## March 22, 2007

### Split Real Forms

The science blogosphere has been all atwitter, this week, about $E_8$, and a purported breakthrough in the representation theory thereof. Most of the posts were not particularly informative. The best of the lot was here, on our sister blog, the n-Category Café.

Not having much intelligent to say, I thought I would take a pass on adding to the frenzy. But, on reconsideration, I thought I might, at least, add $\epsilon$ about the connection with physics.

First of all, you have to realize what is being talked about is not our friend, the compact Lie group, $E_8$, but a distant cousin, the split real form, which I will, henceforth, denote by $\tilde{E}_8$. A complex, simple Lie algebra, $\mathfr{g}_{\mathbb{C}}$, can have several real forms, only one of which is the Lie algebra of a compact Lie group, $G$. At the opposite extreme is the split real form, whose corresponding Lie group, $\tilde{G}$ is “as noncompact as possible.” For example, $so(2n,\mathbb{C})$ has a compact real form $SO(2n)$, and a split real form, $SO(n,n)$ (and intermediate real forms, $SO(2n-k,k)$).

Anyone familiar with the heterotic string will recognize the “E” series of compact Lie groups: $\begin{gathered} E_8,\; E_7,\; E_6,\; E_5=Spin(10),\\ E_4=SU(5),\; E_3=SU(3)\times SU(2) \end{gathered}$

The corresponding split real forms $\begin{gathered} \tilde{E}_8,\;\tilde{E}_7,\; \tilde{E}_6,\; \tilde{E}_5=Spin(5,5),\\ \tilde{E}_4=SL(5,\mathbb{R}),\; \tilde{E}_3=SL(3,\mathbb{R})\times SL(2,\mathbb{R}) \end{gathered}$ appear in the maximal supergravity theories (the dimensional reductions of 11 dimensional supergravity down to $d=11-n$ dimensions). Specifically, the scalars in the supergravity multiplet take values on the homogeneous space $\tilde{E}_n/K_n$, where $K_n$ is the maximal compact subgroup1 of $\tilde{E}_n$. $\begin{gathered} K_8 = Spin(16),\; K_7 = SU(8),\; K_6 = Sp(4),\\ K_5= Spin(5)\times Spin(5),\; K_4=Spin(5),\; K_3 = SU(2)\times SO(2) \end{gathered}$

By construction, $\tilde{E}_n$ acts a a global symmetry group of the supergravity theory.

Alas, with the exception of the $d=3$ (and possibly $d=4$) cases, the supergravity theory is nonrenormalizable, and must be UV-completed. The completion is Type-II string theory (or M-theory) compactified on a torus. The higher dimension operators in the $d$-dimensional effective Lagrangian are not invariant under the continuous $\tilde{E}_n$, but only under a discrete $\tilde{E}_n(\mathbb{Z})$ subgroup, called the U-duality group.

There are massive BPS states in the theory, and these can be organized into representations of $\tilde{E}_n(\mathbb{Z})$. If you are interested in studying the spectrum of such BPS states (say, to write down a U-duality-invariant formula for the entropy of blackholes in this theory), then you are interested in the representation theory of $\tilde{E}_n$.

For $d=3$, that’s $\tilde{E}_8$, and that’s presumably where these latest results might hold some interest for physicists.

1 Looking at this table, I suspect I am not being sufficiently careful about the centers of the respective groups.

Posted by distler at March 22, 2007 12:25 PM

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### Re: Split Real Forms

Last year I got a severe reprimand from M. Wodzicki when I invoked the notation E_5, E_4, E_3

Posted by: Alejandro Rivero on March 22, 2007 4:07 PM | Permalink | Reply to this

### Re: Split Real Forms

Luckily, he doesn’t run the show here.

I should point out that this notation breaks down spectacularly, if you push it one step further.

From the above, you might guess that the global symmetry group of maximal supergravity in 9 dimensions should be $\tilde{E}_2 = SL(2,\mathbb{R})\times SL(2,\mathbb{R})$.

But you would be wrong. The scalars in 9-dimensional maximal supergravity live on $SL(2,\mathbb{R})/SO(2)$, and the global symmetry group is rank-1, rather than rank-2.

Posted by: Jacques Distler on March 22, 2007 4:41 PM | Permalink | PGP Sig | Reply to this

### Re: Split Real Forms

On the other hand, it seems to continue nicely in the other direction: $E_9$ for sugra down in 2 dimensions, $E_{10}$ in $d=1$ and $E_{11}$ in $d = 0$.

Well, I gather that is not settled yet, but there is circumstantial evidence that there is something to this.

By the way, do you know if and how that quantization condition $E_{n} \to \tilde E_{n}(\mathbb{Z})$ appears beyond $n = 8$?

Posted by: urs on March 23, 2007 5:35 AM | Permalink | Reply to this

### Re: Split Real Forms

On the other hand, it seems to continue nicely in the other direction: $E_9$ for sugra down in 2 dimensions, $E_{10}$ in $d=1$ and $E_{11}$ in $d=0$.

I believe it’s the source of these conjectures.

By the way, do you know if and how that quantization condition $\tilde{E}_n\to\tilde{E}_n(\mathbb{Z})$ appears beyond $n=8$?

One way to understand it is from the quantization of charges of BPS particles and branes. The lattice of charges is not preserved by continuous $\tilde{E}_n$ transformations, but is preserved by $\tilde{E}_n(\mathbb{Z})$.

Posted by: Jacques Distler on March 23, 2007 10:44 AM | Permalink | PGP Sig | Reply to this
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