### My Dinner with Garrett

I had a college chum named David. David knew how to live well. He had a perennial twinkle in his eye, as he recounted his last escapade, or told you about his plans for his next. David finished his Senior Thesis a half year early and then spent most of the Spring Semester following the Grateful Dead around the country^{1}.

I was reminded of David, when I sat down recently to lunch with Garrett Lisi. Garrett is one of those free spirits whom many of us (with comparatively humdrum lives) find charming to be around. Garrett seems to also have charmed the folks at FQXi. He received a grant in 2007 to develop a “Theory of Everything” which, as it turns out, has no chiral fermions (and could not possibly have any). Which is fair enough. Most ideas in physics don’t work out, and the only way to find out what works, and what doesn’t, is to *try*. So it makes *total sense* to fund the attempt. It’s not so clear to me that another grant, to “further develop the ‘$E_8$ Theory’” makes sense, but luckily, I’m not on the FQXi selection committee. More remarkable, yet, was that he said he’s organizing a Workshop about his “theory”, and was trying to ascertain whether I would be worth inviting (I suspect the answer was “no”).

So I asked Lisi how he intended to further develop a “Theory of Everything” which, it was *already known*, could not contain chiral fermions. He said that he was still hoping to obtain chiral fermions (somehow or other) and that *complex* $E_8$ was one possibility. Another possibility had something to do with abandoning the whole notion of Lie algebras, but I’m not sure what’s left of the “$E_8$ Theory” then.

I should point out that, in many ways, $E_{8\mathbb{C}}$ is much *simpler*. Proving that a Lisi-esque theory, based on one of the noncompact real forms of $E_8$, cannot contain chiral fermions requires either a somewhat ugly brute-force calculation, or a cleverer, but slightly indirect argument. In the case of complex $E_8$, even a brute force calculation takes only about a page.

So, in the interest of reducing my carbon footprint (and at the cost of boring my readers) …

For a concise review of Lisi’s program, see here. The crucial point to emphasize is that, once one embeds $Spin(3,1)\hookrightarrow G$ where, for Lisi, $G$ is some noncompact form of $E_8$, which generators of $\mathfrak{g}$ are “fermions”, and which are “bosons”, is dictated by the Spin-Statistics Theorem. Spinorial representations of $Spin(3,1)\sim SL(2,\mathbb{C})$ are fermions, tensorial representation are bosons, and which is which in *entirely determined* by the embedding. This distinction, between spinorial and tensorial representations of $SL(2,\mathbb{C})$, yields a $\mathbb{Z}_2$ grading on $\mathfrak{g}$.

The noncompact real forms of $E_8$ afforded some notion of economy — Lisi hoped to get the matter content of the Standard Model without too much extra junk. $E_{8\mathbb{C}}$, being twice as large, will contain a disgusting amount of unphysical extra junk in the bosonic sector. Fortunately, proving that the fermion sector is unsatisfactory is much easier, and we can take a more direct approach and prove a more general result.

Consider *any* embedding $SL(2,\mathbb{C})\hookrightarrow E_{8\mathbb{C}}$. This gives a $\mathbb{Z}_2$ grading on the Lie algebra. The converse is not true; not all $\mathbb{Z}_2$ grading come from an embedding of $SL(2,\mathbb{C})$. If we look at the list of symmetric spaces (in 1-1 correspondence with $\mathbb{Z}_2$ gradings),
$\begin{gathered}
E_{8\mathbb{C}}/E_8\\
E_{8\mathbb{C}}/E_{8(8)}\\
E_{8\mathbb{C}}/E_{8(-24)}\\
E_{8\mathbb{C}}/SO(16,\mathbb{C})\\
E_{8\mathbb{C}}/SL(2,\mathbb{C})\times E_{7\mathbb{C}}
\end{gathered}$
the first three correspond to “outer” automorphisms (complex conjugation); the latter two, as we shall see presently^{2}, arise from embeddings of $SL(2,\mathbb{C})$.

So let’s consider $E_{8\mathbb{C}}/G$, for $G= SO(16,\mathbb{C}),\, SL(2,\mathbb{C})\times E_{7\mathbb{C}}$. Let $H$ be the commutant of $SL(2,\mathbb{C})$ in $E_{8\mathbb{C}}$ and let $H_c$ be the maximal compact subgroup of $H$. We’re not quite interested in any old embedding. When we decompose the adjoint of $E_{8\mathbb{C}}$ under $SL(2,\mathbb{C})$, the only spinorial representation we wish to appear are the $2$ and the $\overline{2}$. This means we want $SL(2,\mathbb{C})\times H \hookrightarrow G$ to be *maximal*.

Up to isomorphism, then, there are two cases to consider.

- $G=SO(16,\mathbb{C})$, $H= SO(13,\mathbb{C})$. Under $\begin{aligned} E_{8\mathbb{C}}&\supset SL(2,\mathbb{C})\times SO(13)\\ 248_{\mathbb{C}} &= (3+\overline{3},1) + (1,78) + (1,78) + (3,13) + (\overline{3},13) + {\color{red} (2,64) + (\overline{2},64)} \end{aligned}$
- $G=SL(2,\mathbb{C})\times E_{7\mathbb{C}}$, $H= E_{7\mathbb{C}}$. Under $\begin{aligned} E_{8\mathbb{C}}&\supset SL(2,\mathbb{C})\times E_7\\ 248_{\mathbb{C}} &= (3+\overline{3},1) + (1,133) + (1,133) + {\color{red} (2,56) + (\overline{2},56)} \end{aligned}$

In both cases, the $64$ and the $56$ are pseudoreal, and the fermion representation is **nonchiral** (for **any** gauge group which is, respectively, a subgroup of $SO(13)$ or of $E_7$).

There … I feel so much greener already.

^{1} The thing about my friend David, though, was that he is *also* really, really, smart. Fun aside, he graduated *Summa Cum Laude* from Harvard.

^{2} A similar “brute force” approach to the noncompact real forms of $E_8$ would be a bit more tedious to carry out, because one would have to study each of the symmetric spaces
$\begin{gathered}
E_{8(8)}/SO(16)\\
E_{8(8)}/SO(8,8)\\
E_{8(8)}/SO^*(16)\\
E_{8(8)}/E_{7(-5)}\times SU(2)\\
E_{8(8)}/E_{7(7)}\times SL(2,\mathbb{R})
\end{gathered}$
and
$\begin{gathered}
E_{8(-24)}/SO(12,4)\\
E_{8(8)}/SO^*(16)\\
E_{8(-24)}/E_7\times SU(2)\\
E_{8(-24)}/E_{7(-5)}\times SU(2)\\
E_{8(-24)}/E_{7(-25)}\times SL(2,\mathbb{R})
\end{gathered}$
individually. Fortunately, that’s unnecessary.

Above, $SO^*(2n)$ is the “additional” real form of $D_n$, in addition to the familiar $SO(p,2n-p)$. It is defined as the subgroup of $GL(2n,\mathbb{C})$ preserving $\sum_{p=1}^n z_p y_p$ and $\sum_{p=1}^n |z_p|^2 - |y_p|^2$. Its maximal compact subgroup is $U(n)$, and for low dimensions, we have the isomorphisms $\begin{aligned} so^*(8) &\simeq so(6,2)\\ so^*(6) &\simeq su(3,1)\\ so^*(4) &\simeq su(2) \times sl(2,\mathbb{R}) \end{aligned}$

## Re: My Dinner with Garrett

“Another possibility had something to do with abandoning the whole notion of Lie algebras, but I’m not sure what’s left of the “E 8 Theory” then.”

Did he say anything else about this?