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:

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 :)