2-Branes in 11 Dimensions
Posted by John Baez
I’m trying to learn a teeny bit more about supersymmetric membrane theories, and I’m so far behind that this old review article is proving helpful:
- Micheal J. Duff, Supermembranes: the first fifteen weeks, Classical and Quantum Gravity 5 (1988), 189–205.
I’m particularly fascinated by the classification of ‘fundamental super $p$-branes’ and its relation to normed division algebras:
- Reals: 2-branes in 4 dimensions, with 1 bosonic and 1 fermionic degree of freedom.
- Complexes: 3-branes in 6 dimensions, with 2 bosonic and 2-fermionic degrees of freedom.
- Quaternions: 5-branes in 10 dimensions, with 4 bosonic and 4 fermionic degrees of freedom.
- Octonions: 2-branes in 11 dimensions, with 8 bosonic and 8 fermionic degrees of freedom.
I talked about this ‘brane scan’ in incredibly elementary terms back in week118, but now I’d like to actually understand it. It’s supposed to follow from something like a classification of closed differential forms on super-Minkowski spacetimes, due to Achúcarro et al. I don’t know how it goes, but it seems potentially quite comprehensible.
If anyone can help me with this, I’d appreciate it a lot. But I really want to say a word about the 11-dimensional case — a brief explanation for complete novices such as myself — and then ask a question about that.
In its so-called ‘Nambu–Goto formulation’, a 2-brane in 11 dimensions is classically just a map
$\phi: \Sigma \to M$
where $\Sigma$ is an oriented 3-manifold and $M$ is ‘11-dimensional super-Minkowski spacetime’. What’s that? Well, we can crassly think of $M$ as a vector space that’s split in two parts, the ‘even’ and ‘odd’ part:
$M = \mathbb{R}^{11} \oplus \mathbb{R}^{32}$
Here $\mathbb{R}^{11}$ is plain old 11-dimensional Minkowski spacetime, while $\mathbb{R}^{32}$ is a certain representation of the double cover of the Lorentz group in 11 dimension: the space of ‘Majorana spinors’.
As usual in classical field theory, to count as a solution of the equations of motion, the map $\phi : \Sigma \to M$ needs to be a critical point of some action. And I guess this action is the sum of two parts.
First, any 3-dimensional manifold mapped into Minkowski spacetime gets a notion of ‘volume’, and I guess the same sort of thing works in the super world. So, the first term in the action is just the ‘supervolume’ of the image of $\Sigma$ in $M$.
The second term is more special: it’s the integral of some 3-form over $\Sigma$ — a 3-form obtained by taking a god-given 3-form on $M$ and pulling it back along $\phi$. This is presumably where the classification of closed differential forms on super-Minkowski spacetimes comes in handy.
Now comes my question. Duff says that something called ‘Siegel symmetry’ cuts the actual number of fermionic degrees of freedom in half — at least ‘on shell’, meaning when the equations of motion hold. I’m wondering if there’s any way to describe the 2-brane so that this cutting in half occurs right from the start. For example, say we take our 11 dimensions and split them into 10 plus 1 extra, thus writing
$M = \mathbb{R}^{10} \oplus \mathbb{R} \oplus \mathbb{R}^{16} \oplus \mathbb{R}^{16}$
where $\mathbb{R}^{10}$ is 10-dimensional Minkowski spacetime, $\mathbb{R}$ is the extra space dimension, and $\mathbb{R}^{32}$ gets split up into $\mathbb{R}^{16} \oplus \mathbb{R}^{16}$, namely the left- and right-handed Majorana-Weyl spinors in 10 dimensions. Can we somehow write down an action for our 2-brane that just involves this portion of the map $\phi$:
$\tilde{\phi}: M \to \mathbb{R}^{10} \oplus \mathbb{R} \oplus \mathbb{R}^{16} \; ?$
This would be cute, because $\mathbb{R}^{10} \oplus \mathbb{R} \oplus \mathbb{R}^{16}$ is just another way of thinking about the exceptional Jordan algebra — that is, the space of $3 \times 3$ self-adjoint octonion matrices!
Any clues would be most greatly appreciated. I’m sure this is pretty basic stuff to those in the know. What I’m hoping for may just not work.
Re: 2-Branes in 11 Dimensions
If you are looking for beautiful algebraic symmetries, I think you will like Peter West’s works on M-Theory and supergravity. I think this is what you want.
He studies an algebra called E11, and has been somewhat successful in trying to prove this conjecture.