### A Little More Group Theory

With a certain reluctance, I wrote a post about Garrett Lisi’s “Theory of Everything,” specifically about Lisi’s claim that he had embedded 3 generations of quarks and leptons in the 248 of $E_8$.

The purported “Theory of Everything” involved embedding
$G = SL(2,\mathbb{C})\times SU(3)\times SU(2)\times U(1)_Y$
in some noncompact form of $E_8$ (as it turns out, the split real form, $E_{8(8)}$), such that the 248 contains 3 copies of
$R = (2, \mathfrak{R}) + (\overline{2}, \overline{\mathfrak{R}})$
where $\mathfrak{R}$ is the $SU(3)\times SU(2)\times U(1)_Y$ representation
$(3,2)_{1/6} + (\overline{3},1)_{-2/3} + (\overline{3},1)_{1/3} + (1,2)_{-1/2} +(1,1)_1$
Note that $\mathfrak{R}$ is a *complex* representation. So $R$, though a real representation of $G$, is *chiral*.

I showed that it is impossible to find an embedding of $G$, which yields 3 copies of $R$, and hence that Lisi’s “Theory of Everything” doesn’t *even* rise to the level of impressive numerology.

And that’s where I left it, *thinking* that this would be enough to settle the matter in the mind of anyone with even a *modicum* of sense. I allowed to slide Lisi’s claim that he “got the first generation right.” After all, what harm could there be, in letting that little bit stand?

Apparently, I was wrong.

So, just so there’s *no ambiguity*, let me go back and point out that Lisi’s proposed embedding of $G$ does not *even* “get the first generation right.”

Despite all the talk, in his paper, about $F_4\times G_2$, I eventually got the following clear statement from Lisi: $G$ is embedded in a $Spin(7,1)\times Spin(8)$ subgroup^{1} of $E_{8(8)}$, and a generation (a copy of $R$) fits in the $(8_v,8_v)$. Now, the spinor representation of $Spin(7,1)$ is complex. So, if there *were* such an embedding of $G$ in $E_{8(8)}$, then it’s possible that this would lead to a chiral “fermion” representation (in particular, to a copy of $R$).

But that’s an “If pigs could fly …” sort of statement. There *is no* such embedding of $Spin(7,1)\times Spin(8)$ and I was too generous (or credulous or whatever) in assenting that there was. If there were, then it would sit inside a $Spin(15,1)$ or $Spin(9,7)$ subgroup, neither of which exists.

In fact, one does not get a chiral fermion representation for *any* embedding based on a subgroup $D_4\times D_4\subset E_8$. Since the group theory might be of moderate interest to someone, let’s go through it.

There are two possible embeddings^{2} of $D_8\subset E_8$, compatible with the real structure $E_{8(8)}$

$\begin{aligned}
& Spin(16)\\
& Spin(8,8)
\end{aligned}$
and one
$Spin(12,4)$
compatible^{3} with the real structure $E_{8(-24)}$.

Correspondingly, there are six possible embeddings of $D_4\times D_4\subset D_8\subset E_8$, compatible with the real structure $E_{8(8)}$: $\begin{gathered} Spin(8,0)\times Spin(8,0)\\ Spin(8,0)\times Spin(0,8),\, Spin(7,1)\times Spin(1,7),\, Spin(6,2)\times Spin(2,6),\, Spin(5,3)\times Spin(3,5),\, Spin(4,4)\times Spin(4,4) \end{gathered}$

and three compatible with the real structure $E_{8(-24)}$:

$Spin(8,0)\times Spin(4,4),\, Spin(7,1)\times Spin(5,3),\, Spin(6,2)\times Spin(6,2)$

Lisi’s choice, $Spin(7,1)\times Spin(8)$, is not on either list.

From the above nine, demanding that $G$ be embedded as a subgroup narrows the choices down to five:

Finding the representation $R$ in the decomposition of the 248 narrows the choices down to two (one appropriate to $E_{8(-24)}$, and one appropriate to $E_{8(8)}$):

$Spin(7,1)\times Spin(5,3),\, Spin(7,1)\times Spin(1,7)$

The spinor representations of $Spin(7,1)$ and $Spin(5,3)$ are complex. The 248 decomposes as

$248 = (28,1) + (1,28) + (8_v,8_v) + (8_s, 8_s) + (\overline{8}_s,\overline{8}_s)$

Under the decomposition

we have $\begin{aligned} 8_v &= 3_{-2} + \overline{3}_{2} + 1_0 + 1_0\\ 8_s &= 3_1 + \overline{3}_{-1} + 1_{-3} + 1_3\\ \overline{8}_s &= 3_1 + \overline{3}_{-1} + 1_{-3} + 1_3\\ 28 &= 8_0 + 1_0 + 1_0 + 3_{-2} + 3_{-2} + 3_4 + \overline{3}_2 + \overline{3}_2 + \overline{3}_{-4} \end{aligned}$ and under $Spin(1,7)\, \text{or}\, Spin(5,3) \supset SL(2,\mathbb{C})\times SU(2) \times SU(2)_B$ we have $\begin{aligned} 8_v &= (\mathbf{4}, 1, 1) + (1, 2, 2)\\ 8_s &= (\mathbf{2}, 2, 1) + (\overline{\mathbf{2}}, 1, 2)\\ \overline{8}_s &= (\overline{\mathbf{2}}, 2, 1) + (\mathbf{2}, 1, 2)\\ 28 &= (\mathbf{Adj}, 1, 1) + (1, 3, 1) + (1, 1, 3) + (\mathbf{4}, 2, 2) \end{aligned}$

Identifying $U(1)_Y = \tfrac{1}{6} U(1)_a + (T_3)_B$, where $(T_3)_B$ is the Cartan generator of $SU(2)_B$, we find

a completely nonchiral representation.

I leave it as an exercise for the reader to work out the remaining cases in (1). The spinor representations of $Spin(6,2)$ are pseudoreal, while those of $Spin(8)$ and $Spin(4,4)$ are real, neither of which lead to a “matter content” remotely resembling that of the Standard Model (in particular, all the “fermions” are in $SU(2)$ doublets).

And, no, I don’t intend to comment on the *rest* of Smolin’s paper. I’ll leave that to Sean or Bee or Steinn. Why should *I* have to do all the hard work around here?

#### Update (12/11/2007):

The fact that one gets a nonchiral “fermion” spectrum seems to be more general than these examples.#### Appendix: Pati-Salam

Lee Smolin complained that I failed to use the phrase “Pati-Salam” in this post. The reason I didn’t is that anyone familiar with the Pati-Salam model could easily fill in the necessary step. Anyone *unfamiliar* with it would not find its invocation the least bit helpful. It would, instead, serve only to clutter the notation.

But, to keep Lee happy (and to correct a small typo), note that the $SU(3)\times U(1)_a$ subgroup in (2) actually sits inside an $SU(4)\subset SO(7,1)$:
$\begin{aligned}
4 &= 3_1 + 1_{-3}\\
\overline{4} &= \overline{3}_{-1} + 1_3\\
6 &= 3_{-2} +\overline{3}_{2}\\
15 &= 8_0 +1_0 + 3_4 +\overline{3}_{-4}
\end{aligned}$
Rather than $G$, above, one can talk, instead, of
$G_{PS} = SL(2,\mathbb{C}) \times SU(4)\times SU(2)\times SU(2)_B$
and a “generation” is
$R_{PS} = (\mathbf{2},\mathfr{R}_{PS}) + (\overline{\mathbf{2}}, \overline{\mathfr{R}}_{PS})$
where
$\mathfr{R}_{PS} = (4, 2, 1)+ (\overline{4}, 1, 2)$
Note that $\mathfr{R}_{PS}$ is a *complex* representation of $SU(4)\times SU(2)\times SU(2)_B$, so this theory is *chiral* (which is no surprise, since $\mathfr{R}_{PS}$ consists of a Standard Model generation plus an extra $SU(3)\times SU(2)\times U(1)_Y$ singlet).

Nothing about the analysis changes, except that one can write (3) as

$(8_s, 8_s) + (\overline{8}_s, \overline{8}_s) = (\mathbf{2},\mathfr{R}_{PS} + \overline{\mathfr{R}}_{PS}) + (\overline{\mathbf{2}},\mathfr{R}_{PS} + \overline{\mathfr{R}}_{PS})$

which, unlike the *desired* result, is every bit as non-chiral as before.

Oh, and I don’t see why the “Euclidean” (compact) case, where one replaces $SL(2,\mathbb{C})$ by $Spin(4)=SU(2)_L\times SU(2)_R$, with $\mathbf{2}\to (2,1)$ and $\overline{\mathbf{2}}\to (1,2)$, is supposed to be any better. Nothing changes, except that you need to use the notion of “chiral” appropriate to Euclidean signature.

^{1} Here, and below, I’m not going to be careful about factors of $\mathbb{Z}_2$. So, *e.g.*, when I talk about $Spin(16)\subset E_{8(8)}$, I really mean $Spin(16)/\mathbb{Z}_2$.

^{2} These are the cases where the spinor, $128$, is a *real* representation. Which $Spin(16-4k,4k)$ group is associated to which real form of $E_8$ is a result of Marcel Berger. I’d like to thank Jeffrey Adams for the reference.

^{3} To get a feeling for *why* $Spin(12,4)$ embeds in $E_{8(-24)}$, but not $E_{8(8)}$, consider the following. We should be able to find an involution of the Lie algebra, which acts as $+1$ on the compact generators and as $-1$ on the noncompact generators. Decompose the $120+128$ of $Spin(12,4)$ under the maximal compact $Spin(12)\times SU(2)\times SU(2)$:

The $(12,2,2)$ are the noncompact generators of $so(12,4)$. The center of the second $SU(2)$ acts as $-1$ on them, as well as on the $(32',1,2)$, and as $+1$ on the others. That makes 112 noncompact generators, and 136 compact ones — the right numbers for $E_{8(-24)}$. And the 136 compact generators clearly line up with the decomposition of the adjoint of $E_7\times SU(2)\supset Spin(12)\times SU(2)\times SU(2)$. But there’s no similar involution of the algebra that acts as $-1$ on the $(12,2,2)$ and on 80 generators in the $128$, which would be what you would need for an embedding in $E_{8(8)}$.

## Re: A Little More Group Theory

Thanks Jacques!

I only wish Smolin and Co would do their homework before giving Lisi’s “proposal” their endorsment.