### Nicolai on E10 and Supergravity

#### Posted by Urs Schreiber

We have had several discussions here on how (parts of) the Lie algebra of the gauge group governing 11-dimensional # and 10-dimensional # supergravity can rightly be thought of in terms of semistrict Lie 3-algebras (equivalently: 3-term $L_\infty$-algebras).

There are various reasons that make some people expect that these various supergravity theories describe certain facets of some essentially unknown single entity. The working title of this unknown structure is “M-theory”. You’ll see one proposal for a precise statement of this “M-theory hypothesis” in a moment.

In our discussions, I had made a remark on how the various Lie 3-algebras that play a role in supergravity might - or might not - be merged into a single structure here.

John rightly remarked that

This M-theory Lie 3-superalgebra should ultimately be something very beautiful and integrated, not a bunch of pieces tacked together, if M-theory is as Magnificent as it’s supposed to be.

There are various indications that the unifying governing structure behind much of the supergravity zoo are **exceptional Kac-Moody algebras**, in particular $e_8$, $e_9$, $e_{10}$
and maybe also $e_{11}$.

In particular, if one takes 11-dimensional supergravity and compactifies it on a 10-dimensional torus, the resulting 1-dimensional field theory exhibits a gauge symmetry under the gauge group $E_{10}/K(E_{10})$, where $E_{10}$ denotes something like the group manifold $\exp(e_{10})$ and $K(E_{10})$ something like the maximal compact subgroup of $E_{10}$.

This, combined with the observation of certain symmetries appearing in the chaotic dynamics of (super)gravity theories near spacelike singularities, has lead a couple of people, most notably H. Nicolai, T. Damour, M. Henneaux, T. Fischbacher and A. Kleinschmidt, to suspect that the classical dynamics encoded in the equations of motion of 11-dimensional supergravity, including its higher order M-theoretic corrections, correspond to geodesic motion on the group manifold of the Kac-Moody group $E_{10}$, or rather the coset $E_{10}/K(E_{10})$.

Since $E_{10}$ is hyperbolic, it is, with current technology, impossible to conceive it in its entirety. Hence all this work is based on a technique, where one uses a certain *level truncation* of the Kac-Moody algebra $e_{10}$ to obtain tractable and useful approximations to the full object.

The idea is that expanding geodesic motion on $E_{10}/K(E_{10})$ in terms of levels this way, corresponds to expanding a supergravity theory in powers of spatial gradients of its fields close to a spacelike singularity.

For several years now, Hermann Nicolai and collaborators have slowly but steadily checked this hypothesis for low levels.

I had once reviewed some basic aspects of this here.

So far, to the degree of detail that has become accessible, the hypothesis has proven to be correct. And, as the *M-theory hypothesis* would suggest, not only can 11-dimensional supergravity be found, level by level (up to level 3, so far), in the geodesic motion on $E_{10}/K(E_{10})$, but higher levels seem to correctly reproduce higher order corrections to supergravity which have been derived by other means. Moreover, depending on how one “slices” $e_{10}$ by means of its subalgebras, one finds that the same geodesic motion also reproduces the other maximal supergravity theories, like 10-dimensional type IIB supergravity and massive 10-dimensional IIA supergravity.

Up to recently, all this work was restricted to the bosonic degrees of freedom of these theories. One of the remarkable aspects of the $E_{10}$ theory was that it gave rise to the various bosonic fields that accompany the graviton field (the Riemannian metric) in supergravity theories, like the supergravity 3-form, and which ordinarily appear only after one requires supersymmetry.

Still, one would like to check the entire program also against the fermionic fields, like the gravitino. The obvious guess is that these appear on the $E_{10}$-side as we pass from the geodesic motion of a spinless particle on $E_{10}/K(E_{10})$ to the motion of a spinning particle.

Results on this part of the project are now also appearing. Today has appeared a new preprint, where further progress in this direction is discussed:

Axel Kleinschmidt, Hermann Nicolai
*K(E9) from K(E10)*

hep-th/0611314.

Among other things, it is discussed how $K(E_{10})$ has certain finite-dimensional spinorial representations under which - on the corresponding supergravity side of things - the equation of motion of the gravitino is covariant.

Today Hermann Nicolai visited Hamburg and gave a talk on this stuff:

E. Nicolai
*$E_10$: Prospects and Challenges*

(slides).

Since the slides of the talk are available online, I will not try to transcribe the notes that I have taken. Instead, I will just focus on certain aspects.

In that previous discussion we had #, John went on to say

However, for all the exceptional Lie algebras (and superalgebras) the simplest description starts by sticking together a bunch of pieces. Only at the end do you see that there are many ways to decompose the same beautiful thing into these pieces. For example, consider the Lie algebra $e_8$. As a vector space, it’s a direct sum $e_8 \simeq \mathrm{so}(16) \oplus S_{16}^+$, where $S_{16}^+$ is the chiral spinor rep of $\mathrm{so}(16)$. But the bracket on this space is a bit subtle, so the above direct sum is not a direct sum of Lie algebras in any way (and indeed, $S_{16}^+$ isn’t even a Lie algebra). In the end, there turns out to be no “preferred” splitting of $e_8$ like this - just a lot of different but equivalent splittings. We’ve taken a beautiful jewel and described it only after cracking it down the middle in an arbitrary way, ruining its symmetry.

I hope the M-theory Lie 3-superalgebra is like this.

Interestingly, H. Nicolai finds that pretty much something like this choice of different splittings of a single jewel is exactly what gives rise to the various flavours of maximal supergravity theories.

This works roughly as follows.

As I mentioned, since $e_{10}$ is a hyperbolic Kac-Moody algebra (I review some basic elements of Kac-Moody algebras here) it is hard to deal with. One way to make progress is to choose a tractable *sub* Kac-Moody algebra of it. Acting with this subalgebra on the rest of $e_{10}$ makes all of $e_{10}$ an infinite dimensional representation of that subalgebra. One can hance get a handle on the structure of $e_{10}$ by decomposing it into irreps under the action of any one of its sub Kac-Moody algebras.

This is also where the *level truncation* comes in, though I realize I would have to look that up again. The idea is that with $\{\alpha_j\}$ the roots of the subalgebra, and $\alpha_0$ some special root, we restrict to the root sub-lattice of roots of the form $\alpha = \sum_j m^j \alpha_j + \ell \alpha_0$. Here $\ell$ is the level.

Now, we get sub KM-algebras of $e_{10}$ be deleting one of the nodes of the Dynkin Diagram of $e_{10}$.

The Dynkin diagram of $e_{10}$ looks like

By deleting the 10th vertex, we get the Dynkin diagram of type $A_9$, corresponding to $\mathrm{sl}(10)$. For the result of decomposing $e_{10}$ with respect to irreps of this $\mathrm{sl}(10)$, and ordered by level $\ell$, Nicolai writes

The claim now is that in levels 0 through 3, we find in $e_{10}$ now the irreps of $\mathrm{sl}(10)$ that correspond to the bosonic field content of 11-dimensional supergravity, namely the graviton ($\ell = 0$), the 3-form ($\ell = 1$), the magnetic dual of the 3-form ($\ell = 2$) and a dual of the graviton $(\ell = 3)$. We can literally think of the $\mathrm{so}(10)$ here as coming from the rotations in tangent space after we have compactified 11-dimensional supergravity on a 10-manifold.

And playing the same game for sub-KM-algebras obtained by deleting other roots leads to a couple of other supergravity theories. For instance,

yields 10-dimensional type IIB supergravity and

yields what is called massive type IIA supergravity. These 10-dimensional supergravity theories contain more types of fields than are present in eleven dimensions, in particular there are the Ramond-Ramond $p$-form fields. All these, the claim is, do appear in the corresponding level decomposition of $e_{10}$.

What is not in the slides linked to above is a remark on the type I and the heterotic flavor of supergravity. In his talk, H. Nicolai remarked that one supergravity theory that does not seem to appear is that corresponding to the heterotic string. On the other hand, type I, which is supposed to be dual to the heterotic thing, was claimed to be obtainable. There was a question concerning this point, but I realize I can’t give a coherent account of the answer that was given.

In any case, the picture emerging here is that various kinds of supergravity theories, all of which are long conjectured to be facets of “M-theory”, all appear from geodesic motion on $E_{10}/K(E_{10})$, the difference between them being merely the difference in the choice of spatial hypersurfaces in tangent space $e_{10}$. A difference only in how we slice a single jewel in order to get it under control.

At the end of the talk I asked if there is anything known about a relation of all this $e_{10}$ technology to $L_\infty$ algebras. After all, while here the 11-d supergravity 3-form arises as a decomposition of $e_{10}$ into irreps of $\mathrm{sl}(10)$, in the FDA approach to supergravity # it appears as a certain Lie algebra 3-cocycle which exists for the super-Poincaré Lie algebra precisely in 11 dimensions, and which can be regarded as defining a 3-term $L_\infty$ structure.

The answer to that question, however, was not known.

## Re: Nicolai on E10 and Supergravity

This is all good and well, but the E series is not exceptional as Kac-Moody algebras, nor is the e_10 grading with g_0 = gl(10). E.g., by looking at the Dynkin diagram one immediately sees that e_10 has another grading with g_0 = sl(6)+sl(5)+C, and e_4711 has a grading with g_0 = gl(4711). It is by no means clear that they are less interesting than the case that Nicolai considers.

Of course, it is SUSY that underlies the interest in e_10/e_11. The physical relevance of these algebras is therefore closely connected to the discovery, nor not, of sparticles at the LHC.