2-Branes in 11 Dimensions

June 15, 2009

Posted by John Baez

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I’m trying to learn a teeny bit more about supersymmetric membrane theories, and I’m so far behind that this old review article is proving helpful:

Micheal J. Duff, Supermembranes: the first fifteen weeks, Classical and Quantum Gravity 5 (1988), 189–205.

I’m particularly fascinated by the classification of ‘fundamental super $p$ -branes’ and its relation to normed division algebras:

Reals: 2-branes in 4 dimensions, with 1 bosonic and 1 fermionic degree of freedom.
Complexes: 3-branes in 6 dimensions, with 2 bosonic and 2-fermionic degrees of freedom.
Quaternions: 5-branes in 10 dimensions, with 4 bosonic and 4 fermionic degrees of freedom.
Octonions: 2-branes in 11 dimensions, with 8 bosonic and 8 fermionic degrees of freedom.

I talked about this ‘brane scan’ in incredibly elementary terms back in week118, but now I’d like to actually understand it. It’s supposed to follow from something like a classification of closed differential forms on super-Minkowski spacetimes, due to Achúcarro et al. I don’t know how it goes, but it seems potentially quite comprehensible.

If anyone can help me with this, I’d appreciate it a lot. But I really want to say a word about the 11-dimensional case — a brief explanation for complete novices such as myself — and then ask a question about that.

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In its so-called ‘Nambu–Goto formulation’, a 2-brane in 11 dimensions is classically just a map

$\phi: \Sigma \to M$

where $\Sigma$ is an oriented 3-manifold and $M$ is ‘11-dimensional super-Minkowski spacetime’. What’s that? Well, we can crassly think of $M$ as a vector space that’s split in two parts, the ‘even’ and ‘odd’ part:

$M = \mathbb{R}^{11} \oplus \mathbb{R}^{32}$

Here $\mathbb{R}^{11}$ is plain old 11-dimensional Minkowski spacetime, while $\mathbb{R}^{32}$ is a certain representation of the double cover of the Lorentz group in 11 dimension: the space of ‘Majorana spinors’.

As usual in classical field theory, to count as a solution of the equations of motion, the map $\phi : \Sigma \to M$ needs to be a critical point of some action. And I guess this action is the sum of two parts.

First, any 3-dimensional manifold mapped into Minkowski spacetime gets a notion of ‘volume’, and I guess the same sort of thing works in the super world. So, the first term in the action is just the ‘supervolume’ of the image of $\Sigma$ in $M$ .

The second term is more special: it’s the integral of some 3-form over $\Sigma$ — a 3-form obtained by taking a god-given 3-form on $M$ and pulling it back along $\phi$ . This is presumably where the classification of closed differential forms on super-Minkowski spacetimes comes in handy.

Now comes my question. Duff says that something called ‘Siegel symmetry’ cuts the actual number of fermionic degrees of freedom in half — at least ‘on shell’, meaning when the equations of motion hold. I’m wondering if there’s any way to describe the 2-brane so that this cutting in half occurs right from the start. For example, say we take our 11 dimensions and split them into 10 plus 1 extra, thus writing

$M = \mathbb{R}^{10} \oplus \mathbb{R} \oplus \mathbb{R}^{16} \oplus \mathbb{R}^{16}$

where $\mathbb{R}^{10}$ is 10-dimensional Minkowski spacetime, $\mathbb{R}$ is the extra space dimension, and $\mathbb{R}^{32}$ gets split up into $\mathbb{R}^{16} \oplus \mathbb{R}^{16}$ , namely the left- and right-handed Majorana-Weyl spinors in 10 dimensions. Can we somehow write down an action for our 2-brane that just involves this portion of the map $\phi$ :

$\tilde{\phi}: M \to \mathbb{R}^{10} \oplus \mathbb{R} \oplus \mathbb{R}^{16} \; ?$

This would be cute, because $\mathbb{R}^{10} \oplus \mathbb{R} \oplus \mathbb{R}^{16}$ is just another way of thinking about the exceptional Jordan algebra — that is, the space of $3 \times 3$ self-adjoint octonion matrices!

Any clues would be most greatly appreciated. I’m sure this is pretty basic stuff to those in the know. What I’m hoping for may just not work.

Posted at June 15, 2009 11:40 AM UTC

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Re: 2-Branes in 11 Dimensions

If you are looking for beautiful algebraic symmetries, I think you will like Peter West’s works on M-Theory and supergravity. I think this is what you want.

He studies an algebra called E11, and has been somewhat successful in trying to prove this conjecture.

Posted by: Daniel de França MTd2 on June 15, 2009 7:24 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

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I’m interested in $E_9$ , $E_{10}$ and $E_{11}$ ; Hermann Nicolai has also written papers on these. But that’s not what I want to know about today!

I have rewritten my blog entry to perhaps make it a bit clearer what I’m wondering about.

Posted by: John Baez on June 15, 2009 7:59 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Correct link: E11

Posted by: Toby Bartels on June 15, 2009 8:32 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Thanks, Toby! I was having trouble properly formatting Daniel’s link to

http://en.wikipedia.org/wiki/En_(Lie_algebra)

because of the underlines and parentheses. You knew the magic trick:

http://en.wikipedia.org/wiki/En_%28Lie_algebra%29

Posted by: John Baez on June 15, 2009 9:42 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Oh, you don’t really want branes from string theory, but you want branes that has something to do with normed division algebras, don’t you?

Posted by: Daniel de França MTd2 on June 15, 2009 8:19 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

I’m talking about the usual supersymmetric membranes that string theorists study. These are known to be related to normed division algebras. In particular, I’m talking about the supersymmetric 2-brane in 11 dimensions, which dimensionally reduces to give the type IIA superstring. Read Duff’s paper if you can access it. It’s a good introduction.

Posted by: John Baez on June 15, 2009 9:23 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

A certain someone said:”16 components of the spinor is that in his optics, such a modified 11-dimensional spacetime looks just like the exceptional Jordan algebra of 3x3 Hermitean octonionic matrices, the algebra whose automorphism group is an E_6.”[…] “There’s no canonical E_6 symmetry in the infinite flat-space 11-dimensional M-theory, so there can’t be any exceptional Jordan algebra, either.”

Now, I terribly confused by that, because you are talking about features that according to that person, are not there. Who is right? Both? No one? I am getting lost…

Posted by: Daniel de França MTd2 on June 16, 2009 2:14 AM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Well, I do not have access to old articles, but I guess this preprint is good enough. :)

Posted by: Daniel de França MTd2 on June 16, 2009 2:32 AM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Thanks for that link to a preprint of Duff’s paper. I’ve added that link to my blog entry.

Posted by: John Baez on June 16, 2009 11:44 AM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

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Daniel wrote:

“…the exceptional Jordan algebra of 3 × 3 Hermitean octonionic matrices, the algebra whose automorphism group is an $E_6$ .”

It’s not very important here, but just to set the record straight:

The automorphism group of the exceptional Jordan algebra is not $E_6$ , but merely a subgroup of that, namely the compact real form of $F_4$ . A certain noncompact real form of $E_6$ acts on the exceptional Jordan algebra, but not in a way that preserves the product or unit: it only preserves a lesser structure, namely the ‘determinant’. Every linear transformation of the exceptional Jordan algebra that preserves the product and unit preserves the determinant, but not conversely.

You can read about the exceptional Jordan algebra and its determinant function here.

“There’s no canonical $E_6$ symmetry in the infinite flat-space 11-dimensional M-theory, so there can’t be any exceptional Jordan algebra, either.”

Now, I terribly confused by that, because you are talking about features that according to that person, are not there. Who is right? Both? No one? I am getting lost…

As far as this particular quote goes the answer could be both. I’m interested to hear that $E_6$ doesn’t act as symmetries on infinite flat-space 11-dimensional M-theory. I’d be happier if someone could point me to a proof — where by ‘M-theory’ we could either mean ‘the theory of the supersymmetric 2-brane’ or ‘perturbative 11d supergravity’.

But if you read my post, you’ll see I was never insisting that every symmetry of the exceptional Jordan algebra (either in $E_6$ or the smaller group $F_4$ ) act as a symmetry of M-theory. I was just asking two specific questions:

How does the ‘classification of closed differential forms on super-Minkowski spacetimes’, to which Michael Duff alludes, actually work?
Is there a way to exploit the Siegel symmetry of the supersymmetric 2-brane in 11 dimensions to give an $so(9,1)$ -invariant formulation of this theory that only uses one handedness of 10d Majorana–Weyl spinors?

And, I’m still eager to know the answers.

For the first question, I probably just need to read this:

A. Achucarro, J.M. Evans, P.K. Townsend, and D.L. Wiltshire, Super $p$ -Branes, Phys. Lett. B198 (1987), 441.

Maybe I’ll do that today.

As for the second question, I hope it’s clear that by ‘ $so(9,1)$ -invariant formulation’ I roughly mean a formulation where we let ourselves ‘break the symmetry’ of 11-dimensional Minkowski spacetime by splitting it into 10-dimensional Minkowski spacetime and one extra spatial dimension. Fun stuff can happen when you do this. For example, check out this paper where the authors take 11d supergravity and break the symmetry by splitting 11-dimensional Minkowski spacetime into 3-dimensional Minkowski spacetime and 8 extra spatial dimensions. They discover a formulation that involves the group $E_8$ !

Posted by: John Baez on June 16, 2009 11:12 AM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Hi John, the person from whom I quoted above also provided this link in response to this comment:
“There is some connection between the division algebras and minimal susy yang-mills theories which was first pointed out many years ago by Kugo and Townsend. The lorentz groups SO for D=3,4,6 and 10 are isomorphic to SL for K=R,C,H and O. This means that spinors in these dimensions can be expressed as 2-component K-vectors.”

Posted by: Daniel de França MTd2 on June 16, 2009 1:10 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

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Yup — none of this is news. I explained how normed division algebras can be used to describe Lorentz groups and spinors in dimensions 3, 4, 6 and 10 in week104, almost exactly 11 years ago. You can read a more detailed account in my octonion review paper. I’ve been thinking about this stuff on and off ever since, and now I’m working on it with my grad student John Huerta.

Posted by: John Baez on June 16, 2009 3:25 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

John Huerta is your student that is the one that is studying GUTs? Well, since you are talking about octonions, branes, E8, etc… Is he trying to expand his cube of GUTs to a hypercube including a hypercube that has an E8 GUT? Coincidentely, lately, there has been a lot of talk about an E8 GUTs created near due a certain singularity from branes of a certain compactification of F-Theory. When I see all the elements of this E8 F-Theory GUT on this very thread, it makes me wonder if there is anything to do with that.

Posted by: Daniel de França MTd2 on June 16, 2009 5:15 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

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Daniel wrote approximately:

John Huerta is your student that is studying GUTs?

Yes, that’s him.

Is he trying to expand his cube of GUTs to a hypercube including a hypercube that has an E8 GUT?

No, right now he’s mainly trying to find the best explanation of why normed division algebras lead to supersymmetric theories in dimensions 3,4,6 and 10. It obviously relies on the isomorphism

$SL(2,K) \cong Spin(n+1,1)$

where $K$ is the $n$ -dimensional normed division algebra, but we’re looking for the maximally elegant explanation of why this isomorphism yields supersymmetric Yang–Mills and superstring Lagrangians. A really elegant explanation might teach us something new someday.

Both this and our expository paper on GUTs are warmups for some more ambitious projects which are, of course, top-secret.

Posted by: John Baez on June 16, 2009 6:14 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Hi John

Duff’s recent work has directly linked Jordan algebras and their Freudenthal triple systems to entropy expressions for d=5 and d=4 extremal black holes and entanglement measures for qubits and qutrits.

Take a look at: arXiv:0809.4685v4 [hep-th], arXiv:0812.3322v3 [quant-ph] and arXiv:0903.5517v3 [hep-th].

Jordan algebras over the split division algebras and their corresponding Freudenthal triple systems are related to M-theory via toroidal compactifications of d=11 supergravity, as compactifying down to d dimensions yields the maximally extended supergravity with global non-compact symmetry group E_{(11-d)(11-d)}. The discrete subgroups of E_{(11-d)(11-d)}, namely E_{(11-d)(11-d)} over the integers, contain the symmetries of the non-perturbative spectra of toroidally compactified M-theory, and are regarded as U-duality groups (arXiv:hep-th/0409263v1).

For example, by compactifying down to 5 dimensions, we recover the group E_{6(6)}, which acts as a symmetry of the Lagrangian of the maximal N=8 supergravity. The entropy of extremal black hole solutions in 5D, N=8 is invariant under E_{6(6)}, as the entropy is proportional to the square root of the determinant in the cubic Jordan algebra over the split octonions.

There is a similar entropy expression in exceptional N=2 supergravity, where the black hole charge space is described by the exceptional Jordan algebra, and the corresponding symmetry group is E_{6(-26)}, the collineation group of OP^2. However, relating this theory to a string compactification is work in progress (arXiv:0712.2976v1 [hep-th]).

Posted by: kneemo on June 21, 2009 4:44 AM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Although this comment will probably be deleted for being stupid and irrelevant, as usual, I feel I must ask: have you ever considered the possibility that gauge theory (especially using classical groups) might not be the right way to understand how the number fields appear in M theory?

Posted by: Kea on June 17, 2009 11:14 AM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

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Kea is right about one thing.

Unlike Dp-branes, the worldvolume theory on an M2-brane is not a gauge theory.

It is believed to be realizable as the strongly-coupled IR fixed point of a gauge theory. But it has rather different physics:

Superconformal symmetry, instead of supersymmetry.
A $Spin(8)$ R-symmetry group, instead of $Spin(7)$ .
Etc.

In some cases, it actually does have a Lagrangian description (surely the names “Bagger-Lambert” and “ABJM” have not totally escaped your attention).

The M5-brane is even further divorced from gauge theory. The worldvolume theory is one of the infamous (2,0) superconformal theories that exist in 6 dimensions. Unlike the M2 case, these cannot be realized as the IR fixed point of anything with a Lagrangian description (for the simple reason that there are no nontrivial Lagrangian QFTs in 6 dimensions).

Instead, if you did the analogous thing that gave you the gauge theory description (which flowed in the infrared to the M2-brane theory), you would end up with a 6-dimensional “little string theory” – a critical string theory in 6 dimensions which does not contain gravity (hence “little”).

…especially using classical groups

Exceptional gauge groups play no role in the above discussion of the M2 brane (though there is an ADE classification of (2,0) superconformal theories in 6 dimensions).

Posted by: Jacques Distler on June 17, 2009 4:07 PM | Permalink | PGP Sig | Reply to this

Re: 2-Branes in 11 Dimensions

Hi Jacques,

I have a questions about M5 branes:

“The worldvolume theory is one of the infamous (2,0) superconformal theories that exist in 6 dimensions.”

I frequently see this theory in Witten’s recent articles on langlands program. Is he trying to use langlands to understand M5 branes?

Can you indicate a nice article showing the weird “missing” properties of these M2 and M5 branes, respectively like:

M5:”**cannot** be realized IR fixed point of anything with a Lagrangian description (for the simple reason that there are no nontrivial Lagrangian QFTs in 6 dimensions).”

M2:”It is **believed** to be realizable as the strongly-coupled IR fixed point of a gauge theory.”

Posted by: Daniel de França MTd2 on June 17, 2009 4:45 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

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surely the names “Bagger-Lambert” and “ABJM””have not totally escaped your attention

Around here we were of course interested in the aspect of Lie 3-Algebras on the Membrane (?) involved with this.

While I do know about various relations of twice categorified algebras related to the membrane, I didn’t quite manage to see if what Bagger-Lambert call a “3-algebra” is really anything of this sort.

Posted by: Urs Schreiber on June 17, 2009 6:02 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

There is a standard reference to Filipov around 1985 whose title includes words similar to “n-algebra”. So if the term can be copyrighted, BL is probably using it in the correct sense.

Posted by: Thomas Larsson on June 19, 2009 6:17 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

are you saying a BL 3-alg is indeed a Filipov 3-alg?

Posted by: jim stasheff on June 19, 2009 10:11 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Hm. I thought so, but after some googling I came across http://arxiv.org/abs/0901.3905v1, which attempts to clarify the relation between BL, Filippov, and shLie algebras. It seems that Filippov n-algebras are a special case of shLie algebras, but I am unclear if BL n-algebras are a further special case.

Posted by: Thomas Larsson on June 20, 2009 9:24 AM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

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are you saying a BL 3-alg is indeed a Filipov 3-alg?

The outcome of our discussion of this last time was summarized on the $n$ Lab at [[n-Lie algebra]], where it says that indeed these are the same.

But whatever these definitions are called, it doesn’t seem to answer my question:

While I do know about various relations of twice categorified algebras related to the membrane, I didn’t quite manage to see if what Bagger-Lambert call a “3-algebra” is really anything of this sort.

Here “anything of this sort” refers to the beginning of the sentence: whether a “BL/Filipov 3-algebra” that is not at the same time a [[Lie 3-algebra]] (notice the difference!) in that it lacks or does not respect the required grading is any genuine higher structure.

Here by “genuine higher structure” I mean something that is not just a random $n$ -ary operation but truly the categorification of some lower structure, or else otherwise conceptually higher, for instance in that it is an algebra for a cofibrant resolution of some operad.

And anything that deserves to be called a higher Lie algebra for instance should have a systematic integration to a higher Lie group. This is the case for Lie $n$ -algebras, and there it is crucially the grading that makes this work.

We know of Lie 3-algebras and higher Lie $n$ -algebras that play a role for membrane physics. The Chern-Simons Lie 3-algebra, foremost, and its incarnation in the AKSZ description of membrane quantizaton in particular.

I wouldn’t be surprised if trinary operations that one sees when writing down the action for the supermembrane have similarly secretly an interpretation as such higher structures. But so far

a) it remains unclear to me to which extent $n$ -Lie algebras (as opposed to Lie $n$ -algebras) are sufficient or even useful for that

b) and then when following the development of the BL idea as recounted for instance here it even began to look like the original idea of the relevance of “3-algebra” somehow evaporated along the way.

All this doesn’t necessarily mean that BL aren’t onto something higher category-wise. It is clear that membrane physics (being 3d QFT) is governed by 3-categorical structures and traces of that are bound to show up even in the most local and coordinate-ridden descriptions. But if the BL “3-algebra” is an example of this seems to not have been made conclusive yet.

At least as far as I have followed this. I’d be glad to be shown otherwise.

Posted by: Urs Schreiber on June 20, 2009 10:08 AM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Urs wrote:

Here by “genuine higher structure” I mean something that is not just a random n-ary operation but truly the categorification of some lower structure, or else otherwise conceptually higher, for instance in that it is an algebra for a cofibrant resolution of some operad.

OK, but then what do we call n-ary structures that are ordinary ungraded algebra?

Posted by: jim stasheff on June 20, 2009 2:07 PM | Permalink | Reply to this

Re: 2-Branes in 11 Dimensions

Could it be that the reason that M2 and M5 branes do not have a lagrangian gauge theoretical description it is that both of them are not treated as the same thing in different referentials or gauges? Just like the electromagnetic fields with magnetic monopoles and electric charges.

Posted by: Daniel de França MTd2 on July 28, 2009 11:08 PM | Permalink | Reply to this

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June 15, 2009