point to IR fixed point. For SU(3) sectors, this leads to four-parameter subspace of the supergravity scalar-gravity action and we identify one of the eigenvalues of A 1 tensor of the theory with a superpotential of scalar potential that governs RG flows on this subspace. We analyze some of the structure of the superpotential and discuss first-order BPS domain-wall solutions, using some algebraic relations between superpotential and derivatives of it with respect to fields, that determine a (super)symmetric kink solution in four-dimensional N = 8 supergravity, which generalizes all the previous considerations. The BPS domain-wall solutions are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N = 8 supergravity. For SO(5) sectors, there exist only nontrivial nonsupersymmetric critical points that are unstable and included in SU(3) sectors. For SO(3) × SO(3) sectors, we construct the scalar potential(never been written) explicitly and study explicit construction of first-order domain-wall solutions. de Wit-Nicolai Potentialde Wit and Nicolai [16,17] constructed a four-dimensional supergravity theory by gauging the SO(8) subgroup of E 7 in the global E 7 × local SU(8) supergravity of Cremmer and Julia [18] by introducing the appropriate couplings by hand and then constructing the supersymmetry model by Noether procedure. In common with Cremmer-Julia theory, this theory contains selfinteraction of a single massless N = 8 supermultiplet of spins (2, 3/2, 1, 1/2, 0 + , 0 − ) but with local SO(8) × local SU(8) invariance. There is a new parameter, the SO(8) gauge coupling constant g besides the gravitational constant. In order to preserve the N = 8 supersymmetry, they modified the Cremmer-Julia Lagrangian and transformation rules by other g-dependent terms. In particular, there was a non-trivial effective potential for the scalars that is proportional to the square of the SO(8) gauge coupling. It is well known [19] that the 70 real, physical scalars of N = 8 supergravity parametrize the coset space E 7 /SU(8)(even though E 7 symmetry is broken in the gauged theory) since 63 fields(133 − 63 = 70) may be gauged away by an SU (8) rotation(maximal compact subgroup of E 7 ) and can be described by an element V(x) of the

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.

Recently, Gaberdiel and Gopakumar proposed that the two-dimensional W A N −1 minimal model conformal field theory in the large N 't Hooft limit is dual to the higher spin theories on the three-dimensional AdS space with two complex scalars. In this paper, we examine this proposal for the W D N 2 and W B N−1 2 minimal models initiated by Fateev and Lukyanov in 1988. By analyzing the renormalization group flows on these models, we find that the gravity duals in AdS space are higher spin theories coupled to two equally massive real scalar fields. We also describe the large N 't Hooft limit for the minimal model of the second parafermion theory.