### Synchronicity

I was visiting the University of Michigan earlier this Spring, where Gordy Kane told me a story. He’d recently given a public lecture, and was somewhat taken aback when, during the question period afterwards, he was asked about the status of Garrett Lisi’s Theory of Everything. The questioner was convinced that Lisi was a key player in the unification 'biz, and was surprised that his theory had not received more attention (which is to say, had not been mentioned at all) in Gordy’s talk.

The very same day, I received an email from a mathematician working in Representation Theory. He was disgruntled that his student was being asked about Lisi’s work in the course of job interviews. He knew I had some blog posts about Lisi’s “theory”, but had I written these up anywhere? I responded that I didn’t see how it could possibly be worthwhile to publish a “refutation” of an unpublished work. And, in any case, what I had done was of such a mind-numbingly trivial nature that no respectable journal (in either Math or Physics) would consent to publish it. But Skip insisted that it would be helpful to someone like his student to be able to cite a paper in which this stuff had been debunked.

By coincidence, an exchange of comments, with the man himself, at the at the n-Category Café, convinced me that my blog posts have been less than efficacious.

So I decided to take Skip up on his suggestion and try to distil the arguments of the aforementioned blog posts and strengthen them into a theorem that some (not necessarily self-respecting) Math journal might publish.

*chiral*if $R$ formed a

*complex*representation of the gauge group. Alas, it is easy to prove

^{1}that $R$ is a pseudoreal (quaternionic) representation of $H$ (and hence, a real or pseudoreal representation of any subgroup of $H$). Thus Lisi’s “theory” is

*always*

**nonchiral**, and cannot have anything to do with that thing we modestly call “the real world.” The proof consists of enumerating all the possible inequivalent embeddings of $SL(2,\mathbb{C})$, subject to our constraint on $m+n$. The result can be summarized in the table below:

Form of $E_8$ | Centralizer ($G$) | Maximal compact subgroup ($H$) | Representation ($R$) |
---|---|---|---|

$E_{8(-24)}$ | $Spin(1,11)$ | $Spin(11)$ | $32$ |

$E_{8(8)}$ | $Spin(5,7)$ | $Spin(5)\times Spin(7)$ | $(4,8)$ |

$E_{8(-24)}$ | $Spin(9,3)$ | $Spin(9)\times Spin(3)$ | $(16,2)$ |

$E_{8\mathbb{C}}$ | $Spin(12,\mathbb{C})$ | $Spin(12)$ | $64$ |

$E_{8\mathbb{C}}$ | $Spin(13,\mathbb{C})$ | $Spin(13)$ | $64$ |

$E_{8\mathbb{C}}$ | $E_{7\mathbb{C}}$ | $E_7$ | $56$ |

In each case, $R$ is a pseudoreal representation of $H$.

In related news, the meltdown of the world’s financial system has prompted Lee Smolin to move from Loop Quantum Gravity into Mathematical Finance. I wish him the best of luck.

^{1} The corresponding “Euclidean” statement, about embeddings of $Spin(4)$ in compact $E_8$ is completely trivial: $G=H=Spin(12)$, whose irreducible spinor representations are pseudoreal.

## Re: Synchronicity

Thanks for the effort Jacques, but it seems that theorems and facts are of little consequence against a PR machine and willful ignorance.