Bosonic solutions of eleven-dimensional supergravity were studied in the 1980s in the context of Kaluza-Klein supergravity. The topic received renewed attention in the mid-to-late 1990s as a result of the branes and duality paradigm and the AdS/CFT correspondence.

One of the earliest solutions of eleven-dimensional supergravity is the maximally supersymmetric Freund--Rubin background with geometry $\mathrm{AdS}_4 \times S^7$ and 4-form flux proportional to the volume form on $\mathrm{AdS}_4$. The radii of curvatures of the two factors are furthermore in a ratio of 1:2. The modern avatar of this solution is as the near-horizon limit of a stack of M2 branes.

Shortly after the original Freund--Rubin solution was discovered, Englert discovered a deformation of this solution where one could turn on flux on the $S^7$; namely, singling out one of the Killing spinors of the solution, a suitable multiple of the 4-form one constructs by squaring the spinor can be added to the volume form in $\mathrm{AdS}_4$ and the resulting 4-form still obeys the supergravity field equations, albeit with a different relation between the radii of curvature of the two factors. The flux breaks the $\mathrm{SO}(8)$ symmetry of the sphere to an $\mathrm{SO}(7)$ subgroup.

My question is whether the Englert solution has a modern avatar, perhaps as the near-horizon limit of some solution.

  • $\begingroup$ Does this solution preserve any supersymmetry? If not then I guess an AdS/CFT interpretation could be problematic if it is unstable. $\endgroup$ – Paul Oct 18 '11 at 18:07
  • $\begingroup$ So it appears that the Englert solution preserves no supersymmetry (Englert, Rooman and Spindel inspirehep.net/record/190303) and is unstable (Page and Pope inspirehep.net/record/201484 and de Witt and Nicolai inspirehep.net/record/191530). I guess stability without supersymmetry is always exceptional though there are the well-known "skew-whiffed" solutions which are non-supersymmetric but stable. $\endgroup$ – Paul Oct 18 '11 at 18:09
  • $\begingroup$ (You beat me to the comment!) All solutions $\mathrm{AdS}_4 \times X^7$ with $X^7$ admitting real Killing spinors and with internal flux obtained by squaring a Killing spinor on $X$ break all supersymmetry. This is proved in the paper of Englert, Rooman and Spindel which you mention. $\endgroup$ – José Figueroa-O'Farrill Oct 18 '11 at 18:16

If the 1998 paper by Castellani et al. below is right,


then all such solutions including Englert's elegant solution – cited as [13] above – are dual to $G/H$ M$p$-branes which are described as solutions interpolating between flat space at infinity and some $G/H$ configuration near the horizon. Let me admit that I don't quite understand the paper – it seems to imply the existence of lots of new objects in ordinary decompactified M-theory.

Of course, as discussed above, the lack of supersymmetry reduces the probability that it makes sense to try to locate the object in any dual description. The stability of such non-SUSY solutions is only restored in some "large radii" limit and is completely lost in the opposite limit. Still, there are some moral examples where it makes sense to trace unstable objects to a dual description although their masses and tensions of course fail to be calculable from SUSY which isn't there. Various authors such as Gauntlett et al.


have mentioned Englert's solution but discarded it because of the instability. You may also want to look at the octonion membrane by Duff et al.:


It's probably not quite equivalent to Englert's solution but seems to be an object of the same dimension with the same algebraic niceties hidden in the core.

Some possibly very similar candidate dual CFTs (also citing Englert) are constructed by Fabbri et al.:


Also, Englert's solution might have a weak $G_2$ holonomy


although this statement of mine could be complete nonsense: I don't have the access to the full paper. In 2008, Klebanov et al. studied dual CFTs to squashed spheres


that are claimed to be of Englert's type.

  • $\begingroup$ Thanks -- actually I knew the paper! Somehow the answer I was hoping for was along the lines of "adding" something to the background of an M2 brane. In this paper what they do is consider M2 branes at a conical singularity with link $G/H$, but I guess what I would like to know is if there is a source for the conical singularity. $\endgroup$ – José Figueroa-O'Farrill Oct 18 '11 at 18:19
  • $\begingroup$ Dear @José, sorry for being redundant then. I also added (into my answer) a link to a seemingly related paper on the octonionic membrane by Duff et al. Conical singularities in general don't solve Einstein's equations at the singular point except that sometimes they can when the UV physics governing the singular loci is favorable. After all, orbifolds are allowed and some of them may also be viewed as deficit angles etc. But I am afraid that if a full-fledged embedding of Englert's solution to a UV complete theory exists, it's not known. $\endgroup$ – Luboš Motl Oct 18 '11 at 18:24
  • $\begingroup$ Dear Luboš, Thanks for the additions. Here are some comments, now with a fuller stomach. First of all, the paper on $G/H$ is only about Freund-Rubin backgrounds, whence does not strictly speaking apply to the Englert solution; their citing it notwithstanding. The geometry of the Englert solution is the round 7-sphere, although the flux breaks the symmetry to $\mathrm{SO}(7)$. This still acts transitively and one can view the round 7-sphere as $\mathrm{SO}(7)/G_2$. The invariant connection in that description has holonomy $G_2$, but it does not agree with the Levi-Civita connection. (TBC) $\endgroup$ – José Figueroa-O'Farrill Oct 18 '11 at 20:52
  • $\begingroup$ There are a parallel 3- and 4-form $C_3$ and $C_4$, say, with respect to the invariant connection and indeed one has that $C_4 = \star C_3$ and $dC_3 \propto C_4$, whence that $G_2$ structure is of the "weak $G_2$ holonomy" type. $\endgroup$ – José Figueroa-O'Farrill Oct 18 '11 at 20:55
  • $\begingroup$ One final comment for now: the geometry in the Klebanov et al. paper is reminiscent of the compactification found by Pope and Warner which consists of viewing the $S^7$ as the total space of a line bundle over $\mathbb{CP}^3$ and introducing a flux on $S^7$ breaking the symmetry down to $\mathrm{SU}(4)$, which still acts transitively. Indeed, $S^7$ is diffeomorphic to $\mathrm{SU}(4)/\mathrm{SU(3)}$. There is (apart from the overall scale) a one-parameter family of $\mathrm{SU}(4)$-invariant metrics on $S^7$: corresponding to a rescaling of the fibre metric. $\endgroup$ – José Figueroa-O'Farrill Oct 18 '11 at 21:03

I wanted to point out this paper of Gauntlett-Sonner-Wiseman


which shows explicitly that the Englert and Pope-Warner solutions (on any Sasaki-Einstein manifold $X^7$) can be obtained via holographic RG flows from the Freund-Rubin $AdS_4\times X^7$ solution. As you pointed out the Englert solution is perturbatively unstable. In this paper


we showed that the Pope-Warner solution on $S^7$ is also pertubatively unstable. This result was later generalized to the Pope-Warner solution on any Tri-Sasakian manifold in


Let me also point out that there are supersymmetric (1/4 BPS) warped $AdS_4$ solutions with flux which describe non-trival IR fixed points of the ABJM-BLG theory. These were discussed here in supergravity


and here (and other references later)


in field theory.

Finally there are even less supersymmetric (1/8 BPS) $AdS_4$ solutions with internal flux which can again be interpreted as the IR fixed point of an RG flow from the ABJM theory




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