### M-theory, Octonions and Tricategories

#### Posted by John Baez

Quite a witches’ brew, eh?

Amazingly, they seem to be deeply related. John Huerta has just finished a paper connecting them… and this concludes a series of papers that makes me very happy, because it fulfills a long-held dream: to connect physics, division algebras, and higher categories.

Let me start with a very simple sketchy explanation. Experts should please forgive the inaccuracies in this first section: it’s hard to tell a story that’s completely accurate without getting bogged down in detail!

### The rough idea

You’ve probably heard rumors that superstring theory lives in 10 dimensions and something more mysterious called M-theory lives in 11. You may have wondered why.

In fact, there’s a nice way to write down theories of superstrings in dimensions 3, 4, 6, and 10 — at least before you take quantum mechanics into account. Of these theories, it seems you can only consistently quantize the 10-dimensional version. But never mind that. What’s so great about the numbers 3, 4, 6 and 10?

What’s so great is that they’re 2 more than 1, 2, 4, and 8.

There are only normed division algebras in dimensions 1, 2, 4, and 8. The real numbers are 1-dimensional. The complex numbers are 2-dimensional. There are also more esoteric options: the quaternions are 4-dimensional, and the octonions are 8-dimensional. When you try to go beyond these, you lose the law that

$|x y| = |x| |y|$

and things aren’t so nice.

I’ve spent decades studying the quaternions and octonions, just because they’re weird and interesting. Why do the dimensions double each time in this game? There’s a nice answer. What happens if you go further, to dimension 16? I’ve learned a bit about that too, though I bet there are big mysteries still lurking here.

Most important, what — if anything — do normed division algebras have to do with physics? The jury is still out on this one, but there are some huge clues. Most fundamentally, a normed division algebra of dimension $n$ gives a nice unified way to describe both spin-1 and spin-1/2 particles in $(n+2)$-dimensional spacetime! The gauge bosons in nature are spin-1 particles, while the fermions are spin-1/2 particles. We’d definitely like a good theory of physics to fit these together somehow.

One cool thing is this. A string is a curve, so it’s 1-dimensional, but as time passes it traces out a 2-dimensional surface. So, if we have a string floating around in some spacetime, we’ve got a 2d surface together with some extra dimensions of spacetime. It turns out to be very good to put complex coordinates on that 2d surface. Then you can describe how the string wiggles in the extra dimensions using equations that have symmetry under conformal transformations.

But for the string to be ‘super’ — for it to have supersymmetry, a symmetry between bosons and fermions — we need a certain special identity to hold, called the 3-$\psi$’s rule. And this holds precisely when we can take the extra dimensions and think of them as forming a normed division algebra!

So, we need 1, 2, 4 or 8 extra dimensions. So the total dimension of spacetime needs to be 3, 4, 6, or 10. *Not at all coincidentally*, these are also the dimensions where spin-1 and spin-1/2 particles can be described using a normed division algebra.

(This is a very rough sketch of a complicated argument, of course. I’m leaving out the details, but later I’ll show you where to find them.)

We can also look at theories of ‘branes’, which are like strings but higher-dimensional. Instead of a curve, a 2-brane is a 2-dimensional surface. As time passes, it traces out a 3-dimensional surface. So, if we have a 2-brane floating around in some spacetime, we’ve got a 3-dimensional surface together with some extra dimensions of spacetime. And it turns out that 2-branes can also have supersymmetry when the extra dimensions can be seen as a normed division algebra!

So now the total dimension of spacetime needs to be 3 more than 1, 2, 4, and 8. It needs to be 4, 5, 7 or 11.

When we take quantum mechanics into account it seems that the 11-dimensional theory works best… but the quantum aspects are still mysterious, murky and messy compared to superstring theory, so it’s called M-theory.

In his new paper, John Huerta has shown that using the octonions we can build a ‘super-3-group’, an algebraic structure that seems just right for understanding the symmetries of supersymmetric 2-branes in 11 dimensions.

I could say a lot more, but if you want more explanation without too much fancy math, try this:

- John Baez and John Huerta, The strangest numbers in string theory.

This is a fun and easy article about this stuff, which we wrote for *Scientific American*.

### The details

The detailed story has four parts.

- John Baez and John Huerta, Division algebras and supersymmetry I, in
*Superstrings, Geometry, Topology, and C*-Algebras*, eds. Robert Doran, Greg Friedman and Jonathan Rosenberg,*Proc. Symp. Pure Math.***81**, AMS, Providence, 2010, pp. 65–80.

Here we explain how to use normed division algebras to describe vectors (and thus spin-1 particles) and spinors (and thus spin-1/2 particles) in spacetimes of dimensions 3, 4, 6 and 10. We use this description to derive the 3-$\psi$’s rule, an identity obeyed by three spinors only in these special dimensions. We also explain how the 3-$\psi$’s rule is important in supersymmetric Yang–Mills theory. This stuff was known before, but not explained all in one place.

- John Baez and John Huerta, Division algebras and supersymmetry II,
*Adv. Math. Theor. Phys.***15**(2011), 1373–1410.

Here go up a dimension and use normed division algebras to derive a special identity that is obeyed by 4 spinors in dimensions 4, 5, 7 and 11. This is called the 4-$\Psi$’s rule, and it’s important for supersymmetric 2-branes.

More importantly, we start studying how the symmetries of superstrings and super-2-branes arise from the normed division algebras. Mathematicians and physicists use Lie algebras to study symmetry, as well as generalizations called ‘Lie superalgebras’, which describe symmetries that mix bosons and fermions. Here we study categorified versions called ‘Lie 2-superalgebras’ and ‘Lie 3-superalgebras’. It turns out that the 3-$\psi$’s rule is a ‘3-cocycle condition’ — just the thing you need to build a Lie 2-superalgebra extending the Poincaré Lie superalgebra! Similarly, the 4-$\Psi$’s rule is a ‘4-cocycle condition’ which lets you build a Lie 2-superalgebra extending the Poincaré Lie superalgebra.

Next, try this:

- John Huerta, Division algebras and supersymmetry III.

At this point John Huerta sailed off on his own!

In this paper John cooked up the ‘Lie 2-supergroups’ that govern classical superstrings in dimensions 3, 4, 6 and 10. Just as a group is a category with one object and with all its morphisms being invertible, a 2-group is a bicategory with one object and with all its morphisms and 2-morphisms being weakly invertible. A Lie 2-supergroup is a bicategory internal to the category of supermanifolds. John shows how to derive the pentagon identity for this bicategory from the 3-$\psi$’s rule!

And here’s his new paper, the last of the series:

- John Huerta, Division algebras and supersymmetry IV.

Here John built the ‘Lie 3-supergroups’ that govern classical super-2-branes in dimensions 3, 4, 6 and 10. A 3-group is a tricategory with one object and with all its morphisms, 2-morphisms and 3-morphisms being weakly invertible. John shows how to derive the ‘pentagonator identity’ — that is, a commutative diagram shaped like the 3d Stasheff polytope — from the 4-$\Psi$’s rule.

In case you’re wondering: I believe this game stops here. I’m pretty sure there isn’t a nontrivial 5-cocycle (valued in the trivial representation) which gives a Lie 4-superalgebra extending the Poincaré superalgebra in 12 dimensions. But I hope someone proves this, or has proved it already!

Of course, Urs Schreiber and collaborators have done vastly more general things using a more intensely modern point of view. For example:

Hisham Sati, Urs Schreiber and Jim Stasheff,

*L*_{∞}algebra connections and applications to String- and Chern-Simons*n*-transport.Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie

*n*-algebra extensions, higher WZW models, and super*p*-branes with tensor multiplet fields.Urs Schreiber, Differential cohomology in a cohesive infinity-topos.

But one thing the ‘Division Algebras and Supersymmetry’ series has to offer is a focus on the way *normed division algebras* help create the exceptional higher algebraic structures that underlie superstring and super-2-brane theories. And with the completion of this series, I can now relax and forget all about these ideas, confident that at this point, the minds of a younger generation will do much better things with them than I could.

I should add that Layra Idarani has been ‘live-blogging’ his reading of John Huerta’s new paper:

Layra Idarani, Liveblogging “Division Algebras and Supersymmetry IV” by John Huerta, part 0 .

Layra Idarani, Liveblogging “Division Algebras and Supersymmetry IV” by John Huerta, part 1 .

Layra Idarani, Liveblogging “Division Algebras and Supersymmetry IV” by John Huerta, part 2 .

So, you can get another perspective there.

## Re: M-theory, Octonions and Tricategories

three cheers for this! Now let’s look for spaces with interesting bundles for these structures ;-)