Chern-Simons Actions for (Super)-Gravities

March 12, 2008

Chern-Simons Actions for (Super)-Gravities

Posted by Urs Schreiber

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Just as

electromagnetism is a theory of line 1-bundles with connection coupled to electric 1-particles and magnetic 1-particles,

we have that

supergravity # in eleven dimensions is a theory of line 3- and line 6-bundles with connection coupled to electric 3-particles and magnetic 6-particles.

(There is a beautiful discussion of essentially this statement by D. Freed, which I talked about here, and here. Freed doesn’t say “ $n$ -bundle with connection”, but instead says “differential cocycle”. But it’s the same kind of thing.)

Wonders never cease, and hence there are indications that there is more to 11-dimensional supergravity than meets the eye. The question is: what? What is 11-dimensional supergravity really about?

One idea is: it is really about 1-particles on the “ $E_{10}$ -group manifold”. This we talked about before.

Another idea is: it is really about the higher Chern-Simons theory # of an invariant degree 6-polynomial on a super Lie algebra not unlike super- $so(n,m)$ #.

This speculation was put forward in

Petr Hořava
M-Theory as a Holographic # Field Theory
(arXiv)

The jargon in the title is such as to make certain physicists excited. A completely different, but possibly just as exciting jargon would be: it is speculated here that, very fundamentally, physics is about those representations of extended cobordism categories which are naturally induced from Chern-Simons $n$ -bundles with connection.

I was reminded of that by the appearance of the very nicely written basic review

Jorge Zanelli
Lecture notes on Chern-Simons (super-)gravities
(arXiv)

which was updated a few days ago. (Thanks to It’s equal but It’s different for noticing.)

This reviews the action functionals for theories of gravity one obtains by picking a $d = 2k +1$ -dimensional manifold $X$ , a structure group $G$ like $SO(d-1,1) \hookrightarrow \left\lbrace \array{ SO(d,1) \\ (ISO(d-1,1)) \\ SO(d-1,2) &\hookrightarrow& OSP(m|N) } \right.$ together with a degree $(d+1)/2$ invariant polynomial $\langle \cdots \rangle$ on its Lie algebra; and takes the action functional to be the corresponding Chern-Simons integral which sends $g$ -valued 1-forms $A$ on $X$ to $A \mapsto \int_X \mathrm{CS}(A) \,,$ where the Chern-Simons $d$ -form # $CS(A)$ satisfies $d CS(A) = \langle F_A \wedge F_A \wedge \cdots \wedge F_A \rangle$ .

For $d=3$ this yields, famously, the ordinary (super) Einstein-Hilbert action in that dimension. For higher (odd) $d$ , this yields the (super) Einstein-Hilbert action with higher curvature contributions.

Hořava gave arguments suggesting that and how for $d=11$ the Chern-Simons gravity action reduces to that of ordinary supergravity in the appropriate limit.

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More later.

Posted at March 12, 2008 6:55 PM UTC

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Re: Chern-Simons Actions for (Super)-Gravities

Whenever I start to read this post I hear the ‘like’ in Americanese – “Like electromagnetism is so a theory of line 1-bundles…”

Staying with the trivial, “wonders never cease” is more common than “wonders never end”.

Now for something more serious. Recasting M-theory as you have, do we get any clearer idea of what’s so special about it? Is there some happenstantial fluke occurring in this dimension? Or is it part of that greater body of theory studied by ‘exceptionology’?

Here’s John from a while back

Anyway, I don’t understand the d=11 Fierz identity and why it just luckily happens to be the Jacobiatorator identity in disguise.

Any clearer idea now?

Posted by: David Corfield on March 13, 2008 11:05 AM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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Thanks for the editorial comments. Have fixed that now.

Recasting M-theory as you have

Wait. The term “M-theory” appears in the title of Hořava’s article which I cited, but here I just said:

There is an interesting class of “theories of gravity” (meaning: action functionals generalizing the Einstein-Hilbert action) which are of Chern-Simons-type: their Lagrangian is simply a Chern-Simons form for the deSitter ( $SO(d,1)$ ) or anti-deSitter (SO(d-1,2)) group, or some supergroup extending that.

Since these Chern-Simons gravity theories have very nice formal properties, one is naturally lead to wonder, as Zanelli does from p. 62 on, if and how they relate to more conventional gravity theories.

In particular, since ordinary (meaning here: no higher powers of Riemann curvature terms in the action) 11-dimensional supergravity is supposed to be unique, one is lead to wonder how 11-dimensional super-Chern-Simons gravity relates to that.

My impression is that nobody has much of a clue concerning details, but Hořava at least presented some rough, hand-waving arguments which suggest that 11-dimensional Chern-Simons gravity should be related to ordinary 11-dimensional supergravity – and its expected refinement.

I’d like to emphasize that this is an issue which is noteworthy and sensible in its own right. Zanelli never ever mentions “M-theory”.

You see, there are a couple of mathematical structures here which are rather neat, and then there are physicists trying to find places for these structure in their models of (more or less) reality. But it pays to distinguish between what is actually being done with the structure, and which stories are told for motivating that.

As Zanelli emphasizes throughout his article, another aspect to emphasize here is that Chern-Simons theories of gravity fully realize the concept of gravity as a gauge theory. That aspect might inspire different stories to be told here.

I tried, maybe in vain, to relate this to a story which should resonate well with readers of this blog here: Chern-Simons-like TQFTs are among the best understood (or maybe conversely: least mysterious) classes of representations of higher cobordism categories. Should it be true that gravity is ultimately a Chern-Simons-like TQFT itself, that would mean that current $\infty$ -categorical endeavours are rather close to some “last questions” in physics.

Posted by: Urs Schreiber on March 13, 2008 11:44 AM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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Here’s John from a while back

Anyway, I don’t understand the d=11 Fierz identity and why it just luckily happens to be the Jacobiatorator identity in disguise.

Any clearer idea now?

Maybe more concretely formulated the question is:

The super-Poincaré algebra in $d=11$ dimensions happens to have a nontrivial Lie algebra 4-cocycle. To which degree does the existence of that 4-cocycle depend on $d$ ?

We had discussed a slightly indirect, but useful way to look at it:

the polyvector super-Poincaré Lie (1-)algebras have been classified. The “polyvector” part of these is computed there as a function of the dimension (see the summary at the end of that entry).

Moreover, Castellani essentially shows that the Lie 1-algebra of inner derivations of the sugra Lie 3-algebra is the polyvector Lie 1-algebra in 11-dimensions. The polyvector part arises directly from the 4-cocycle.

Posted by: Urs Schreiber on March 13, 2008 12:17 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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I mentioned

the Lie 1-algebra of inner derivations of the sugra Lie 3-algebra

Maybe I should recall that we talked about this in Derivation Lie 1-Algebras of Lie $n$ -Algebras:

for every Lie $n$ -algebra there is a Lie 1-algebra of inner derivations (there are also higher Lie $n$ -algebras of inner derivations, as I discuss there, but in particular there is always this Lie 1-algebra).

Castellani’s result indicates that the famous polyvector super-Poincaré Lie algebras – which played and play such an important role in the theory of supersymmetric physics, and which, historically, gave rise to much of the $d$ -brane science – arise as inner derivations of certain Lie $n$ -algebras.

(Well, he shows this just for one example, but it is clear that this generalizes. Somebody should sit down and work it out.)

But what I don’t know is to which degree the map from Lie $n$ -algebras to Lie 1-algebras obtained from taking inner derivations is surjective and/or injective.

There is a a lot of literature on closely related issues, and it feels like with more effort on unifying languages, much more could be said from knowledge already scattered in the literature.

For instance, there is an article

Bando, Azcárraga, Picón, Varela, On the formulation of $D=11$ supergravity and the composite nature of its three-from field

in which the authors start with the supergravity Lie 3-algebra and then (I think) try to trivialize the 4-cocycle by embedding the super-Poincare Lie 1-algebra into a larger super Lie algebra which they address by the nice name

$\tilde E^{(528|32+32)}(s) \,.$

Posted by: Urs Schreiber on March 13, 2008 12:56 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

Urs wrote:

the very nicely written basic review

Jorge Zanelli

Lecture notes on Chern-Simons (super-)gravities

That’s putting it mildly - not only does it communicate very well indeed, but it is beautifully, literately written

AND he believes in the philosophy I lerned from Yuri Rainich in his book
“Mathematics of Relativity”:
The last thing you want to do is write it down in coordiantes (double entendre I’m sure was intentional)

Posted by: jim stasheff on March 13, 2008 1:22 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

Yes, very beautifully written.

Zanelli says (p. 4),

Unfortunately, general relativity doesn’t qualify as a gauge theory, except for a remarkable accident in three spacetime dimensions.

I remember Tom Leinster telling me how odd he finds it when mathematicians use the word ‘unfortunately’ about inconvenient truths of mathematics messing up a neat theory. Presumably, it would seem equally odd in physics.

We’re always happy to draw an oddity within a larger scheme:

In these lectures we attempt to shed some light on this issue, and will show how the three dimensional accident can be generalized to higher dimensions.

Elsewhere Zanelli calls this a ‘miracle’ rather than an ‘accident’, which gives a different gloss to it. I mention ‘miracles’ in mathematics here.

Posted by: David Corfield on March 13, 2008 2:34 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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I remember Tom Leinster telling me how odd he finds it when mathematicians use the word ‘unfortunately’ about inconvenient truths of mathematics messing up a neat theory. Presumably, it would seem equally odd in physics.

I suppose in math one is after understanding all the true statemens, so it is paradoxical to say that a given statement is “unfortunately true”.

Even though I suppose what is really usually meant is that “this implies that my original way of thinking about it was wrong, which is unfortunate (for me)”.

So this issue whether or not any truth is “unfortunate” is closely related to the mixed feelings one has when discovering a mistake:

a) one is furious to discover that one made a mistake.

b) one can be glad that it didn’t take even longer to realize the mistake.

I seem to remember that Feynman said something like: it is not about not making mistakes – but about finding the mistakes you inevitably make as soon as possible.

Anyway, what I really wanted to say is: the inconvenience of a truth is a different matter in math and in theoretical physics.

In math, once I decide which mathematical structure to study, none of the truths I will find can ever really be inconvenient. I’ll be glad about every truth I discover.

In theoretical physics, it’s a little different. There, the mathematical structure I choose to start with is always a mean to an end: I start with the desire to model a given phenomenon, then I try to choose the right mathematical structure.

After some investigations it may turn out that the mathematical structure chosen has properties which make it unsuited to model what it was supposed to model.

Whether or not you want to call such a situation “unfortunate”, in physics a statement may be true and still worthless. Whereas in math, a true fact is never worthless, but one more piece in the whole edifice.

This is what happens from Zanelli’s point here: he finds that the standard structure for the description of gravity lacks a property which he believes such a description should have to be a promising basis for further development.

Well, all I want to say is: whether you want to call that “unfortunate” or not, the situation is a little different than with pure math.

Posted by: Urs Schreiber on March 13, 2008 6:04 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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Whereas in math, a true fact is never worthless, but one more piece in the whole edifice.

I’m not sure I should let that go unchallenged. Who can come up with the least worthwhile mathematical fact?

Let me go first:

The fourth triangular number is equal to the sum of the first and third square numbers.
The eighth digit of the dimension of the fifteenth largest irrep of the monster group written in base 34 is $n$ .
The associated conditional (axioms imply conclusion) for the six billionth line emitted by my copy of the OTTER automated theorem prover when I enter such and such axioms in a certain syntactic way.

Or were you taking any such choice as already filtered by an earlier decision as to the importance of a structure – “once I decide which mathematical structure to study”?

Posted by: David Corfield on March 14, 2008 10:15 AM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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I’m not sure I should let that go unchallenged. Who can come up with the least worthwhile mathematical fact?

But notice the difference to physics:

while there are billions of true facts in math that are rather unintersting, they all are answers to some question which might arise and would be answered by math.

Some student comes to your office and asks if the the fourth triangular number is equal to the sum of the first and third square numbers. You say: “I don’t find that very interesting but, yes, I can guaratee you that it is true.”

In physics, there may be facts, even by themselves interesting facts, which are completely worthless as long as they don’t say anything about those particular models which one finds one needs to look at.

To stay roughly within the context we started with: suppose you have a great mathematical machinery for quantizing systems of kind $X$ . But then it turns out that the kind of system which you are actually studying is not of that kind. Then that’s unfortunate, right? It means that all this nice machinery is rather worthless.

Whereas if you feel like a mathematician (or like a theoretical physicists who agrees to study anything that has at least a remote resemblance to something like physics (not so rare)), that cool method for quantization of systems of kind $X$ might make for lots of intersting articles, studying it just in its own right.

Zanelli in his review roughly says something like: the only good machinery for quantization we have is for systems that are gauge theories. Unfortunately gravity in its metric-based formulation is not a gauge theory. Hence all that machinery does not apply and is rather worthless.

(Of course then he goes on to emphasize the fact that gravity may well be reformulated to be a gauge theory after all.)

Posted by: Urs Schreiber on March 14, 2008 11:18 AM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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Well I’d go for something along the lines of ( $2.3678\ne 8.9923$ or $12.55 \ne 82.3$ ) as a “not worthwhile” fact as it not only is not only not an interesting proposition (that at least one of two paris of two numbers we have no reason to believe ought to be equal are in fact not equal), but it’s one of an infinitude of such facts selected without any specific rationale for the choice.

Being more serious, one might change Urs’ assertion to “Whereas in math, a true deep fact is never worthless…”. That has its own problems, since presumably that’s a judgement dependent on the time at which it’s made in the sense that depth is somewhat dependent on what proofs/analyses we’ve got at that time. For example, is the four colour theorem “deep” or a “shallow happenstance”? My understanding of the current “best” proof is that it still involves computer verification of a large ensemble of a set of basic configurations, so in a sense it looks more like a “happenstance” than, say , unique prime factorizability of natural numbers. It’s known to be equivalent to other statements, eg, something about Lie algebras. Suppose someone proves such a statement using “deeper, fundamental” arguments, what’s happened to the status of the four colour theorem? Is it now a more worthwhile mathematical fact?

Posted by: bane on March 14, 2008 11:26 AM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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Your examples show that I might have to look for a better formulation of my point, but I still do think there is a point.

Let me try again:

the statement:

“In Hamiltonian mechanics, a system of energy 2.3678 differs from a system of energy 8.9923.”

is true, but boring.

That’s the kind of worthless fact you mentioned, but trivially restated in a physics context just for the sake of the argument.

But now compare this to the following fact:

“In twisted octonionic nonlinear quantum mechanics, a state with energy expectation value 2.3678 differs from a state with energy expectation value 8.9923.”

That’s really worthless in physics, since there is no indication whatsoever that twisted octonionic nonlinear quantum mechanics is of physical interst, even though it may exists as an interesting mathematical structure in itw own right.

Posted by: Urs Schreiber on March 14, 2008 12:20 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

I did get your point. Physics worthiness is harder to earn than mathematics worthiness, unless you’re Tegmarkian. It’s just I get nervous that people will understand that there are no degrees of worthiness in maths.

Then again, other people get nervous when the issue of one piece of maths being more important than another gets raised.

Posted by: David Corfield on March 14, 2008 1:22 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

I wrote too soon - Zanelli soon retreats to `centipede tensor calculus’ so I lose track of what the entities `are’. but I’ll read on as later enlightenment reoccurs.

Posted by: jim stasheff on March 14, 2008 1:20 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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Zanelli has lecture slides covering essentially the material in his lecture notes available here:

J. Zanelli, (Super) Gravities of a different sort, talk at Constrained Dynamics and Quantum Gravity, Sardegna, 2005

(watch out: huge power point file)

Posted by: Urs Schreiber on March 13, 2008 9:01 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

The purpose of physics is to think up explanations for what you observe. In physics, you try to think up mathematical models that are consistent with what you observe. If you were in the process of trying to develop a mathematical model that you hoped would turn out to be consistent with observation, but then you realized that it was not, after all, consistent with observation, then that would be “unfortunate” in the sense that you failed to achieve what you set out to do.

However, the discovery of something new, whether in math or physics, should never be discovered “unfortunate”. For instance, the discovery of relativity and quantum mechanics, and that Newtonian mechanics was wrong, should not have been considered “unfortunate” by physicists at the time.

Jeffery Winkler

http://www.geocities.com/jefferywinkler

Posted by: Jeffery Winkler on March 14, 2008 7:02 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

Let’s say you were to plot the accuracy of our view of the Universe at a given time on a graph where the view of the universe possessed by the earliest most primative stone age tribes of nomadic hunter-gatherers on the left hand side, and “actual reality”, meaning the real Universe as it really is on the right hand side.

earliest —————————- reality

Throughout most of history, most physicists or other scientists have assumed that their current view was close to the right hand side.

earliest ————————-X– reality

The ancient Greeks thought they were close to the truth. In the Renaissance, they thought they were close to the truth.
In the late 19th Century, when Newtonian mechanics and Maxwell’s electromagnetism seemed to explain almost everything, except for a few details like blackbody radiation, the photoelectric effect, and the Michelson-Moray experiment, they also assumed they were close to the truth. In the 1930’s, when they seemed to have worked out quantum mechanics, special and general relativity, and all matter seemed to be composed of only three particles, the proton, neutron, and electron, they also assumed they were close to the truth. In each of these cases, they turned out to be totally wrong. They weren’t close to the truth. On what grounds were they assuming the following graph?

earliest ————————-X– reality

They are basing it on the extent to which current theorizes could explain what is currently observed. However, that doesn’t tell you anything about how far “reality” is to the right of the graph. The graph could be like this.

earliest -X————————– reality

or in fact, the graph could be like this

earliest -X————————–>

where on this scale, reality is 100 miles to the right. However, even here, you might say that our current view is measurably more accurate than the earliest view. However, what if the graph is like this

earliest -X————————–>

where on this scale, reality is 100 megaparsecs to the right? Then the extent to which our current view of the Universe, based on our most advanced theories, is more accurate than the view held by the earliest tribes of hunter-gathers is neglible. How do we know the process than the human species has made so far in understanding the Universe isn’t neglible?

The purpose of physics is not to get to the right hand of graph. There is no way to know if we’re close to the right hand side. The purpose of physics is to increase the extent to which our current theories could explain what we observe. That extent of consistency between what our current theories predict and what we currently observe is something we can measure, and something we can reduce by thinking up better theories.

Therefore if, say, at the Large Hadronic Collider, we do not observe the Higgs particle, or supersymmetric particles, that would be very unfortunate, since it would reduce the extent of consistency between what our current theories predict and what we currently observe.

Jeffery Winkler

http://www.geocities.com/jefferywinkler

Posted by: Jeffery Winkler on March 18, 2008 10:23 PM | Permalink | Reply to this

Re: Chern-Simons Actions for (Super)-Gravities

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As you say, certainly it is not clear that we have any good right to expect that there is not just always better physical theory, but actually a final, optimal physical theory.

And even if there were have we little right to expect that we can find it at the pace at which we are proceeding before the end of all days.

Despite of all this, I do think there is always the struggle for the next best approximation. That’s what theoretical physics is about. Wherever it comes from, there are certain properties you want your physical theory to satisfy which the current theories do not. So you need to refine. And try. And play around.

And lots of mathematical structures you may test in that process, and many of them will be quite interesting in their own right. But few will match your desiderata for the best next approximation to that part of physical reality which you intended to describe.

In fact, the notion of “reality” here is rather irrelevant for my main point: even if you set out to describe the physical laws of some dreamed up universe: while you are doing it, you will have to evaluate mathematical structures by measures which don’t rest entirely in their intrinsic nature, but which are put on them from outside of math itself.

And if the structure you’ve been spending two years on to understand turns out not to match those desiderata after all then that’s well, unfortunate…

Posted by: Urs Schreiber on March 19, 2008 11:33 PM | Permalink | Reply to this

Read the post What has happened so far
Weblog: The n-Category Café
Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
Tracked: March 27, 2008 6:41 PM

Re: Chern-Simons Actions for (Super)-Gravities

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I should have also pointed out this article, relevant to the above discussion:

Horatiu Nastase, Towards a Chern-Simons M theory of $OSp(1|32)\times OSp(1|32)$

Abstract: A possible way of defining M theory as the CS theory for the supergroup $OSp(1|32)\times OSp(1|32)$ is investigated, based on the approach by Horava in hep-th/9712130. In the high energy limit (expansion in M), where only the highest ( $R^5$ ) terms survive in the action, the supergroup contracts to the D’Auria-Fré M theory supergroup. Then the contracted equations of motion are solved by the usual 11d supergravity equations of motion, linearized in everything but the vielbein. These two facts suggest that the whole nonlinear 11d sugra should be obtainable somehow in the contraction limit. Type IIB also arises as a contraction of the $OSp(1|32)\times OSp(1|32)$ theory. The presence of a cosmological constant in 11d constraints the parameter M experimentally to be of the order of the inverse horizon size, $1/L_0$ . Then the 11d Planck mass $M_{P,11}\sim 10GeV$ (hopefully higher: $\ge TeV$ due to uncertainties). Unfortunately, the most naive attempt at cosmological implications for the theory is excluded experimentally. Interestingly, the low energy expansion (high M) of the CS theory, truncated to the gravitational sector, gives much better phenomenology.

Posted by: Urs Schreiber on April 7, 2008 2:39 PM | Permalink | Reply to this

Read the post E8 Quillen Superconnection
Weblog: The n-Category Café
Excerpt: On Quillen superconnections on E8-bundles and on the mathematical interpretation of the connection appearing in an article by A. Lisi.
Tracked: May 10, 2008 9:54 AM

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