Musings

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.