**M-theory and octonions**My former student

+John Huerta (at left) has just finished an amazing paper that I've

*got* to tell you about!

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 define superstrings in dimensions 3, 4, 6, and 10 - at least before you take quantum mechanics into account. Of these theories, 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.

If you try to set up a nice number system where you can add, multiply, subtract and divide, it only works in dimensions 1, 2, 4, and 8. The

**real numbers** form a line, and that's 1-dimensional. The

**complex numbers** form a plane, and that's 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

|xy| = |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? I've learned the answer, and I could tell you - but it might shatter your brain. 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.

I also learned that being an expert on this stuff does not make you popular at parties.

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 2-dimensional surface together with some

*extra* dimensions of spacetime.

But for the string to be 'super' - for it to have supersymmetry, a symmetry between bosons and fermions - we need a certain special equation to be true. And it's true precisely when we can take the extra dimensions and think of them as one of our nice number systems.

So, we need 1, 2, 4 or 8 extra dimensions. So the total dimension of spacetime needs to be 3, 4, 6, or 10.

(That's 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 one of our nice numbers systems!

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.

There's stuff we don't understand, and stuff we do. In his new paper, John Huerta has pushed forward the line separating the two. He's 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 it's better if you read this:

• John Baez and John Huerta, The strangest numbers in string theory,

http://math.ucr.edu/home/baez/octonions/strangest.htmlThis is a fun and easy article about this stuff, which we wrote for

*Scientific American*.

Then, if that's too easy, try this:

• John Baez and John Huerta, Division algebras and supersymmetry I,

http://arxiv.org/abs/0909.0551.

Here we get into the details, and explain the special equation that makes superstrings work nicely in 3, 4, 6, and 10 dimensions - and how it follows from having a nice number system in dimensions 1, 2, 4 and 8. This stuff was known before, but not explained all in one place.

Next, try this:

• John Baez and John Huerta, Division algebras and supersymmetry II,

http://arxiv.org/abs/1003.3436.

Here we explain the special equation that makes supersymmetric 2-branes work in dimension 4, 5, 7 and 11. More importantly, we start studying how the

*symmetries* of superstrings and super-2-branes come out of the nice number systems. Physicists use gadgets called 'Lie algebras' to study symmetry... so they should like these generalizations, called 'Lie 2-superalgebras' and 'Lie 3-superalgebras'.

Next, try this:

• John Huerta, Division algebras and supersymmetry III,

http://arxiv.org/abs/1109.3574.

At this point John Huerta sailed off on his own!

Physicists like Lie algebras, but what they really

*love* are 'Lie groups'. Lie algebras are just a trick for studying Lie groups: it's the groups that directly describe symmetry. 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 special sort of category, a 2-group is a special sort of 2-category. So at this point John got into 'higher category theory' - one of my favorite subjects.

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

• John Huerta, Division algebras and supersymmetry IV,

http://arxiv.org/abs/1409.4361.

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 special sort of 3-category.

I really love how the math of superstrings and M-theory emerge nicely from combining the octonions with higher category theory. In case you're wondering: I have no strong opinion about whether these ideas apply to our physical universe. I see no convincing experimental evidence in favor of string theory or M-theory. All I know is that they're beautiful.

Maybe they apply to some other universes that are less messed-up than ours. Maybe we're in some sort of purgatory for species who still need to learn basic math. If so, John Huerta just placed out. :-)

#spnetwork arxiv:1409.4361

#octonions #superstrings #Mtheory