The n-Category Café

Good question! A composition algebra is one which satisfies

without requiring $\mid -{\mid }^2$ to be positive definite. It’s a classic theorem that the only real composition algebras have dimension 1, 2, 4, and 8, just like normed division algebras. In fact, the only real composition algebras are either the normed division algebras for which the norm is positive definite, or the split composition algebras, for which the norm has signature $(n/2,n/2)$, where $n$ is the dimension of our composition algebra.

I think the answer to your question is: they do work, but not for physical applications. In dimensions $n+2$, vectors are built from $2\times 2$ Hermitian matrices over $𝕂$. This is nice, because determinant gives the norm:

If $𝕂$ is normed of dimension $n$, you’ll see the above formula gives signature $(1,n+1)$. If $𝕂$ is split, it gives a split signature!

Most of the formulas (all of them, I bet — but I’d need to check) would still work in the split case, so you’d still get vectors, spinors and intertwiners, all for spacetimes with split signature. These might be worth thinking about, but they’re not the spacetimes one usually considers for superstrings or more general membranes.

[John Baez: I’m still stuck inside the computer here, so I can’t reply like a normal person. Here’s an explanatory remark for the lurking layfolk: ‘spacetimes with split signature’ is jargon for spacetimes that have an equal number of space and time dimensions! These have mathematically beautiful properties — but few physicists work on them, for an obvious reason: nobody wants to be called a ‘low-down dirty two-timing rat’.]

Thanks for the discussion of exact sequences, Urs. I have at times worried a lot about why one would ever be interested in “strict” exact sequences of the sort we’re using here. It’s nice to get a clear general picture of this, and I may add your comments — citing you, of course — to our paper. I guess whenever we construct a Lie $n$-algebra by extending a Lie algebra using a cocycle, we actually get a “strict” exact sequence of the sort mentioned in our paper. But yes, I see that this is a “coincidence”.

And yes, we should talk about Fierz identities. We just forgot! And thanks for catching all those typos.

Can you maybe see aspects of what makes these cocycles special compared to other cocycles that the Poincaré super Lie algebra has? What other cocycles that involve the spinors are there? Maybe there are a bunch of generic cocycles and then some special ones that depend on the dimension?

Well, of course these cocycles are special because they come from Fierz identities that hold only in certain special dimensions, and we’ve tried to “explain” that by giving a proof using normed division algebras.

But alas, we don’t know reallly good answers to any of the questions you are asking here. Of course we’ve considered these questions. I think they are utterly fascinating. At times I’ve wanted John Huerta to do his thesis on these questions! But right now, other questions seem a bit easier, and perhaps interesting to more people. The questions you’re asking require a highly developed expertise in representation theory.

Is there any indication from the math to which extent $(3,4,6,10)$ and $(4,5,7,11)$ are the first two steps in a longer sequence of sequences? I might expect another sequence $(7,8,10,14)$ and $(11,12,14,18)$ corresponding to the fivebrane and the ninebrane.

If John Huerta gets really good at representation theory we could work out the full story, but right now we are mainly happy to see that the techniques we’re using do not give a 3-brane sequence $(5,6,8,12)$. I.e., they do not give a ‘5-$\Psi$’s rule’ in all these dimensions. And that’s what one would expect, given the apparent lack of a supersymmetric 3-brane theory in 12d Minkowski spacetime.

David wrote:

Aren’t the first two sequences [strings in dimensions $3,4,6,10$ and 2-branes in dimension $4,5,7,11$] supposed to be from the columns in Duff’s chart in the post?

Yes, and you’ll see that he explicitly links this chart to the normed division algebras, but in a somewhat mysterious way, which we wanted to clarify.

Is the specialness of $5$ and $9$ connected to their being $(1+4)$ and $(1+8)$?

I’m not sure what you think is special about $5$ and $9$.

I was just wondering about Urs’ fivebrane and ninebrane as in the quoted portion. He then replied about the sequence (string, fivebrane, ninebrane) of dimension $4k+1$.

Okay, David I get what you mean now. I think the cocycles governing Urs’ fivebrane and ninebrane are “purely bosonic”, unlike the ones John Huerta and I are considering. I now realize that’s why I’m having trouble making any connection between what Urs is talking about now and what we did. But they should be part of the same story.

In other words: I’m guessing all $p$-brane theories involve cocycles on the Poincaré Lie superalgebra. This superalgebra is a Z/2-graded vector space with a bracket. The even part, or “bosonic part”, is an ordinary Lie algebra, namely the Lie algebra of the Poincaré group. The odd part, or “fermionic part”, is the space of spinors. I think Urs is implicitly getting cocycles on the Poincaré Lie superalgebra from cocycles on its bosonic part.

(Checking that this is possible requires a tiny calculation, which I am alas too busy to do right now).

What are cocycles on the Poincaré Lie algebra like? Well, it should include the cohomology of the rotation Lie algebra, and in fact that could even be all there is.

The Lie algebra of the rotation group has a bunch of interesting cocycles, related to Pontryagin classes.

If I’m not getting mixed up, the Lie algebra of the rotation group has a nontrivial 3-cocycle, a nontrivial 7-cocycle, a nontrivial 11-cocycle… and so on up to a certain cutoff — and if you work with rotations in high enough dimensions, you can make this cutoff as high as you like.

So, we get cocycles of degree $4k+3$ for $k$ below a certain cutoff. These give Lie $(4k+2)$-algebras, which in turn can be used to describe the parallel transport of $(4k+1)$-branes.

Oh, good — the calculation is working — it matches Urs’ claim! I was worried until the end there.

But anyway, all this stuff is “purely bosonic”. John Huerta and I were focusing on cocycles that only exist on the Poincaré Lie superalgebra.

I will need to think about this more someday.

I do not know if you noticed the recent article

• Fei Han, Kefeng Liu, Gravitational anomaly cancellation and modular invariance, arxiv/1007.5295

…we obtain general anomaly cancellation formulas of any dimension. For $4k+2$ dimensional manifolds, our results include the gravitational anomaly cancellation formulas of Alvarez-Gaumé and Witten in dimensions 2, 6 and 10 as special cases. In dimension $4k+1$, we derive anomaly cancellation formulas for index gerbes. In dimension $4k+3$, we obtain certain results about eta invariants, which are interesting in spectral geometry.

Of course, it is well known that the eta invariants are related to the cancellation of anomalies, but here a rather complete picture is claimed for the dimensions discussed in this cafe post, some aspects of which are directly related to the work of Urs with Hisham Sati. Unfortunately the above article is dense with complicated formulas, hence hard to read; fortunately the classical formulas from Bismut, Lott, Freed and others are recalled.

What’s ‘our construction’? Lie 2-superalgebras and Lie 3-superalgebras extending the Poincaré superalgebra? No, nobody has considered this in characteristic other than 0. After all, we only revealed the construction for characteristic 0 this weekend! Math moves fast these days, but not that fast.

There’s a book that discusses composition algebras in nonzero characteristic, though: The Book of Involutions.

Urs wrote:

I am wondering what the most general abstract way would be to understand that division algebras are related to supersymmetry.

Me too!

I am guessing it must be all related to the fact that the four division algebras are twistings by a 2-cocycle of the group algebras over $ℝ$ of the groups $({\mathbb{Z}}_2{)}^0=\{0\}$, $({\mathbb{Z}}_2{)}^1={\mathbb{Z}}_2$, $({\mathbb{Z}}_2{)}^2$ and $({\mathbb{Z}}_2{)}^3$, as described in the last paragraph of John Baez: The Fano plane.

I wish that were true. But this construction produces lots of algebras, not just the four normed division algebras. For example, I can get any Clifford algebra by taking the group algebra of $({\mathbb{Z}}_2{)}^n$ and twisting it by a 2-cocycle.

Furthermore, the octonions are obtained by taking the group algebra of $({\mathbb{Z}}_2{)}^3$ and twisting it by a 2-cochain that’s not a 2-cocycle. That’s why the octonions are nonassociative.

And if I allow myself to twist group algebras of the groups $({\mathbb{Z}}_2{)}^n$ by 2-cochains that aren’t cocycles, I can get all sorts of crazy nonassociative algebras. For example: if we keep applying the Cayley-Dickson construction starting with the real numbers, we get the complex numbers, the quaternions, the octonions, the sedenions, and so on — an infinite sequence of algebras! After the quaternions, they’re all nonassociative. And they can all be obtained by twisting the group algebras of the groups $({\mathbb{Z}}_2{)}^n$ by certain 2-cochains.

I would love it if these nonassociative algebras were all related to supersymmetry, but I don’t see any sign of that.

From the papers John Huerta and I have written, it seems quite clear that supersymmetric field theories beloved by physicists have a lot to do with composition algebras. The real numbers, complex numbers, quaternions and octonions are all composition algebras. And, the properties of composition algebras, such as the alternative law, give all the cocycles that we need to construct superstring theories in dimensions 3,4,6, and 10, and super-2-brane theories in dimensions 4,5,7, and 11.

As further evidence for the importance of composition algebras, note that besides the real numbers, complex numbers, quaternions and octonions, there are precisely three more composition algebras over $ℝ$: the split complex numbers, the split quaternions and the split octonions. And, I believe we get classical superstring and super-2-brane Lagrangians from these as well! These theories live in spacetimes with more than one time dimension — so people don’t think about these theories very much. But, I think they exist.

So, I believe we should think about composition algebras and their special properties. Anyone who likes composition algebras and category theory should read this:

• Markus Rost, On the dimension of a composition algebra.

It gives a purely diagrammatic proof that the dimension of a composition algebra must be 1, 2, 4, or 8!

Thanks for pointing out those papers! For some reason I hadn’t known about them. Crazy people like me have hoped for a long time that the octonionic projective plane would be important in physics. Octonionic structures are showing up all over in string theory and $M$-theory, so why not this?

But, it will take me a long time to say something intelligent about these papers.