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

Abstract

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 =...

Holographic {\cal N}=1 supersymmetric membrane flows

“…The 11-dimensional description preserving N = 2 SU(2) × U(1) × U(1) R symmetry is found in [22] and the smaller N = 2 U(1) × U(1) × U(1) R symmetry flow is discussed in [23]. Further study on [8] is done in [24] recently.…”

“…The L is the radius of the round 7-sphere S 7 . We focus on the possible 11-dimensional lifts of the RG flows around N = 2 SU(3) × U(1) R critical point in this paper and those around N = 1 G 2 critical point have been described in [24] recently.…”

Towards New Membrane Flow From De Wit–nicolai Construction

“…Therefore, as soon as we impose the SO(7) + constraint b(r) = 1 a(r) into these coefficient functions, they vanish. What remains for the 4-forms is exactly the components of F AI 1234 and F AI 1235 in [7,26] where the 5-th direction is the direction of θ. We observe that they are the same as the ones (3.19) and (3.18) exactly.…”

“…Now we move on the matrix elements of [D 4 , D 5 ] in (E.6). The nonzero expressions of them are summarized by the following matrix elements (1, 2), (2,17), (3,4), (4,19), (5,22), (6, 5), (7,24), (8, 7), (9, 10), (9,26), (11,12), (11,28), (14,13), (14,29), (16,15), (16,31), (17,18), (18,1), (19,20), (20,3), (21,6), (22,21), (23,8), (24,23), (25,10), (25,26), (27,12), (27,28), (30,...…”

“…(E.8) Furthermore, the following nonzero matrix elements (1,23), (2,24), (3,21), (4,22), (5, 3), (6, 4), (7, 1), (8,2), (13,11), (13,27), (14,12), (14,28), (15,9), (15,25), (16,10), (16,26), (17,7), (18,8), (19,5), (20,6), (21,19), (22,20), (23,17), (24,18), (29,11), (29,27), (30,12), (30,28), (31,…”