Musings

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 presently², 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.