Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

Please welcome our new moderators!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

122 submissions , 103 unreviewed
3,497 questions , 1,172 unanswered
4,543 answers , 19,337 comments
1,470 users with positive rep
407 active unimported users
More ...

Modern avatar of Englert's solution?

+ 15 like - 0 dislike
14 views

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.

This post has been migrated from (A51.SE)
asked Oct 18, 2011 in Theoretical Physics by José Figueroa-O'Farrill (2,125 points) [ no revision ]
retagged Apr 19, 2014 by dimension10
So it appears that the Englert solution preserves no supersymmetry (Englert, Rooman and Spindel http://inspirehep.net/record/190303) and is unstable (Page and Pope http://inspirehep.net/record/201484 and de Witt and Nicolai http://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.

This post has been migrated from (A51.SE)
(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.

This post has been migrated from (A51.SE)
Does this solution preserve any supersymmetry? If not then I guess an AdS/CFT interpretation could be problematic if it is unstable.

This post has been migrated from (A51.SE)

1 Answer

+ 5 like - 0 dislike

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

http://arxiv.org/abs/hep-th/9803039

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.

http://arxiv.org/abs/hep-th/0505207

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.:

http://arxiv.org/abs/hep-th/9706124

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.:

http://arxiv.org/abs/hep-th/9907219

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

http://www.sciencedirect.com/science/article/pii/S0550321302000421

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

http://arxiv.org/abs/0809.3773

that are claimed to be of Englert's type.

This post has been migrated from (A51.SE)
answered Oct 18, 2011 by Luboš Motl (10,178 points) [ no revision ]
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.

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)
I see, thanks for the clarrified connections etc., @José.

This post has been migrated from (A51.SE)
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)

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsO$\varnothing$erflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...