## November 29, 2006

### 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

(1)$\array{ && && && && && && 10 && && \\ && && && && && && | && && \\ 1 &-& 2 &-& 3 &-& 4 &-& 5 &-& 6 &-& 7 &-& 8 &-& 9 } \,.$

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

(2)$\array{ && && && && && && (10,\ell) && && \\ && && && && && && | && && \\ 1 &-& 2 &-& 3 &-& 4 &-& 5 &-& 6 &-& 7 &-& 8 &-& 9 } \,.$

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,

(3)$\array{ && && && && && && 10 && && \\ && && && && && && | && && \\ 1 &-& 2 &-& 3 &-& 4 &-& 5 &-& 6 &-& 7 &-& (8,\ell) &-& 9 }$

yields 10-dimensional type IIB supergravity and

(4)$\array{ && && && && && && 10 && && \\ && && && && && && | && && \\ 1 &-& 2 &-& 3 &-& 4 &-& 5 &-& 6 &-& 7 &-& 8 &-& (9,\ell) }$

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.

Posted at November 29, 2006 3:47 PM UTC

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### 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.

### Re: Nicolai on E10 and Supergravity

This is interesting stuff! I’ve never gotten around to understanding what Nicolai was doing with $\mathrm{e}_10$, but I’m getting an inkling of it now. I especially like the idea of taking the obvious $\mathrm{sl}(10)$ Lie subalgebra of $\mathrm{e}_{10}$ and using it to decompose `levels’ of $\mathrm{e}_{10}$ into irreps of $\mathrm{sl}(10)$:

(1)$\array{ && && && && && && (10,\ell) && && \\ && && && && && && | && && \\ 1 &-& 2 &-& 3 &-& 4 &-& 5 &-& 6 &-& 7 &-& 8 &-& 9 }$

Just to be clear: are you saying that each level $\ell = 0,1,2,3$ consists of a single irrep?

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.

Do you mean the $\mathrm{so}(10)$ that’s the maximal compact Lie subalgebra of the $\mathrm{sl}(10)$ you’re talking about? This is a bit funny, since while the bosonic fields you mentioned (graviton, 3-form, …) are reps of $\mathrm{sl}(10)$, I think the gravitino is just a rep of $\mathrm{so}(10)$, not of $\mathrm{sl}(10)$. Right?

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.

Since the heterotic string is related to $\mathrm{e}_8$, I’d naively want to look at the $\mathrm{e}_8$ Dynkin diagram obtained by deleting two dots in the diagram for $\mathrm{e}_{10}$:

(2)$\array{ && && && && && && 10 && && \\ && && && && && && | && && \\ & & && 3 &-& 4 &-& 5 &-& 6 &-& 7 &-& 8 &-& 9 }$

But, if Nicolai is playing a game where he removes one dot at time, en route to this $\mathrm{e}_8$ he’ll run into the Dynkin diagram for the affine version of $\mathrm{e}_8$:

(3)$\array{ && && && && && && 10 && && \\ && && && && && && | && && \\ & & 2 &-& 3 &-& 4 &-& 5 &-& 6 &-& 7 &-& 8 &-& 9 }$

Then he could consider levels like this:

(4)$\array{ && && && && && && 10 && && \\ && && && && && && | && && \\ (1,\ell) &-& 2 &-& 3 &-& 4 &-& 5 &-& 6 &-& 7 &-& 8 &-& 9 }$

Did he talk about this at all? Of course the levels will now probably be infinite-dimensional reps of affine $\mathrm{e}_8$… but maybe these show up in heterotic string theory?

Posted by: John Baez on November 29, 2006 8:46 PM | Permalink | Reply to this

### Re: Nicolai on E10 and Supergravity

Urs wrote:

After all, while here the 11-d supergravity 3-form arises as a decomposition of $\mathrm{e}_10$ into irreps of $\mathrm{sl}(10)$, in the FDA approach to supergravity it appears as a certain Lie algebra 4-cocycle which exists for the super-Poincaré Lie algebra precisely in 11 dimensions…

Actually, I thought about that 4-cocycle a little, and I got the impression that it exists, and is a cocycle, in lots of different dimensions. Nothing I read dispelled that impression.

It would be good to understand this better. It may be that the Lie 3-superalgebra exists quite generally, but admits certain supersymmetries only in this special dimension.

Posted by: John Baez on November 30, 2006 8:15 AM | Permalink | Reply to this

### Re: Nicolai on E10 and Supergravity

Just to be clear: are you saying that each level $\ell = 0,1,2,3$ consists of a single irrep?

No, there are many irreps at a given level. On slide 11 you can see Nicolai mentioning that the $A_9$ decomposition of $E_{10}$ has precisely

(1)$4 400 752 653$

irreps in levels $\ell \leq 28$.

I was once shown part of the computer output that contained results like that. Quite impressive. Several telephone books full of irreps.

Do you mean the $\mathrm{so}(10)$ that’s the maximal compact Lie subalgebra of the $\mathrm{sl}(10)$ you’re talking about? This is a bit funny, since while the bosonic fields you mentioned (graviton, 3-form, …) are reps of $\mathrm{sl}(10)$, I think the gravitino is just a rep of $\mathrm{so}(10)$, not of $\mathrm{sl}(10)$. Right?

Yes, right. While I am far from being fully acquainted with all the details of what Nicolai is talking about, this is one point he emphasized, because it is the topic of his latest paper, which I linked to above.

Everything that had been done in this $M = E_{10}/K(E_{10})$-business so far was almost exclusively only about the bosonic degrees of freedom of supergravity.

For others reading this, I might maybe briefly comment what this means: when one takes a theory of bosonic fields - like ordinary Einstein gravity, for instance - and tries to add superparners to all fields, such that the resulting theory exhibits supersymmetry, one may find oneself to be forced to also add new bosonic fields for this to work.

This way one finds that, for instance 11-dimensional supergravity contains not just the metric (graviton) and its superpartner (gravitino) but also a 3-form field.

One quite interesting aspect of the program of extracting supergravity fields from $e_{10}$ was that even without adding any supersymmetry on the $e_{10}$ side one found in it not just the ordinary bosonic sugra fields, like the graviton, but also all the other bosonic fields, like the 11d 3-form, or the RR-forms in 10d sugra.

This is nice, and for a while attention has focused only on these bosonic fields. Only rather recently, Nicolai et. al. have begun to also look for the fermionic field content of supergravity inside $e_{10}$.

As John rightly remarks, these fermions usually transform in the maximal compact sub-algebra. Accordingly, Nicolai says that for the supergravity fermions we should not look at $e_{10}$ but at its maximal compact subalgebra $k(e_{10})$.

Towards the end of his talk he sketched how this does seem to work. I cannot quite give you the details, since I haven’t read the corresponding paper. But the message was that $k(e_{10})$ does have finite dimensional reps, that these can be seen to act on the gravitino, and that the equation of motion of the gravitino is covariant under these actions.

Since the heterotic string is related to $e_8$, I’d naively want to look at the $e_8$ Dynkin diagram obtained by deleting two dots

[…]

No, he didn’t. I can’t quite tell if he really meant to say that the heterotic sugra is definitely not in there, or if it’s just that it hasn’t been found yet. He did, however, close his remark about the failure of the heterotic sugra to show up with “Well, too bad for the heterotic string, then.”

Of course the levels will now probably be infinite-dimensional reps of affine $e_8$ but maybe these show up in heterotic string theory?

The internal part of the heterotic worldsheet theory is an $e_8$ current algebra, so this definitely shows up there. In which form this is visible in the corresponding sugra fields I am not sure.

Posted by: urs on November 30, 2006 9:53 AM | Permalink | Reply to this

### Re: Nicolai on E10 and Supergravity

Actually, I thought about that 4-cocycle a little, and I got the impression that it exists, and is a cocycle, in lots of different dimensions. Nothing I read dispelled that impression.

I haven’t checked, but it is claimed that the cocycle property depends on a Fierz identity that holds in eleven dimensions, but not generally.

Posted by: urs on November 30, 2006 10:00 AM | Permalink | Reply to this

### Kac-Moody n-algebras

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.

[…]

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

We have learned from the example of the String Lie 2-algebra # that weak Lie $n$-algebras coming from Lie algebra 3-cocycles may be transmuted into strict Lie $n$-algebras, where now the cocycle defines a strict - but affine - Lie algebra.

This gives rise to the vague hope that something analogous might be at work behind the scenes here.

Is there any reason why the semistrict Lie 3-algebras $\mathrm{sugra}_{11}$ and $\mathrm{cs}(g)$ # could not have a strictification, i.e. be equivalent to a non-skeletal but strict “Lie” 3-algebra?

If $\mathrm{sugra}_{11}$ has a strictification, it should involve something like a Lie algebra naturally associated to a 4-cocycle.

To a Lie algebra 3-cocycle we have naturally associated affine Kac-Moody algebras : “current algebras”, the infinite-dimensional Lie algebras of centrally extended loop groups.

Based on that, it does not look entirely implausible, naively at least, that a Lie algebra 4-coycle might give rise to a Kac-Moody algebra beyond the affine case. Or does it?

Can anyone provide evidence for or against the following

Conjecture: The semistrict Kac-Moody 3-algebra $\mathrm{sugra}_{11}$ is equivalent to a strict Kac-Moody 3-algebra that involves, in top degree, an ordinary hyperbolic KM-algebra - or maybe a coset of such.

I say KM-$n$-algebra here, where I simply imagine an $n$-term $L_\infty$-algebra. As far as I can see, there is nothing in the definition of $L_\infty$ algebras that would exclude hyperbolic Kac-Moody structures. Or is there? I mean, any Kac-Moody algebra can in particular be regarded as a 1-term $L_\infty$ algebra.

Posted by: urs on November 30, 2006 3:02 PM | Permalink | Reply to this

### Re: Kac-Moody n-algebras

Based on that, it does not look entirely implausible, naively at least, that a Lie algebra 4-coycle might give rise to a Kac-Moody algebra beyond the affine case. Or does it?

A standard way to settle such issues is growth considerations. A hyperbolic KM algebra grows exponentially, so cannot be equivalent to something that grows polynomially, e.g. current algebras in higher dimensions. Do you know how fast your 4-cocycle thing grows?

### centrally extended torus groups

Do you know how fast your 4-cocycle thing grows?

See, my conjecture says that this 4-cocycle thing is a hyperbolic KM-algebra (or a coset thereof).

However, it’s just a conjecture.

But consider this:

Why is it that a 3-cocycle defines an affine algebra? Because the 3-cocycle can be regarded as giving a class in $H^3(G,\mathbb{Z})$ on the group, this classifies a gerbe on the group and this in turn gives - by transgression - a line bundle on the loop space of the group. This line bundle on $L G$ is nothing but the central extension of $L G$.

Naively, this suggests an obvious generalization to 4-cocycles. These should classify 2-gerbes on $G$, which should gives rise to gerbes on the loop space of $G$, which in turn give rise to line bundles on the double loop space or torus space $L L G$ of $G$.

By analogy, this line bundle one might expect to have a multiplicative structure which gives rise to a central extension of the torus group of $G$.

Is anything known about torus groups and their central extensions?

It seems that my conjecture implies that I conjecture that centrally extended torus groups are cosets of hyperbolic KM algebras.

Maybe this version of the conjecture can be ruled out more easily.

(By the way, from the physics point of view this sounds rather plausible: we sort of do know that $\mathrm{sugra}_{11}$ is related to $\mathrm{string}(g)$ as the membrane is related to the string. In H. Nicolai’s previous life he did all that he does now from the perspective of the supermembrane. In the end, it should all come together.)

Posted by: urs on November 30, 2006 3:31 PM | Permalink | Reply to this

### Re: centrally extended torus groups

Posted by: brano on November 30, 2006 3:56 PM | Permalink | Reply to this

### Re: centrally extended torus groups

Thanks!

I have skimmed through this. I guess the answer I am looking for is: is the centrally extended torus Lie algebra still an affine Lie algebra, i.e. is its Cartan matrix still positive semidefinite? Or is it indefinite?

Posted by: urs on November 30, 2006 4:10 PM | Permalink | Reply to this

### Re: centrally extended torus groups

Is anything known about torus groups and their central extensions?

Did I ever mention the multi-dimensional Virasoro and affine algebras?

Current algebras on a torus basically have two types of abelian extensions: the central extension and the Mickelsson-Faddeev extension, which pertains to chiral anomalies. They were contrasted in math-ph/0501023.

A simple argument why these extensions exhaust all possibilities is this: A current algebra is a diffeomorphism algebra - a subalgebra of diffeomorphisms on the total manifold of the bundle, preserving vertical fibers. Askar Dzhumadildaev has classified abelian extensions (by tensor modules) of the (polynomial) diff algebra. Hence one obtains candidate current algebra cocycles by restriction of Dzhumadildaev cocycles. Check case by case for non-triviality.

In particular, the central extension of current algebra a la Etingof-Frenkel (the oldest reference I know of is Kassel 1985) is obtained by restriction of the multi-dimensional Virasoro cocycles.

I have skimmed through this. I guess the answer I am looking for is: is the centrally extended torus Lie algebra still an affine Lie algebra, i.e. is its Cartan matrix still positive semidefinite? Or is it indefinite?

Neither. Toroidal Lie algebras don’t have a Cartan matrix, unless the torus is a circle.

Recall that a Cartan matrix is tied to a particular presentation. There are many natural Lie algebras which do not allow such a presentation, e.g. the diffeomorphism algebra in n dimensions, vect(n). And I learned from Nicolai’s slides that K(e_10) is not a Kac-Moody algebra, thus has no Cartan matrix.

For Lie superalgebras Cartan matrices are even less useful. Even for finite-dimensional Lie superalgebras, some have no Cartan matrix, e.g. vect(0|m), and some have several, cf. hep-th/9607161.

Non-affine KM algebra have no obvious relations to current algebra in higher dimension. Victor Kac warns for this common mistake in the introducion to his book, I think. As I indicated in my previous post, this follows immediately from growth considerations.

### Re: centrally extended torus groups

Is anything known about torus groups and their central extensions?

Did I ever mention the multi-dimensional Virasoro and affine algebras?

Right, you might have mentioned that before. :-)

Non-affine KM algebra have no obvious relations to current algebra in higher dimension. Victor Kac warns for this common mistake in the introducion to his book, I think. As I indicated in my previous post, this follows immediately from growth considerations.

Okay, good. So what do these growth considerations tell us about the coset of a KM algebra by a maximal compact subalgebra? Any chance to map that to a torus algebra, somehow?

Posted by: urs on December 1, 2006 9:43 AM | Permalink | Reply to this

### Re: centrally extended torus groups

Okay, good. So what do these growth considerations tell us about the coset of a KM algebra by a maximal compact subalgebra? Any chance to map that to a torus algebra, somehow?

I don’t understand. A coset g/h is not generally a Lie algebra at all, unless h is an ideal.

Assume that we wonder whether two graded Lie algebras g and g’ are isomorphic (the level decomposition gives e10 a grading). They are isomorphic if gn = g’n for all n. We can not conclude that they are different if this is not the case, because the same algebra may have many different gradings (e10 has at least 10, and in fact many more). However, if

dim gn ~ nd-1 cn

then the numbers d and c do not depend on the chosen grading but only on the algebra. If they differ, g and g’ are different.

Current algebra in d dimensions has d=d and c=1 (polynomial growth). Non-affine KM algebras have c >1 (exponential growth).

### Re: centrally extended torus groups

The estimate for dim gn holds of course only asymptotically for large n.

### Re: centrally extended torus groups

A coset $g/h$ is not generally a Lie algebra

Sure, that’s why it’s called a set. Sorry, I am just looking for any clues at all, like any relation that might be known. But I guess the answer is simply: no, there is no known hint that what Nicolai is talking about has any relation to toroidal algebras.

Posted by: urs on December 1, 2006 10:21 AM | Permalink | Reply to this

### Re: centrally extended torus groups

Current algebras on a torus basically have two types of abelian extensions

Hm, no central extension coming from a Lie algebra 4-cocycle then?

Maybe those 2-gerbes on a group $G$ are not multiplicative?

Posted by: urs on December 1, 2006 9:58 AM | Permalink | Reply to this

### Re: centrally extended torus groups

I must confess that I have not even understood what 3-cocycles have to do with affine algebras. Abelian extensions are classified by second Lie algebra cohomology, and the central current algebra extension is related to first de Rham homology*, but I thought that the third cohomology is related to other things (non-abelian extensions, non-associativity, ..).

*: The number of independent current group extensions on a manifold equals its first Betti number. I learned this fact from Karl-Hermann Neeb. In math.RA/0511260, Karl-Hermann and Friedrich Wagemann, whose work was recently discussed here, discuss the central current algebra extension.

### Re: centrally extended torus groups

what 3-cocycles have to do with affine algebras

By continuing the 3-cocycle left- (or right-)invariantly over the group, we get an element in $H^3(G)$ to which is associated a gerbe, which gives a line bundle over the loop space of the group, which has itself the structure of a group, whose Lie algebra is the loop algebra.

Posted by: urs on December 1, 2006 12:37 PM | Permalink | Reply to this

### Re: centrally extended torus groups

I must confess that I have not even understood what 3-cocycles have to do with affine algebras.

There are lots of answers to this question; Urs gave one but I’ll give another.

The 3rd cohomology of a semisimple Lie algebra $\mathrm{g}$ is isomorphic to the 2nd cohomology of its loop Lie algebra $C^\infty(S^1, \mathrm{g})$.

An $(n+1)$-cocycle on a Lie algebra allows us to centrally extend it to a Lie $n$-algebra — this is a result in HDA6.

So, ways of centrally extending $C^\infty(S^1,\mathrm{g})$ to get a Lie algebra are the same as ways of centrally extending $\mathrm{g}$ to get a Lie 2-algebra.

This gives a relation between central extensions of loop Lie algebras, and Lie 2-algebras, which is further explained and exploited here.

Right now I don’t remember if, more generally, the $(n+1)$st cohomology of a semisimple Lie algebra $\mathrm{g}$ is isomorphic to the $n$th cohomology of $C^\infty(S^1, \mathrm{g})$. Nor do I remember if semisimplicity is really required here. Grrr!

Posted by: John Baez on December 1, 2006 7:05 PM | Permalink | Reply to this

### Re: centrally extended torus groups

Let me see if I understand this. The third g cohomology is spanned by

fabdkdc,

and the second Lg cohomology by the central extension

kab m δm+n,0.

So they are related in the sense that both depend on the Killing metric kab.

The central extension of Tdg reads, in the notation of math-ph/0501023,

kab mi Si(m+n),

where mi are the components of the momentum, and Si(m) commutes with g and satisfies

mi Si(m) = 0.

Since the extension still only depends on the Killing metric, and you cannot build anything with four antisymmetric indices from it, it cannot be related to fourth cohomology. Perhaps the MF extension

dabc mi nj Tijc(m+n),

is related to fourth cohomology, since

defg khi fabe fchf fdig

could be antisymmetrized in abcd. Not sure if that thing is nonzero, though.

### Re: centrally extended torus groups

My early training under Rainich prejudices me in favor of invariant description, i.e.
without using a basis. Urs and I are reworking his fda lab to give both presentations. If there is interest,
could attempt this example also.

Posted by: jim stasheff on December 2, 2006 3:24 PM | Permalink | Reply to this

### Re: centrally extended torus groups

If there is interest, could attempt this example also.

I would certainly be very interested, if we could get this under control.

To my mind, the central question is if one can show that a skeletal Lie 3-algebra/3-term $L_\infty$ algebra with an ordinary Lie algebra in first degree and the only other nontrivial bracket being the 4-ary one, and coming from a Lie algebra 4-cocycle, is equivalent to a strict Lie 3-algebra coming from a toroidal Lie algebra.

As a first step, we should try to guess a natural candidate for a strict Lie 3-algebra cooked up from an ordinary toroidal Lie algebra.

Once we have that, there should be more or less obvious choices for the data that would establish an equivalence to a skeletal Lie 3-algebra.

So let’s try to guess the structure of the strict Lie 3-algebra coming from a toroidal Lie algebra, following the example of the $\mathrm{string}(g)$ Lie 2-algebra:

The toroidal Lie algebra $[S^1 \times S^1,g]$ itself should be sitting in degree 3. Probably in degree 2 we need the algebra of maps from the cylinder $[S^1 \times [0,1],g]$ into the Lie algebra $g$ and in degree 1 the Lie algebra $[[0,1]^2,g]$. There are obvious maps from level $n$ into level $n-1$ then, simply by embedding, and there are obvious actions (by conjugation) of elements in level $n-1$ on level $n$.

The only nontrivial step, as for $\mathrm{string}_g$, should be the check that this natural action can be lifted to the case where we have the central extension in the game.

Posted by: urs on December 3, 2006 4:35 PM | Permalink | Reply to this

### sugra configuration space

Maybe I can give a better version of that conjecture.

What is the $(M = E_{10}/K(E_{10}))$-proposal really saying: it says in particular that the configuration space of 11-dimensional supergravity is (a subspace of) $E_{10}/K(E_{10})$.

But, as we have discussed # the italian supergravity school is fond of the fact that a field configuration of classical supergravity is nothing but - now my paraphrase: - a 3-connection with values in the Lie 3-algebra $\mathrm{sugra}_{11}$.

There should be quantum corrections to that. It seems that we want at least something like

(1)$\mathrm{sugra}_{11}^{e_8} := \mathrm{sugra}_{11} \oplus \mathrm{cs}(e_8)$

#

Let me assume for the moment this is all there is, keeping in mind that possibly we need not quite this Lie 3-algebra $\mathrm{sugra}_{11}^{e_8}$, but something along these lines.

So merging the age-old italian-style supergravity wisdom with my functorial reformulation, we get this nice statement:

A configuration of 11-dimensional supergravity on $X^{10}\otimes \mathbb{R}$ is (at least locally, but never mind for the moment) a pseudofunctor

(2)$\mathrm{tra} : P_1(X^{10}) \to \Sigma( \exp (\mathrm{inn}(\mathrm{sugra}_{11}^{e_8}))) := \Sigma( \mathrm{INN}(\mathrm{SUGRA}_{11}^{E_8})) \,.$

from the pair groupoid to the (weak) 4-group (if any) integrating the Lie 4-algebra of inner derivation of the supergravity Lie 3-algebra.

I should say that here

(3)$\mathrm{par} = P_1(X)$

is the parameter space and

(4)$\mathrm{P} = \Sigma(\mathrm{INN}(\mathrm{SUGRA}_{11}^{E_8}))$

is the target space. Then we should think of this as some formidable generalization of something like a Chern-Simons theory.

The 4-category of all these pseudofunctors,

(5)$\mathrm{conf} = [P_1(X^{10}), \Sigma(\mathrm{INN}(\mathrm{SUGRA}_{11}^{E_8}))] ]\,,$

is our full configuration space. Equivalence classes of functors herein correspond to gauge equivalent configurations.

Now, the classical evolution of the system is some trajectory in configuration space. To get a true space, we might try to form the geometric realization of the nerve of $\mathrm{conf}$ in the sense of nerves of $n$-categories

(6)$|\mathrm{conf}| = |[P_1(X^{10}), \mathrm{INN}( \mathrm{SUGRA}_{11}^{E_8} )]| \,.$

Good. So then the conjecture might be:

Conjecture: $|\mathrm{conf}|$ is like $E_{10}/K(E_{10})$.

Posted by: urs on November 30, 2006 4:39 PM | Permalink | Reply to this

### Re: Nicolai on E10 and Supergravity

This Mtheory 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.

I just want to record one useful reference which I ran across while thinking about how to unify the sugra Lie 3-algebra with the two Chern-Simons Lie 3-algebras for $E_8$ and $\mathrm{spin}(10,1)$:

Greg Moore has useful lecture notes here, which summarize the main properties that a 3-connection with values in such a beast should have.

Posted by: urs on May 10, 2007 12:09 PM | Permalink | Reply to this
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