${\mathcal N}=8$ GAUGED SUPERGRAVITY THEORY AND ${\mathcal N}=6$ SUPERCONFORMAL CHERN–SIMONS MATTER THEORY

2009

J. High Energy Phys.

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We construct the 11-dimensional lift of the known N = 2 supersymmetric RG flow solution in 4-dimensional N = 8 gauged supergravity. The squashed and stretched 7-dimensional internal metric preserving SU(2) × U(1) × U(1) R symmetry contains an Einstein-Kahler 2fold which is a base manifold of 5-dimensional Sasaki-Einstein Y p,q space found in 2004. The nontrivial r(transverse to the domain wall)-dependence of the AdS 4 supergravity fields makes the Einstein-Maxwell equations consistent not only at the critical points but also along the supersymmetric whole RG flow connecting two critical points. With an appropriate 3form gauge field, we find an exact solution to the 11-dimensional Einstein-Maxwell equations corresponding to the above lift of the SU(2)×U(1)×U(1) R -invariant RG flow. The particular limits of this solution give rise to the previous solutions with SU(3)×U(1) R or SU(2)×SU(2)× U(1) R .

Section: Introductionmentioning

confidence: 99%

New 𝒩 = 2 supersymmetric membrane flow in eleven-dimensional supergravity

2009

J. High Energy Phys.

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“…The holographic RG flow equations connecting G 2 point to SU(3) × U(1) R point have been found in [11]. Moreover, the other holographic supersymmetric RG flows have been studied and further developments on the 4-dimensional gauged supergravity have been done in [12]. The spin-2 Kaluza-Klein modes around a warped product of AdS 4 and a seven-ellipsoid which has above global G 2 symmetry are discussed in [13].…”

Section: Introductionmentioning

confidence: 99%

The eleven-dimensional uplift of four-dimensional supersymmetric RG flow

Journal of Geometry and Physics

2012

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The squashed and stretched 7-dimensional internal metric preserving U(1) × U(1) × U(1) R symmetry possesses an Einstein-Kahler 2-fold which is a base manifold of 5-dimensional Sasaki-Einstein L p,q,r space. The r(transverse to the domain wall)-dependence of the two 4-dimensional supergravity fields, that play the role of geometric parameters for squashing and stretching, makes the 11-dimensional Einstein-Maxwell equations consistent not only at the two critical points but also along the whole N = 2 supersymmetric RG flow connecting them. The Ricci tensor of the solution has common feature with the previous three 11-dimensional solutions. The 4-forms preserve only U(1) R symmetry for other generic parameters of the metric. We find an exact solution to the 11-dimensional Einstein-Maxwell equations corresponding to the lift of the 4-dimensional supersymmetric RG flow.

“…The holographic N = 1 supersymmetric SU(3)-invariant RG flow equations connecting N = 1 G 2 point to N = 2 SU(3) × U(1) R point in 4-dimensions have been studied in [13]. Moreover, the other holographic supersymmetric RG flows have been found and further developments on the 4-dimensional gauged supergravity (see also [14,15]) have been made in [16,17].…”

Section: Introductionmentioning

confidence: 99%

Towards an N=1SU(3)-invariant supersymmetric membrane flow in eleven-dimensional supergravity

Journal of Geometry and Physics

Woo

2012

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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 =...