Friday, May 06, 2005

E11 in the Reference Frame

It is the election here in the UK tonight, I have poured myself a glass of wine and am watching Peter Snow leap joyously between his swingometer and his triangulated battleground (ITV are using ELVIS - their ELection VISualiser) and listening to pundits make extraordinary use of statistics (in one case representing the entire election result from a result in just one seat!). But, as the counting drags out, the gleam of the graphics fade and the studio discussions are recycled, I find myself needing a secondary activity to keep me occupied. So I have decided to write something in defense of poor old E11 after reading Lubos Motl despair that almost nothing is known about its properties. I should add that Lubos was making a passing comment in an article about Hermann Nicolai's promising cosmological billiards programme using E10, which certainly deserves attention and it is pleasing that it is being discussed in his Reference Frame.

First let me quote Victor Kac who once wrote, "It is a well-kept secret that the theory of Kac-Moody algebras has been a disaster." Those familiar with E11 will know that it has a Kac-Moody algebra and by extension is a disaster, so Lubos is quite right in what he says, nevertheless there has been some progress in uncovering its association with eleven dimensional supergravity.

E11 is the result of extending the E8 dynkin diagram by adding three nodes to it, giving:

Dynkin diagram of E11
The algebra that results is a Kac-Moody algebra, and infinite dimensional. This means that, as I overheard my supervisor in an echo of Kac, Peter West, telling a visiting speaker in the coffee room, "E11 is a complete mess". Since Peter has had me work on nothing but E11 since I started my PhD this is a favourite quote of mine, and perhaps I will put it on the front of my thesis, if I get there.

There has been plenty of good news about E11 since Peter made his E11 conjecture (that E11 is the symmetry group giving rise to the M-theory dualities). But first some groundwork. The connection with 11-dimensional supergravity comes through decomposing its algebra into representations of A10 i.e. SU(11). The longest line of ten roots in the E11 Dynkin diagram is the A10 used, and is often called the gravity line. A restriction to the real form of SU(11) i.e. SL(11,R) is made, and then by including the eleventh generator from E11 (the one from the node that sticks out on the Dynkin diagram) this algebra is enlarged to GL(11,R). For the usual (1,10) signature of low energy M-theory/supergravity, the vielbein is an element of a coset of this group, namely of GL(11,R)/SO(1,10).

The decomposition gives an infinite number of generators, classified by their Dynkin labels, and in particular the Dynkin label of the eleventh root which is called the level. Low level tables of these generators can be found here, back when E11 was known as E8+++. It was noted that a truncation of the algebra, to generators of level 3 and less, leads to eleven dimensional supergravity fields. Furthermore, and this is the nicest result I have seen so far, you can find the 10-dimensional theories from E11 too. In this case the vielbein is a member of GL(10,R)/SO(1,9), and the decomposition is of E11 into A9 representations. In this case there are two distinct ways to pick A9, which is a straight line of nine nodes on the Dynkin diagram:

1. using the first nine nodes along the horizontal of the E11 diagram above
2. using the first eight horizontal nodes and the one orthogonal node on the above diagram

In the first choice the IIA theory is found, and in the second choice we find the IIB theory. More about this can also be read here. This is a bit more than one would expect from dimensional reduction, because chirality appears.

Further results relating D=11, IIA and IIB, which are "central to string theory" are given in hep-th/0407088. My single piece of work has been concerned with a group element of E11(although the group element is also applicable to all the oxidised supergravity theories built from any of the very-extensions of the semisimple Lie groups) that encodes the vielbein for the half BPS cases, so that a generator resulting from a decomposition is associated with a brane solution. For the low level generators which coincide with 11-dimensional supergravity and M-theory, the M2, M5 and the pp-wave are found, as you would expect. The real question is what role do the other generators, which extend off to infinity in number, play?

I have presented a slightly skewed presentation of E11 research, so let me redress this and point out that significant research has also been carried out on E11 by Englert, Houart, Keurentjes and the Mkrtchyans, amongst others.

So, while almost nothing is known about E11, the little that is known is very promising and quite exciting. Of course the next challenge is to find some fermions...

Footnote: This was posted a few days after starting it, as my graphics card broke. The very last recounts of the votes in the election have been completed, and the results almost perfectly matched the exit polls. Nevertheless the whole counting process, and crazy graphics that goes with it, was quite good fun, and I think we should keep doing it :)


Urs said...


maybe you can help me with the following questions:

The E_10 conjecture says that the dynamics of M-theory is encoded in a geodesic on the group manifold of E_10/K(E_10).

What would be the analogous statement for E_11? Is this geodesic mapped to a point in some group manifold of E_11/K(E_11) or something?

Hermann Nicolai is thinking about how to incorporate SUSY into the E_10 scenario. Is anything known about susy from the E_11 persepctive?

Others have attempted to quantize the motion of a point on the E_10 group manifold and have claimed to find brane degrees of freedom this way. What would be the analogue of this quantization step from the E_11 point of view?

P.P. Cook said...

Hi Urs,

Tough questions :) - I haven't read any of the E_10 papers yet, although I do keep a copy of the first "Cosmological Billiards" paper on my desk, and hope to get to grips with it soon. So excuse me if my answers aren't very well informed.

The E_10 conjecture, as you say, proposes that a null geodesic motion in E_10/K(E_10) encodes the bosonic field equations of M-theory, where K(E_10) is the maximal compact subalgebra. There is, as far as I know (and this is a significant caveat), no exactly equivalent approach for E_{11}. But it is very similar, and, I believe, almost exactly as you suggest. Although without reading through the cosmological billiards work in detail this is hard to say :)

For E_{11} the conjecture is that M-theory has a non-linear E_{11} symmetry, encoded in E_{11}/H_{11}. So anything of dynamical importance comes from this coset. Now, and this is where the approaches must differ, taking the maximal compact subalgebra as H_{11} gives a Euclidean 11-dimensional background. So H_{11} is chosen to be non-compact. The recipe for picking which generators form a basis of H_{11} is sometimes referred to as the "temporal involution". The coset may be chosen to correspond to a (1,10) theory, if you have picked the H_{11} relevant to M-theory. There's a very good overview of compact and non-compact local subalgebras in E_{11} in Keurentjes exciting paper hep-th/0402090 - see section 3.1, which has some surprising consequences, such as that picking (1,10)~M, automatically leads to (2,9)~M* and (5,6)~M' too. The vielbein/elfbein corresponding to the usual static brane solutions is found from the group element referred to in the post above, which is an element of E_{11}/K(E_{11}). So, I think the analog is as you suggest.

SUSY is the most worrying absentee from the E_{11} picture. I know of no positive news in this direction. The only person I have spoken to about it was very doubtful that it was straightforward. Since I have only thought about it briefly, I have the opposite opinion, but when I know more I expect to come round to their point of view :)

As to your final question, I don't know the answer! But could you let me know where I could read about the extra brane degrees of freedom from E_{10}? It sounds very interesting.

Best wishes,

P.P. Cook said...

Oops, I made a mistake. The sentence:

"...the group element referred to in the post above, which is an element of E_{11}/K(E_{11})."

should end


The difference being as I mentioned that H_{11} is not generally the maximal compact subgroup.

Best wishes,

Thomas Larsson said...

West's work on E_11 was discussed on Google a year ago. I find it quite amusing that I seem to know more about this part of the stringy literature than Urs and Lubos.

My post even contains the Kac quote. I cannot speak for Kac, of course, but I doubt that the recent physics works really changes his assessment. Remember that he complains about the lack of explicit realizations for non-affine Kac-Moody algebras. The works of Nicolai, West and others realize E_10 and E_11 as polynomial vector fields on certain non-holonomic manifolds. However, these manifolds are very infinite-dimensional - the number of coordinates grows exponentially with level and only finitely many coordinates in the beginning are known.

Moreover, the existence of these realizations are not special to E_10 and E_11. E.g. E_4711 admits an analogous grading with g_0 = A_4710 + C. E_11 also admits other realizations, e.g. with g_0 = D_10 + C or g_0 = A_1 + A_2 + A_7 + C. So I don't think that there is anything mathematically special about the E_11 grading with g_0 = A_10 + C. As for physics, I'll believe in 11D SUGRA when sparticles and extra-dimensions are detected experimentally.

Urs said...

Hi P. P.,

thanks for the reply!

The paper on quantization of the free point motion on E_10 that I had in mind was

Brown, Ganor and Helfgott
M-theory and E10: Billiards, Branes, and Imaginary Roots

I have once written a little summary of the basic ideas of Nicolai's proposal on the SCT.

I haven't checked if there are any new developments along these lines.
I wish I had more time studying this stuff. If you keep us posted about progress in this area you are sure to have a couple of grateful readers.

Roslyn said...

Really effective data, thank you for the article.

John Baez said...

Hi! I've been thinking about E11 lately, and your thesis is being very helpful.

I notice the E11 Dynkin diagram in this blog article has disappeared.

The comment above mine, by "Roslyn", is almost surely spam.