## December 9, 2007

### 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 $Spin(7,1)\times Spin(8)$ subgroup1 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 embeddings2 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 $Spin(12,4)$ compatible3 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:

(1)$\begin{gathered} Spin(8,0)\times Spin(4,4),\, Spin(7,1)\times Spin(5,3),\, Spin(6,2)\times Spin(6,2),\,\\ Spin(7,1)\times Spin(1,7),\, Spin(6,2)\times Spin(2,6) \end{gathered}$

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

(2)$Spin(7,1)\supset SU(3)\times U(1)_a$

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

(3)$(8_s, 8_s) + (\overline{8}_s + \overline{8}_s) = (\mathbf{2},\mathfrak{R} + \overline{\mathfrak{R}} + 1_0 + 1_0) + (\overline{\mathbf{2}},\mathfrak{R} + \overline{\mathfrak{R}} + 1_0 + 1_0)$

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 $Spin(12)\times SU(2)\times SU(2)$:

\begin{aligned} 120 &= (66,1,1) + (1,3,1) + (1,1,3) + (12,2,2) \\ 128 &= (32,2,1) + (32',1,2) \end{aligned}

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 $E_{8(8)}$.

Posted by distler at December 9, 2007 11:17 PM

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### 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.

Posted by: mark on December 10, 2007 6:53 PM | Permalink | Reply to this
Read the post Garrett and Smolin, to boldly go…
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Excerpt: I should warn the reader that all that follows is a personal interpretation from a curious person that is attempting to learn new subjects. Your corrections are welcome. —– As I have previously mentioned, Garrett Lisi’s work sounds ex...
Tracked: December 13, 2007 4:26 AM

### The Rest of Smolin’s Paper

There may be some discussion here.

Posted by: mitchell porter on December 13, 2007 5:32 AM | Permalink | Reply to this

### Re: The Rest of Smolin’s Paper

By the way, Lee Smolin makes some comments here and it seems that he finds nothing wrong with Lisi’s proposal 8-)

Posted by: wolfgang on December 13, 2007 12:47 PM | Permalink | Reply to this

### Re: A Little More Group Theory

Garrett makes some comments here saying:
The Pati-Salam GUT I’m embedding in E8 is a left-right symmetric model, and therefore not chiral. This symmetry would have to be broken down to the chiral standard model – but ways to do this are well established.

Am I reading it right? The last paragraph in this blog, in Appendix, contains a suggestion that it cannot be correct either?

Posted by: observer on December 15, 2007 9:11 AM | Permalink | Reply to this

### Pati-Salam

No, it’s not correct.

1. $\mathfr{R}_{PS}$ is a complex representation of $SU(4)\times SU(2)\times SU(2)_B$. Pati-Salam is, like the Standard Model, a chiral gauge theory.
2. Garrett’s embedding in $E_8$ yields a generation and an anti-generation (bits and pieces of each, being contained in the $(8_s,8_s)$ and the $(\overline{8}_s,\overline{8}_s)$). This is a non-chiral spectrum.
3. It is possible, without breaking electroweak gauge symmetry, to “pair up” the generation and the anti-generation, giving them both a large mass.
4. But it is not possible to give a mass just to the anti-generation, without breaking electroweak gauge symmetry. You can’t turn a non-chiral spectrum into a chiral one.

I don’t know why Garrett said what he said. It seems to indicate a confusion about things much more basic than the (admittedly rather esoteric) representation theory of non-compact real forms of $E_8$.

Posted by: Jacques Distler on December 15, 2007 9:42 AM | Permalink | PGP Sig | Reply to this

### Re: Pati-Salam

Can’t you solve the chirality problem with orbifolds?

Posted by: John G on December 18, 2007 11:43 AM | Permalink | Reply to this

### Re: A Little More Group Theory

I think your last equation has a typo in it: a “+” that should be a comma.

Posted by: TimmyT on December 15, 2007 9:28 PM | Permalink | Reply to this

### Thanks!

Whoops! Fixed.

And I took the liberty of adding a footnote on the embedding of $Spin(20,4)$, which confused the heck out of me, at first.

Posted by: Jacques Distler on December 15, 2007 9:33 PM | Permalink | PGP Sig | Reply to this
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