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

We consider the most general SU(3) singlet space of gauged N = 8 supergravity in four-dimensions. The SU(3)-invariant six scalar fields in the theory can be viewed in terms of six real four-forms. By exponentiating these four-forms, we eventually obtain the new scalar potential. For the two extreme limits, we reproduce the previous results found by Warner in 1983. In particular, for the N = 1 G 2 critical point, we find the constraint surface parametrized by three scalar fields on which the cosmological constant has the same value. We obtain the BPS domain-wall solutions for restricted scalar submanifold. We also describe the three-dimensional mass-deformed superconformal Chern-Simons matter theory dual to the above supersymmetric flows in four-dimensions.

By studying already known extrema of non-semi-simple Inonu-Wigner contraction CSO(p, q) + and non-compact SO(p, q) + (p + q = 8) gauged N = 8 supergravity in 4-dimensions developed by Hull sometime ago, one expects there exists nontrivial flow in the 3-dimensional boundary field theory. We find that these gaugings provide first-order domain-wall solutions from direct extremization of energy-density.We also consider the most general CSO(p, q, r) + with p + q + r = 8 gauging of N = 8 supergravity by two successive SL(8, R) transformations of the de Wit-Nicolai theory, that is, compact SO(8) gauged supergravity. The theory found earlier has local SU(8) × CSO(p, q, r) + gauge symmetry as well as local N = 8 supersymmetry. The gauge group CSO(p, q, r) + is spontaneously reduced to its maximal compact subgroup SO(p) + × SO(q) + × U(1) +r(r−1)/2 . The T-tensor we obtain describes a two-parameter family of gauged N = 8 supergravity from which one can construct A 1 and A 2 tensors. The effective nontrivial scalar potential can be written as the difference of positive definite terms. We examine the scalar potential for critical points at which the expectation value of the scalar field is SO(p) + ×SO(q) + ×SO(r) + invariant. It turns out that there is no new extra critical point. However, we do have flow equations and domain-wall solutions for the scalar fields are the gradient flow equations of the superpotential that is one of the eigenvalues of A 1 tensor. IntroductionOne of the interesting issues in recent research is the domain wall(DW)/quantum field theory(QFT) correspondence initiated by [1] between supergravity, in the near horizon region of the corresponding supergravity brane solution, compactified on domain wall spacetimes that are locally isometric to Anti-de Sitter(AdS) space but different from it globally and quantum(nonconformal) field theories describing the internal dynamics of branes and living on the boundary of such spacetimes. DW/QFT correspondence was motivated by the fact that the AdS metric in horospherical coordinates is a special case of the domain wall metric [2,3]. Rsymmetry of the supersymmetric QFT on the boundary of domain worldvolume should match with the gauge group of the corresponding gauged supergravity. Compact gaugings are not the only ones for extended supergravities but there exists a rich structure of non-compact and nonsemi-simple gaugings (Note that the unitarity property is preserved since in all extrema of scalar potential, non-compact gauge symmetry is reduced to some residual compact subgroup). Such a theory plays a fundamental role in the description of the DW/QFT correspondence as the maximally compact gauged supergravity has played in the AdS/conformal field theory(CFT) duality [4,5,6] that is a correspondence between certain compact gauged supergravities and certain conformal field theories. It would be interesting to identify the appropriate non-compact and non-semi-simple gauged supergravities corresponding to each choice of brane configuration.One of the que...

Corrado, Pilch and Warner in 2001 have found the second 11-dimensional solution where the deformed geometry of S 7 in the lift contains S 2 × S 2 . We identify the gauge dual of this background with the theory described by Franco, Klebanov and Rodriguez-Gomez recently. It is the U(N)×U(N)×U(N) gauge theory with two SU(2) doublet chiral fields B 1 transforming in the (N, N, 1), B 2 transforming in the (1, N, N), C 1 in the (1, N, N) and C 2 in the (N, N, 1) as well as an adjoint field Φ in the (1, adj, 1) representation.By adding the mass term for adjoint field Φ, the detailed correspondence between fields of AdS 4 supergravity and composite operators of the IR field theory is determined. Moreover, we compute the spin-2 KK modes around a warped product of AdS 4 and a squashed and stretched seven-manifold. This background with global SU(2) × SU(2) × U(1) R symmetry is related to a U(N) × U(N) × U(N) N = 2 superconformal Chern-Simons matter theory with eighth-order superpotential and Chern-Simons level (1, 1, −2). The mass-squared in AdS 4 depends on SU(2) × SU(2) × U(1) R quantum numbers and KK excitation number. The dimensions of spin-2 operators are found.√ 79 4 −1

We compute the spin-2 Kaluza-Klein modes around a warped product of AdS 4 and a seven-ellipsoid. This background with global G 2 symmetry is related to a U(N) × U(N) N = 1 superconformal Chern-Simons matter theory with sixth order superpotential. The mass-squared in AdS 4 is quadratic in G 2 quantum number and KK excitation number. We determine the dimensions of spin-2 operators using the AdS/CFT correspondence. The connection to N = 2 theory preserving SU(3) × U(1) R is also discussed.

By studying the previously known holographic N = 4 supersymmetric renormalization group flow (Gowdigere-Warner) in four dimensions, we find the mass deformed Chern-Simons matter theory which has N = 4 supersymmetry by adding the four mass terms among eight adjoint fields. The geometric superpotential from the 11 dimensions is found and provides the M2-brane probe analysis. As second example, we consider known holographic N = 8 supersymmetric renormalization group flow (Pope-Warner) in four dimensions. The eight mass terms are added and similar geometric superpotential is obtained. a There are other related works 24-29 which discuss about the different supersymmetric theories in the context of Chern-Simons matter theory. Int. J. Mod. Phys. A 2010.25:3407-3444. Downloaded from www.worldscientific.com by MICHIGAN STATE UNIVERSITY on 02/06/15. For personal use only.

The M-theory lift of N = 2 SU(3) × U(1) R -invariant RG flow via a combinatorical use of the 4-dimensional flow and 11-dimensional Einstein-Maxwell equations was found previously. By taking the three internal coordinates differently and preserving only SU(3) symmetry from the CP 2 space, we find a new 11-dimensional solution of N = 1 SU(3)-invariant RG flow interpolating from N = 8 SO(8)-invariant UV fixed point to N = 2 SU(3) × U(1) Rinvariant IR fixed point in 4-dimensions. We describe how the corresponding 3-dimensional N = 1 superconformal Chern-Simons matter theory deforms. By replacing the above CP 2 space with the Einstein-Kahler 2-fold, we also find out new 11-dimensional solution of N = 1 SU(2) × U(1)-invariant RG flow connecting above two fixed points in 4-dimensions.

The M-theory lift of N = 1 G 2 -invariant RG flow via a combinatoric use of the 4dimensional RG flow and 11-dimensional Einstein-Maxwell equations was found some time ago. The 11-dimensional metric, a warped product of an asymptotically AdS 4 space with a squashed and stretched 7-sphere, for SU(3)-invariance was found before. In this paper, by choosing the 4-dimensional internal space as CP 2 space, we discover an exact solution of N = 1 G 2 -invariant flow to the 11-dimensional field equations. By an appropriate coordinate transformation on the three internal coordinates, we also find an 11-dimensional solution of N = 1 G 2 -invariant flow interpolating from N = 8 SO(8)-invariant UV fixed point to N = 1 G 2 -invariant IR fixed point. In particular, the 11-dimensional metric and 4-forms at the N = 1 G 2 fixed point for the second solution will provide some hints for the 11-dimensional lift of whole N = 1 SU(3) RG flow connecting this N = 1 G 2 fixed point to N = 2 SU(3) × U(1) R fixed point in 4-dimensions. the RG flows [5], we focus on the G 2 -invariant sector (which is a subset of above SU(3)invariant sector) in 4-dimensional viewpoint by constraining the four arbitrary supergravity fields together with the particular condition. There exist a supersymmetric N = 1 G 2 critical point and two nonsupersymmetric SO(7) ± critical points [29,30] in this sector. If we take the different constraints on the supergravity fields, then we are led to the SU(3) × U(1) Rinvariant sector where there are a supersymmetric N = 2 SU(3) × U(1) R critical point and a nonsupersymmetric SU(4) − critical point [31].The 11-dimensional metric is found by [26] where the compact 7-dimensional metric and warp factor are completely determined in local frames. The two geometric parameters, by constraints, that are nothing but the above supergravity fields, depend on the AdS 4 radial 2 coordinate and are subject to the RG flow equations [7,5] in 4-dimensional gauged supergravity. The global coordinates for S 7 appropriate for the base six-sphere S 6 ≃ G 2 SU (3) preserve the Fubini-Study metric on CP 2 and this describes the ellipsoidally deformed S 7 [25]. On the other hand, by using the relation to the Hopf fibration on CP 3 ≃ SU (4) SU (3)×U (1) explicitly and keeping only the CP 2 space, one can change the remaining three local frames in 7-dimensional manifold via an orthogonal transformation [26]. Each global coordinates depends on each base six-spheres they use.What is the 11-dimensional lift of holographic N = 1 supersymmetric SU (3)-invariant RG flow [13] connecting from N = 1 G 2 critical point to N = 2 SU(3) × U(1) R critical point? At each critical point, the 4-forms are known in different background. Along the supersymmetric RG flow, one expects that there exist the extra 4-forms which should vanish at the critical points. In order to describe the whole RG flow, one needs to have the consistent background.Around N = 2 SU(3) × U(1) R IR fixed point, the use of global coordinates for Hopf fibration on CP 3 was done in [8]. Around N =...