\documentclass{amsart}

\usepackage{times}
\usepackage{hyperref}

%\input skip-thm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Special theorems
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsthm}

%\swapnumbers

% \message{Loading skip-thm.tex!}

\newtheorem{thm}[equation]{Theorem}
\newtheorem{lem}[equation]{Lemma}
\newtheorem{cor}[equation]{Corollary}
\newtheorem{prop}[equation]{Proposition}
\newtheorem{conj}[equation]{Conjecture}
\newtheorem{MT}[equation]{Main Theorem}
\newtheorem{blank}[equation]{}

% the * form has no number attached to it
\newtheorem*{thm*}{Theorem}
\newtheorem*{prop*}{Proposition}
\newtheorem*{cor*}{Corollary}
\newtheorem*{lem*}{Lemma}
\newtheorem*{MT*}{Main Theorem}

% the slick way to get "Theorem A", "Theorem B", etc.
\newtheorem{specthm}{Theorem}
\renewcommand{\thespecthm}{\Alph{specthm}}

\newtheorem{speccor}[specthm]{Corollary}
\renewcommand{\thespeccor}{\Alph{specthm}}


\theoremstyle{definition} %
\newtheorem{defn}[equation]{Definition}
\newtheorem*{defn*}{Definition}
\newtheorem{ques}[equation]{Question}

\newtheorem{eg}[equation]{Example}
\newtheorem{egs}[equation]{Examples}
\newtheorem{app}[equation]{Application}
\newtheorem{prob}[equation]{Problem}
\newtheorem{notation}[equation]{Notation}
\newtheorem{obs}[equation]{Observation}
\newtheorem{fact}[equation]{Fact}

\theoremstyle{remark} %
\newtheorem{rmk}[equation]{Remark}
\newtheorem{rmks}[equation]{Remarks}
\newtheorem*{rmk*}{Remark}
\newtheorem*{rmks*}{Remarks}

%
% here we define the "exercise" environment, which goes by "exs"
%
\newtheoremstyle{exercise}% name
  {3pt}%      Space above
  {3pt}%      Space below
  {\small}%         Body font
  {}%         Indent amount (empty = no indent, \parindent = para indent)
  {\sc\small}% Thm head font
  {.}%        Punctuation after thm head
  {.5em}%     Space after thm head: " " = normal interword space;
   {}     %       \newline = linebreak
  {}%         Thm head spec (can be left empty, meaning `normal')


\theoremstyle{exercise}

\newtheorem{ex}[equation]{Exercise}

%\newtheoremstyle{steinberg}% name
%  {3pt}%      Space above
%  {3pt}%      Space below
%  {\it}%         Body font
%  {}%         Indent amount (empty = no indent, \parindent = para indent)
%  {\bf}% Thm head font
%  {.}%        Punctuation after thm head
%  {.5em}%     Space after thm head: " " = normal interword space;
%   {}     %       \newline = linebreak
%  {}%         Thm head spec (can be left empty, meaning `normal')
%  
%\theoremstyle{steinberg}
%\newtheorem{blank}[equation]{}  
  
%
% Make it so that we get
%     Theorem 3.1
%
%\numberwithin{equation}{section}

%
% \begin{speceqn}{tag}
% x^2 + 2\pi
% \end{speceqn}
%
% Gives you a displayed equation tagged with (tag)
%
\makeatletter
\newenvironment{speceqn}[1]%
{\renewcommand{\theequation}{#1}}%
{\renewcommand{\theequation}{\arabic{equation}}\addtocounter{equation}{-1}\global\@ignoretrue}
\makeatother

\makeatletter
\newenvironment{speceqnp}[1]%
{\renewcommand{\theequation}{#1}\begin{eqnarray}}%
{\end{eqnarray}\renewcommand{\theequation}{\arabic{equation}}\addtocounter{equation}{-1}\global\@ignoretrue}
\makeatother

% \begin{borel}{Stupid title}
% blah blah blah
% \end{borel}
%
% Gives you something like:
%   1.1.1.  Stupid title.  blah blah blah
%
\makeatletter
\newenvironment{borel}[1]%
{\smallskip \refstepcounter{equation}\noindent{\textbf{\theequation.} }{{\textbf{#1.}}}}%
{\smallskip \global\@ignoretrue}
\makeatother

% \begin{boreln}{Stupid title}
% blah blah blah
% \end{boreln}
%
% Gives you something like:
%   1.1.1.  Stupid title  blah blah blah
%
\makeatletter
\newenvironment{boreln}[1]%
{\smallskip \refstepcounter{equation}\noindent{\textbf{\theequation.} }{{\textbf{#1}}}}%
{\smallskip \global\@ignoretrue}
\makeatother

% \begin{borelp}{Stupid title}
% blah blah blah
% \end{borelp}
%
% Gives you something like:
%   1.1.1 (Stupid title).  blah blah blah
%
\makeatletter
\newenvironment{borelp}[1]%
{\smallskip \refstepcounter{equation}{\sc \theequation}{\sc (#1).}}%
{\smallskip \global\@ignoretrue}
\makeatother

% \begin{boreli}{Stupid title}
% blah blah blah
% \end{boreli}
%
% Gives you something like (stupid title in slant)
%   1.1.1. Stupid title.  blah blah blah
%
\makeatletter
\newenvironment{boreli}[1]%
{\smallskip \refstepcounter{equation}\noindent{\textbf\theequation.}{\textsl{ #1.}}}%
{\smallskip \global\@ignoretrue}
\makeatother

%
% The *-form omits the stupid title.
%
\makeatletter
\newenvironment{borel*}%
{\smallskip \refstepcounter{equation}\noindent{\textbf{\theequation.}}}%
{\global\@ignoretrue}
\makeatother

%
% This guy does the right thing to start a theorem with an enumerate
%
\newcommand{\flist}[1]{\hangindent\leftmargini\textup{(1)}\hskip\labelsep {#1}%
\begin{enumerate}%
\setcounter{enumi}{1}%
}




%\input qformsl-def

\renewcommand{\labelenumi}{(\theenumi)}
\renewcommand{\labelenumii}{(\theenumii)}

\newcommand{\ot}{\otimes}

\newcommand{\op}{\mathrm{op}}

\renewcommand{\P}{\mathbb{P}}

\DeclareMathOperator{\spn}{span}

\newcommand{\darkrad}{0.115}
\newcommand{\lrad}{0.25}

\setlength{\unitlength}{.4in}

\newcommand{\eand}{\quad\text{and}\quad}

\DeclareRobustCommand{\upstrut}[1]{\rule{0mm}{#1 ex}}
\DeclareRobustCommand{\dnstrut}[1]{\rule[-#1ex]{0mm}{#1 ex}}

\DeclareMathOperator{\Lie}{Lie}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Special characters
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\C}{{\mathbb{C}}}        % the complexes
\newcommand{\I}{{\sqrt{-1}}}      % the square root of -1
\newcommand{\Q}{{\mathbb{Q}}}        % the reationals
\newcommand{\R}{{\mathbb{R}}}        % the reals
\newcommand{\F}{{\mathbb{F}}}        % finite fields
\newcommand{\Z}{{\mathbb{Z}}}        % the integers
\newcommand{\N}{{\mathbb{N}}}      % the natural numbers 
\renewcommand{\H}{{\mathcal{H}}}  % the hyperbolic plane
\newcommand{\hyp}{\mathcal{H}}

\newcommand{\QZt}{\mathbb{Q}/\mathbb{Z}(2)}

\newcommand{\Zm}[1]{\Z/{#1}\Z}

\newcommand{\e}{\varepsilon}

\newcommand{\La}{\Lambda}
\newcommand{\la}{\lambda}
\newcommand{\D}{\Delta}

\newcommand{\G}{{\Gamma}}       % a Galois group

\newcommand{\proj}{{\mathbb{P}}}     % projective space
\newcommand{\aff}{{\mathbb{A}}}      % affine space

\newcommand{\oddots}{{\mathinner{\mkern1mu\raise1pt\vbox{\kern7pt\hbox{.}}\mkern2mu\raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Algebras and fields
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% The algebra (A, sigma)
%
\newcommand{\s}{\sigma}
\newcommand{\us}{\underline{\s}}
\newcommand{\As}{{(A,\sigma)}}
\newcommand{\Ass}{{(A,\sigma)_s}}
\newcommand{\AsL}{{(A_L,\sigma_L)}}
\newcommand{\AsK}{{(A_K,\sigma_K)}}

%
% The algebra (E, theta)
%
\newcommand{\Eth}{{(E,\theta)}}

%
% The algebra (B, tau)
%
\newcommand{\Bt}{{(B,\tau)}}

%
% The fields F, L, their multiplicative groups and their squares
%
\newcommand{\ksep}{k_{{\mathrm{sep}}}}
\newcommand{\Fsep}{F_{{\mathrm{sep}}}}

\newcommand{\Fx}{{F^{\times}}}
\newcommand{\Fxsq}{{F^{\times2}}}
\newcommand{\Fsx}{{F_s^{\times}}}
\newcommand{\Fsq}{{\Fx/\Fxsq}}

\newcommand{\Lx}{{L^{\times}}}
\newcommand{\Lxsq}{{L^{\times2}}}
\newcommand{\Lsx}{{L_s^{\times}}}
\newcommand{\Lsq}{{\Lx/\Lxsq}}

\newcommand{\Kx}{{K^{\times}}}
\newcommand{\Kxsq}{{K^{\times2}}}
\newcommand{\Ksx}{{K_s^{\times}}}
\newcommand{\Ksq}{{\Kx/\Kxsq}}

\newcommand{\supast}[1]{{{#1}^\times}}
\newcommand{\xsq}[1]{{{#1}^{\times2}}}
\newcommand{\sq}[1]{{\supast{#1}/\xsq{#1}}}

\newcommand{\kx}{k^\times}
\newcommand{\ksq}{\sq{k}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Quadratic forms
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\quat}[3]{{\fracwithdelims(){#1, #2}{#3}}}     % a quaternion algebra
\newcommand{\Fquat}[2]{{\quat{#1}{#2}{F}}}                   % over F

\newcommand{\bquat}[3]{{\fracwithdelims[]{#1, #2}{#3}}}    % the Brauer class
\newcommand{\Fbquat}[2]{{\bquat{#1}{#2}{F}}}                 % over F

\newcommand{\qform}[1]{{\left\langle{#1}\right\rangle}}                   % a quadratic form
\newcommand{\pform}[1]{{\langle\!\langle{#1}\rangle\!\rangle}} % a Pfister form
\newcommand{\qpform}[1]{{\langle\!\langle{#1}\rrbracket}}   % a characteristic 2 Pfister form

\newcommand{\IF}[1]{{I^{#1}\negthinspace F}}                   % the principal ideal
\newcommand{\IK}[1]{{I^{#1}\negthinspace K}}
\newcommand{\IL}[1]{{I^{#1}\negthinspace L}}
\newcommand{\Ifield}[2]{I^{#2}\negthinspace #1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Algebraic groups
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% oldbasemu:   \mu\hskip -.2truecm \mu
\newcommand{\basemu}{\boldsymbol{\mu}}
\newcommand{\mmu}[1]{\basemu_{#1}}     % the {#1}th roots of unity
\newcommand{\mmut}[1]{{\basemu^{\otimes 2}_{#1}}}     % the {#1}th roots of unity
\newcommand{\mmutrio}[1]{{\basemu^{\times 3}_{#1}}}     % the {#1}th roots of unity

\DeclareMathOperator{\Spin}{Spin}           % the Spin group
\newcommand{\Sp}{\mathrm{Sp}}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\PGL}{PGL}
\newcommand{\SO}{\mathrm{SO}}
\newcommand{\SU}{\mathrm{SU}}

\newcommand{\so}{\mathfrak{so}}

\DeclareMathOperator{\Ad}{Ad}

\newcommand{\Ga}{\mathbb{G}_a}
\newcommand{\Gm}{\mathbb{G}_m}

\newcommand{\oE}{\ensuremath{^1\!E_6}}
\newcommand{\dE}{\ensuremath{^2\!E_6}}

\newcommand{\iiiD}{^3\!D_4}
\newcommand{\viD}{^6\!D_4}
\newcommand{\oD}{^1\!D_4}
\newcommand{\dD}{^2\!D_4}
\newcommand{\iiD}{^2\!D_4}

\DeclareMathOperator{\Stab}{Stab}

\DeclareMathOperator{\Spec}{Spec}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Operators
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Gal}{Gal}  

\DeclareMathOperator{\im}{im}

\DeclareMathOperator{\coker}{coker}

\DeclareMathOperator{\trdeg}{trdeg}

%\newcommand{\sqplus}{\oplus}
%\newcommand{\semid}{\rtimes}

\DeclareMathOperator{\rank}{rank}
%\newcommand{\rankarg}[1]{\mathrm{rank}_{#1}\,}
%\newcommand{\rankA}{\rankarg{A}}
%\newcommand{\rk}{{\mathrm{rk}\,}}

%\newcommand{\Skew}{\mathrm{Skew}}

%\newcommand{\op}[1]{#1^\mathrm{op}}

% \newcommand{\ind}{{\mathrm{ind}\,}}
\DeclareMathOperator{\Int}{Int}
\DeclareMathOperator{\chr}{char}
\DeclareMathOperator{\disc}{disc}
\DeclareMathOperator{\Id}{Id}
\newcommand{\id}{\Id}

\DeclareMathOperator{\cd}{cd}
\DeclareMathOperator{\vcd}{vcd}

\DeclareMathOperator{\res}{res}
\DeclareMathOperator{\cores}{cor}

\DeclareMathOperator{\diag}{diag}

%
% Brauer groups
%
\DeclareMathOperator{\Br}{Br}

%
% Endomorphism rings
%
\DeclareMathOperator{\End}{End}

\DeclareMathOperator{\aut}{Aut}
\DeclareMathOperator{\Aut}{Aut}

\DeclareMathOperator{\Iso}{Iso}


\newcommand{\Hom}{{\mathrm{Hom}}}

\newcommand{\osum}{\oplus}
\newcommand{\lt}{<}
\newcommand{\gt}{>}

%
% Reduced norms
%
\newcommand{\Nrd}{{\mathrm{Nrd}}}
\newcommand{\NrdA}{{\mathrm{Nrd}_A}}
\newcommand{\NrdAs}{{\mathrm{Nrd}_{A_s}}}
\newcommand{\NrdB}{{\mathrm{Nrd}_B}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Arrows
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\longto}{\longrightarrow}
\newcommand{\iso}{\xrightarrow{\sim}}
\newcommand{\injects}{\hookrightarrow}

\newcommand{\ra}{\rightarrow}

\newcommand{\rat}{\dashrightarrow}

%
% logical operations
%
\newcommand{\Iff}{{\ \Longleftrightarrow\ }}
\newcommand{\Implies}{{\ \Longrightarrow\ }}
\newcommand{\If}{{\ \Longleftarrow\ }}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Jordan algebra stuff
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\tbtmat}[4]{\left( \begin{array}{cc} #1&#2 \\ #3&#4 \end{array} \right) }
\newcommand{\tcolvec}[2]{\left( \begin{array}{c} #1 \\ #2 \end{array} \right) }
\newcommand{\stcolvec}[2]{\left( \begin{smallmatrix} #1 \\ #2 \end{smallmatrix} \right) }
\newcommand{\stbtmat}[4]{\left( \begin{smallmatrix} #1&#2 \\ #3&#4 \end{smallmatrix} \right) }
\newcommand{\thbthmat}[9]{\left( \begin{array}{ccc} #1&#2&#3 \\ #4&#5&#6 \\ #7&#8&#9 \end{array} \right) }
\newcommand{\sthbthmat}[9]{\left( \begin{smallmatrix} #1&#2&#3 \\ #4&#5&#6 \\ #7&#8&#9 \end{smallmatrix} \right) }

\newcommand{\jordmat}[6]{\left( \begin{array}{ccc} #1&#6&\cdot \\ \cdot&#2&#4 \\ #5&\cdot&#3 \end{array} \right) }
\newcommand{\sjordmat}[6]{\left( \begin{smallmatrix} #1&#6&\cdot \\ \cdot&#2&#4 \\ #5&\cdot&#3 \end{smallmatrix} \right) }

\newcommand{\basjord}{\jordmat{\e_0}{\e_1}{\e_2}{c_0}{c_1}{c_2}}
\newcommand{\sbasjord}{\sjordmat{\e_0}{\e_1}{\e_2}{c_0}{c_1}{c_2}}

\newcommand{\sbasmat}{\stbtmat{\alpha}{j}{j'}{\beta}}
\newcommand{\basmat}{\tbtmat{\alpha}{j}{j'}{\beta}}




\renewcommand{\D}{\mathrm{D}}

\newcommand{\B}{\mathrm{B}}
\newcommand{\E}{\mathrm{E}}
\renewcommand{\F}{\mathrm{F}_4}
\renewcommand{\G}{\mathrm{G}_2}
%\newcommand{\SU}{\mathrm{SU}}
\newcommand{\U}{\mathrm{U}}

\renewcommand{\H}{\mathbb{H}}

\newcommand{\cE}{\E}

\newcommand{\SM}{\mathrm{SM}}
\newcommand{\toe}{\mathrm{ToE}}
\newcommand{\GUT}{\mathrm{GUT}}

\renewcommand{\s}{\mathfrak{s}}
\newcommand{\g}{\mathfrak{g}}
\newcommand{\z}{\mathfrak{z}}
\renewcommand{\sl}{\mathfrak{sl}}
\renewcommand{\sp}{\mathfrak{sp}}
\newcommand{\spin}{\mathfrak{spin}}

\newcommand{\h}{\mathfrak{h}}
\renewcommand{\e}{\mathfrak{e}}

\newcommand{\at}{\tilde{\alpha}}

\newcommand{\ach}{\alpha^\vee}
\newcommand{\bch}{\beta^\vee}
\newcommand{\Qch}{Q^\vee}

\newcommand{\mx}{{\mathrm{max}}}

\numberwithin{equation}{section}

\begin{document}

\title{There is no ``Theory of Everything" inside $\E_8$}
\author{Jacques Distler}
\address{(Distler) Theory Group, Department of Physics, and Texas Cosmology Center, University of Texas, Austin, TX 78712}
\email{distler@golem.ph.utexas.edu}
\thanks{Report \#: UTTG-05-09, TCC-020-09}

\author{Skip Garibaldi}
\address{(Garibaldi) Department of Mathematics \& Computer Science, 400 Dowman Dr., Emory University, Atlanta, GA 30322}
\email{skip@member.ams.org}

\subjclass[2000]{Primary 22E47; secondary 17B25, 81R05, 83E15}

%\thanks{{\tt Version of \today.}}

\begin{abstract}
We analyze certain subgroups of real and complex forms of the Lie group $\E_8$, and deduce that any ``Theory of Everything" obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of $\E_8$ lacks certain representation-theoretic properties required by physical reality.  The arguments themselves amount to representation theory of Lie algebras along the lines of Dynkin's classic papers and are written for mathematicians.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Recently, the preprint \cite{Lisi} by Garrett Lisi has generated a lot of popular interest. It boldly claims to be a sketch of a ``Theory of Everything", based on the idea of combining the local Lorentz group and the gauge group of the Standard Model in a real form of $\E_8$ (necessarily not the compact form, because it contains a group isogenous to $\SL(2, \C)$).  The purpose of this paper is to explain some reasons why an entire class of such models---which include the model in \cite{Lisi}---cannot work, using mostly mathematics with relatively little input from physics.

%Below, the even-numbered sections are purely mathematical and the odd-numbered sections have more physics content.  We first consider real forms of $\E_8$ in sections \ref{real.math.sec} and \ref{real.phys.sec} and then consider the complex $\E_8$ (as a real Lie group of dimension $2 \times 248$) in sections \ref{c.math.sec} and \ref{c.phys.sec}.

The mathematical set up is as follows.  Fix a real Lie group $\cE$.  We are interested in subgroups $\SL(2, \C)$ and $G$ of $\cE$ so that:
\[
\parbox{4in}{$G$ is connected, reductive, compact, and centralizes $\SL(2,\C)$} \tag{ToE1}
\]
We complexify and decompose $\Lie(\cE) \ot \C$ as a direct sum of representations of $\SL(2,\C)$ and $G$.
We identify $\SL(2, \C) \times \C$ with $\SL_{2,\C} \times \SL_{2,\C}$ and write
\begin{equation} \label{decomp}
\Lie(\cE) = \bigoplus_{m,n \ge 1} m \ot n \ot V_{m,n}
\end{equation}
where $m$ and $n$ denote the irreducible representation of $\SL_{2,\C}$ of that dimension and $V_{m,n}$ is a complex representation of $G \times \C$.  (Physicists would usually write $2$ and $\bar{2}$ instead of $2 \ot 1$ and $1 \ot 2$.)  Of course,
\[
\overline{m \ot n \ot V_{m,n}} \simeq n \ot m \ot \overline{V_{m,n}}
\]
and since the action of $\SL(2,\C).G$ on $\Lie(\cE)$ is defined over $\R$, we deduce that $\overline{V_{m,n}} \simeq V_{n,m}$.
We further demand that
\begin{gather*}
\parbox{4in}{$V_{2,1}$ is a complex representation of $G$, and} \tag{ToE2} \\
\parbox{4in}{$V_{m,n} = 0$ if $m + n > 4$.} \tag{ToE3}
\end{gather*}
We recall the definition of complex representation and explain the physical motivation for these hypotheses in the next section.  Roughly speaking, (ToE1) is a trivial requirement based on trying to construct a Theory of Everything along the lines suggested by Lisi, (ToE2) is the statement that the gauge theory (with gauge group $G$) is \emph{chiral}, as required by the Standard Model, and (ToE3) is the requirement that the model not contain any ``exotic" higher-spin particles. In fact, physics requires slightly stronger hypotheses on $V_{m,n}$, for $m+n=4$. We will not impose the stronger version of (ToE3).

\begin{defn} \label{toe.def}
A \emph{ToE subgroup} of a real Lie group $\cE$ is a subgroup generated by a copy of $\SL(2, \C)$ and a subgroup $G$ such that (ToE1), (ToE2), and (ToE3) hold.
\end{defn}

Our main result is:

\begin{thm} \label{MT}
There are no ToE subgroups in (the transfer of) the complex $\E_8$ nor in any real form of $\E_8$.
\end{thm}

\subsection*{Notation}
Unadorned Lie algebras and Lie groups mean ones over the real numbers.  We use a subscript $\C$ to denote complex Lie groups---e.g., $\SL_{2,\C}$ is the (complex) group of 2-by-2 complex matrices with determinant 1.  We can view a $d$-dimensional complex Lie group $G_\C$ as a $2d$-dimensional real Lie group, which we denote by $R(G_\C)$.  (Algebraists call this operation the ``transfer" or ``Weil restriction of scalars".)  We use the popular notation of $\SL(2, \C)$ for the transfer $R(\SL_{2,\C})$ of $\SL_{2, \C}$; it is a double covering of the ``restricted Lorentz group", i.e., of the identity component $\SO(3,1)_0$ of $\SO(3,1)$.

\begin{borel}{Strategy and main results}\label{mainresults}
Our strategy for proving Theorem \ref{MT} will be as follows. We will first catalogue, up to conjugation, all possible embedding of $\SL(2,\C)$ satifying the hypotheses of (ToE3). The list is remarkably short.  Specifically, every ToE subgroup of $\cE$ is contained in $\SL(2, \C)\cdot G_\mx$, where $G_\mx$ is the maximal compact, connected, reductive subgroup of the centralizer of $\SL(2, \C)$ in $\cE$.  The proof of Theorem \ref{MT} shows that the only possibilities are:
\[
\begin{array}{cc} 
\cE & G_\mx \\ \hline
\E_{8(8)}&\Spin(5)\times \Spin(7)\\
\E_{8(-24)}&\text{$\Spin(11)$ or $\Spin(9)\times \Spin(3)$} \\
R(\E_{8,\C})&\text{$\E_7$, $\Spin(13)$ or $\Spin(12)$.}
\end{array}
\]
We then note that the representation, $V_{2,1}$, of $G_{\text{max}}$ (and hence, of any $G\subseteq G_{\text{max}}$) has a self-conjugate structure. In other words, (ToE2) fails.
\end{borel}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Physics background} \label{phys.sec}

One of the central features of modern particle physics is that the world is described by a \emph{chiral gauge theory}.

\begin{borel*} \label{fermions}
Let $M$ a four dimensional pseudo-Riemannian manifold, of signature $(3,1)$, which we will take to be oriented, time-oriented and spin. Let $G$ be a compact Lie group. The data of a \emph{gauge theory on $M$, with gauge group $G$} consists of a connection, $A$, on a principal $G$-bundle, $P\to M$, and some ``matter fields" transforming as sections of vector bundle(s) associated to unitary representations of $G$.

Of particular interest are the \emph{fermions} of the theory. The orthonormal frame bundle of $M$ is a principal $\SO(3,1)_0$ bundle. A choice of spin structure defines a lift to a principal $\Spin(3,1)_0 = \SL(2,\mathbb{C})$ bundle.  Let $S_\pm \to M$ be the irreducible spinor bundles, associated, via the defining two-dimensional representation and its complex conjugate, to this $\SL(2,\mathbb{C})$ principal bundle.

The \emph{fermions of our gauge theory} are denoted
\[
  \psi \in \Gamma(S_+\otimes V),\qquad \overline{\psi} \in \Gamma(S_-\otimes \overline{V})
\]
where $V\to M$ is a vector bundle associated to a (typically reducible) representation, $R$, of $G$.
\end{borel*}

\begin{defn}\label{physreps}
A \emph{real structure} on a representation $V$ (over $\mathbb{C}$) is an antilinear map, $J: V\to V$, satisfying $J^2 =1$. Physicists call a representation possessing a real structure \emph{real}.  

A \emph{quaternionic structure} on a representation $V$ (over $\mathbb{C}$) is an antilinear map, $J: V\to V$, satisfying $J^2 = -1$. Physicists call a representation possessing a quaternionic structure \emph{pseudoreal}.  

Subsuming these two subcases, we will say that a representation $V$ (over $\mathbb{C}$) has a \emph{self-conjugate structure} if  there is an antilinear map $J: V\to V$, satisfying $J^4 =1$.
Physicists call a representation $V$, which does not possess a self-conjugate structure, \emph{complex}.

For an alternative view, we suppose that $V$ is an irreducible representation (over $\C$) of a real reductive Lie group $G$ and exploit \cite[\S7]{Ti:R}.  If $V$ has a real structure $J$, then the subset $V'$ of elements of $V$ fixed by $J$ is a real vector space that is a representation of $G$ such that $\End_G(V') = \R$ and $V' \ot \C$ is canonically identified with $V$.  A quaternionic structure on $V$ defines a real structure on $V \oplus V$ via $(v_1, v_2) \mapsto (Jv_2, -Jv_1)$ such that $(V \oplus V)'$ is irreducible with $\End_G((V \oplus V)') \cong \H$.  If $V$ is complex, 
then $V \oplus \overline{V}$ has an essentially unique real structure and $\End_G((V \oplus \overline{V})') \cong \C$. 
\end{defn}

\begin{defn}\label{chiral}
A gauge theory, with gauge group $G$, is said to be \emph{chiral} if the representation, $R$ by which the fermions \eqref{fermions} are defined, is complex in the above sense. By contrast, a gauge theory is said to be \emph{nonchiral} if the representation $R$ in \ref{fermions} has a self-conjugate structure.

Note that whether a gauge theory is chiral depends crucially on the choice of $G$. A gauge theory might be chiral for gauge group $G$, but \emph{nonchiral} for a subgroup $H\subset G$, because there exists a self-conjugate structure on $R$, compatible with $H$, even though no such structure exists, compatible with the full group $G$.

Conversely, suppose that a gauge theory is nonchiral for the gauge group $G$. It is also necessarily nonchiral for any gauge group $H\subset G$.
\end{defn}

\begin{borel}{GUTs}
The Standard Model is a chiral gauge theory with gauge group 
\[
G_\SM := (\SU(3) \times \SU(2) \times \U(1)) / (\Zm6)
\]
Various grand unified theories (GUTs) proceed by embedding $G_\SM$ is some (usually simple) group, $G_\GUT$. Popular choices for $G_\GUT$ are  $\SU(5)$ \cite{GeorgiGlashow}, $\Spin(10)$, $\E_6$, and the Pati-Salam group, $(\Spin(6)\times \Spin(4))/ (\Zm2)$ \cite{PatiSalam}.

It is easiest to explain what the fermion representation of $G_\SM$ is after embedding $G_\SM$ in $G_\GUT := \SU(5)$. Let $W$ be the five-dimensional defining representation of $\SU(5)$. The representation $R$ from \ref{fermions} is the direct sum of three copies of
\[
   R_0=\wedge^2 W \oplus \overline{W}
\]
Each such copy is called a ``generation" and is 15-dimensional.  One identifies each of the 15 weights of $R_0$ with left-handed fermions: 6 quarks (two in a doublet, each in three colors), two leptons (e.g., electron and its neutrino), 6 antiquarks, and a positron.  With three generations, $R$ is 45-dimensional.

For the other choices of GUT group, the analogue of a generation ($R_0$) is higher-dimensional, containing additional fermion which are not seen at low energies. When decomposed under $G_\SM\subset G_\GUT$, the representation decomposes as $R_0 + R'$, where $R'$ is a real representation of $G_\SM$. In $\Spin(10)$, a generation is the 16-dimensional half-spinor representation. In $\E_6$, it is the 27, and for the Pati-Salam group, it is the $(4,1,2)\oplus (\overline{4},2,1)$ representation. In each case, these representations are complex representations (in the above sense) of $G_\GUT$, and the complex-conjugate representation is called an ``anti-generation."
\end{borel}

So far, we have described a chiral gauge theory in a fixed (pseudo) Riemannian structure on $M$. Lisi's proposal \cite{Lisi} is to try to combine the spin connection on $M$, and the gauge connection on $P$ into a single dynamical framework.  This motivates Definition \ref{toe.def} of a ToE subgroup.

Fix a ToE subgroup---say, with $G = G_\SM$---in some real Lie group $\cE$. The action of central element $-1 \in \SL(2, \C)$ provides a $\mathbb{Z}/2\mathbb{Z}$ grading on the Lie algebra of $\cE$. This $\mathbb{Z}/2\mathbb{Z}$ grading allows one to define a sort of superconnection associated to $\cE$ (precisely what sort of superconnection is explained in a blog post by the first author \cite{Distler:SuperConnections}.)  In the proposal of \cite{Lisi}, we are supposed to identify each of the generators of $\Lie(\cE)$ as either a boson or a fermion. The Spin-Statistics Theorem \cite{StreaterWightman:PCT} says that fermions transform as spinorial representations of $\Spin(3,1)$; bosons transform as ``tensorial" representations (representation which lift to the double cover, $\SO(3,1)$). To be consistent with the Spin-Statistics Theorem, we must, therefore, require that the fermions belong to the $-1$-eigenspace of the aforementioned $\mathbb{Z}/2\mathbb{Z}$ action, and the bosons to the $+1$-eigenspace.

In fact, to agree with \ref{fermions}, we should require that the $-1$ eigenspace (when tensored with $\mathbb{C}$) decompose as a direct sum of two-dimensional representations (over $\C$) of $\SL(2,\C)$, corresponding to ``left-handed" and ``right-handed" fermions, in the sense of \ref{fermions}.

\begin{rmk}
In the language of (ToE3), $m+n=\text{odd}$ are fermions and $m+n=\text{even}$ are bosons. In Lisi's setup, the bosons are 1-forms on $M$, with values in a vector bundle associated to the aforementioned $\Spin(3,1)_0$ principal bundle via the $m\otimes n$ representation (with $m+n$ even). The case $m+n=4$ is special; these correspond to the gravitational degrees of freedom in Lisi's theory. $(3\otimes 1) \oplus (1\otimes 3)$ is the adjoint representation; these correspond to the spin connection. The 1-form with values in the $2\otimes 2$ representation is the vierbein. It is a serious result from physics (see sections 13.1, 25.4 of \cite{Weinberg:QFT}) that a unitary interacting theory is incompatible with massless particles in higher representation ($m + n\ge 6$). But, in light of the difficulties in making physical sense of the bosonic sector of Lisi's theory, it would be cleaner --- meaning demanding less input from physics--- to focus on the fermionic sector and forbid the presence of gravitinos ($m+n=5$) or yet-higher spin fermionic fields. (ToE3), as stated, forbids both. In \S\ref{supergravity}, we will revisit the possibility of admitting gravitinos.
\end{rmk}


\begin{borel}{Dimension considerations}  \label{dim}
Elaborating on the discussion above, in a Theory of Everything one wishes to identify weight vectors in $V_{2,1}$ and $V_{1,2}$ with left- and right-handed fermions.  As there are $3\times 15= 45$ known fermions of each chirality, we find that the $-1$-eigenspace must have dimension at least $2 \times 2 \times 45 = 180$.

In case $\cE$ is a real form of $\E_8$, the $-1$-eigenspace has dimension 112 or 128 (this is implicit in Elie Cartan's classification of real forms of $\E_8$ as in \cite[p.~518, Table V]{Helgason}),\footnote{Alternatively, the marvelous bound on the trace from \cite[Th.~3]{Se:trace} implies that for every element $x$ of order 2 in a reductive complex Lie group $G$, the $-1$-eigenspace of $\mathrm{Ad}(x)$ has dimension $\le (\dim G + \rank G) / 2$.  In particular, when $G$ is a real form of $\E_8$, the $-1$-eigenspace has dimension $\le 128$.} so no identification of the fermions as distinct weight vectors in $\Lie(\cE)$ (as in Table 9 in \cite{Lisi}) can be compatible with the Spin-Statistics Theorem and the existence of three generations.
\end{borel}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\sl_2$ subalgebras and the Dynkin index}

\begin{borel*}
In \cite[\S2]{Dynk:ssub}, Dynkin defined the \emph{index} of an inclusion $f \!: \g_1 \injects \g_2$ of simple complex Lie algebras as follows.  Fix a Chevalley basis of the two algebras, so that the Cartan $\h_1$ of $\g_1$ is contained in the Cartan $\h_2$ of $\g_2$.  The Chevalley basis identifies $\h_i$ with the complexification $\Qch_i \ot \C$ of the coroot lattice $\Qch_i$ of $\g_i$, and the inclusion $f$ gives an inclusion $\Qch_1 \ot \C \injects \Qch_2 \ot \C$.  Fix the Weyl-invariant inner product $(\,,\,)_i$ on $\Qch_i$ so that $(\ach, \ach)_i = 2$ for short coroots $\ach$.  Then the \emph{Dynkin index} of the inclusion is the ratio $(f(\ach), f(\ach))_2 / (\ach, \ach)_1$ where $\ach$ is a short coroot of $\g_1$.  For example, the irreducible representation $\sl_2 \ra \sl_n$ has index $\binom{n+1}{3}$ by \cite[Eq.~(2.32)]{Dynk:ssub}. 
\end{borel*}

\begin{borel*} \label{sl2.h}
We now consider the case where $\g_1 = \sl_2$ and write simply $\g$ and $\Qch$ for $\g_2$ and $Q_2^\vee$.  In \S8 of that same paper (or see \cite[\S{VIII.11}]{Bou:g7}), Dynkin proved that after conjugating by an element of the automorphism group of $\g$, one can assume that the Cartan subalgebra of $\sl_2$ is contained in the given Cartan subalgebra of $\g$ and that the image $h$ of a simple root of $\sl_2$ in $\Qch \ot \C$ satisfies the strong restrictions:
\[
h = \sum_{\delta \in \Delta} p_\delta \delta^\vee \quad \text{for $p_\delta$ real and non-negative \cite[Lemma 8.3]{Dynk:ssub},}
\]
where $\Delta$ denotes the set of simple roots of $\g$ and further that
\[
\delta(h) \in \{ 0, 1, 2 \} \quad \text{for all $\delta \in \Delta$.}
\]
But note that for each simple root $\delta$, the fundamental irreducible representation of $\g$ with highest weight dual to $\delta$ has weight $p_\delta$ (as a representation of $\sl_2$), hence \emph{$p_\delta$ is an integer.} 

As a consequence of this and specifically \cite[Lemma 8.2]{Dynk:ssub}, one can identify an $\sl_2$-subalgebra of $\g$ up to conjugacy by writing the Dynkin diagram of $\g$ and putting the number $\delta(h)$ at each vertex; this is the \emph{marked Dynkin diagram} of the $\sl_2$ subalgebra.

Here is an alternative formula for computing the index of an $\sl_2$ subalgebra from  its marked Dynkin diagram.  Write $\kappa_\g$ and $m^\vee$ for the Killing form and dual Coxeter number of $\g$.  We have:
\begin{equation} \label{index.alt}
(\text{Dynkin index}) = \frac12 (h, h) = \frac{1}{4m^\vee} \kappa_\g(h, h) = \frac{1}{2m^\vee} \sum_{\text{positive roots $\alpha$ of $\g$}} \alpha(h)^2,
\end{equation}
where the second equality is by, e.g., \cite[\S5]{GrossNebe}, and the third is by the definition of $\kappa_\g$.  One can calculate the number $\alpha(h)$ by writing $\alpha$ as a sum of positive roots and applying the marked Dynkin diagram for $h$.
\end{borel*}

\begin{lem} \label{conj1}
For every simple complex Lie algebra $\g$, there is a unique copy of $\sl_2$ in $\g$ of index $1$, up to conjugacy.
\end{lem}

\begin{proof}
The index of an $\sl_2$-subalgebra is $(h,h)/2$, where the defining vector $h$ belongs to the coroot lattice $\Qch$.  If $\g$ is not of type B, then the coroot lattice is not of type C, and the claim amounts to the statement that the 
vectors of minimal length in the coroot lattice are actually roots.  This follows from the constructions of the root lattices in \cite[\S12.1]{Hum:LA}. 

Otherwise $\g$ has type B and is $\so_n$ for some odd $n \ge 5$.  The conjugacy class of an $\sl_2$-subalgebra is determined by the restriction of the natural $n$-dimensional representation; they are parameterized by partitions of $n$ (i.e., $\sum n_i = n$) so that $n_1 \ge \cdots \ge n_s > 0$, $n_1 > 1$, and there are an even number of even $n_i$'s  \cite[\S6.2.2]{OV3}.  The index of the composition $\sl_2 \ra \so_n \ra \sl_n$ is then $\sum \binom{n_i+1}{3}$; we must classify those partitions such that this sum equals the Dynkin index of $\so_n \ra \sl_n$, which is 2.  The unique such partition is $2 \ge 2 \ge 1 \ge \cdots \ge 1 > 0$.
\end{proof}

If $\g$ has type C, $\F$, or $\G$, then the same argument shows that there is up to conjugacy a unique copy of $\sl_2$ in $\g$ with index 2, 2, or 3 respectively.  We will see below that for $\g$ of type $\B_6$, there are two conjugacy classes of $\sl_2$-subalgebras of index 2.  This amounts to the fact that there are vectors in the $\mathrm{C}_6$ root lattice that are not roots but have the same length as a root, cf.~Exercise 5 in \S12 of \cite{Hum:LA}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Copies of $\sl_{2,\C}$ in the complex $\E_8$} \label{sl2.e8.sec}

We first prove some facts about copies of $\sl_{2,\C}$ in the complex Lie algebra $\e_8$ of type $\E_8$.  The 69 conjugacy classes of such are listed in Table 20 on p.~182 of \cite{Dynk:ssub}; we are interested in just a few of these, without using Dynkin's table.

Fix a pinning for $\e_8$; this includes a maximal toral subalgebra $\h$, a set of simple roots $\D := \{ \alpha_i \mid 1 \le i \le 8 \}$ (numbered 
\begin{equation} \label{e8.num}
\begin{tabular}{ccccccc}
1&3&4&5&6&7&8\\
&&2
\end{tabular}
\end{equation}
as in \cite{Bou:g4}), and fundamental weights $\omega_i$ dual to $\alpha_i$. As all roots of the $\E_8$ root system have the same length, we can and do identify the root system with its inverse root system.

\begin{eg} \label{sl2.eg1}
Taking any root of $\E_8$, one can generate a copy of $\sl_{2,\C}$ in $\e_8$ with index 1.  Doing this with the highest root gives an $\sl_{2,\C}$ with marked Dynkin diagram
\[
\begin{tabular}{cc}
index 1:&
\begin{tabular}{ccccccc}
0&0&0&0&0&0&1\\
&&0
\end{tabular} 
\end{tabular}
\]
Every index 1 copy of $\sl_2$ in $\e_8$ is conjugate to this one by Lemma \ref{conj1}.
\end{eg}

\begin{eg} \label{sl2.eg2}
One can find a copy of $\sl_{2,\C} \times \sl_{2,\C}$ in $\e_8$ by taking the  first copy to be generated by the highest root of $\E_8$ and the second copy to be generated by the highest root of the obvious $\E_7$ subsystem.  If you embed $\sl_{2,\C}$ diagonally in this algebra, you find a copy of $\sl_{2,\C}$ with index 2 and marked Dynkin diagram
\[
\begin{tabular}{cc}
index 2:&
\begin{tabular}{ccccccc}
1&0&0&0&0&0&0\\
&&0
\end{tabular}
\end{tabular}
\]
\end{eg}

\begin{prop} \label{sl2}
The following collections of copies of $\sl_{2,\C}$ in $\e_8$ are the same:
\begin{enumerate}
\item copies such that $\pm 1$ are weights of $\e_8$ (as a representation of $\sl_{2,\C}$) and no other odd weights occur.
\item copies such that every weight of $\e_8$ is in $\{ 0, \pm 1, \pm 2 \}$.
\item copies such that the inclusion $\sl_{2,\C} \subset \e_8$ has Dynkin index $1$ or $2$.
\item copies of $\sl_{2,\C}$ conjugate to one of those defined in Examples \ref{sl2.eg1} or \ref{sl2.eg2}.
\end{enumerate}
\end{prop}

The equality of (3) and (4) says that $\sl_{2,\C}$'s of index 1 or 2 are determined (up to conjugacy) by their index.  This is a small part of the data contained in Dynkin's Table 20, which is not used in our proof.

\begin{proof}[Proof of Proposition \ref{sl2}]
One easily checks that (4) is contained in (1)--(3); we prove the opposite inclusion.

For (3), we identify $\h$ with the complexification $Q \ot \C$ of the (co)root lattice $Q$, hence $h$ with $\sum \alpha_i(h) \omega_i$.  By \eqref{index.alt}, the index of $h$ satisfies:
\[
\frac{1}{60} \sum_\alpha \alpha(h)^2 = \frac{1}{60} \sum_\alpha \left(\sum_i \alpha_i(h) \qform{\omega_i, \alpha}\right)^2 \ge \sum_i \left( \alpha_i(h)^2 \sum_\alpha \frac{\qform{\omega_i, \alpha}^2}{60} \right)
\]
where the sums vary over the positive roots.
We calculate for each fundamental weight $\omega_i$ the number $\sum_\alpha \qform{\omega_i, \alpha}^2/60$:
\begin{equation} \label{e8.indexes}
\begin{tabular}{rrrrrrr}
2&7&15&10&6&3&1 \\
&&4
\end{tabular}
\end{equation}
As the numbers $\alpha_i(h)$ are all 0, 1, or 2, the numbers \eqref{e8.indexes} show that $h$ for an $\sl_{2,\C}$ with Dynkin index 1 or 2 must be $\omega_1$ (index 2) or $\omega_8$ (index 1).

For (2), the highest root $\at$ of $\E_8$ is $\at = \sum_i c_i \alpha_i$, where $c_1 = c_8 = 2$ and the other $c_i$'s are all at least 3.  As $\at(h)$ is a weight of $\e_8$ relative to a given copy of $\sl_{2,\C}$, we deduce that an $\sl_{2,\C}$ as in (2) must have $h = \omega_1$ or $\omega_8$, as claimed.

\medskip
Suppose now that we are given an $h$ for a copy of $\sl_{2,\C}$ as in (1).  As $\pm 1$ occur as weights, there is at least one 1 in the marked Dynkin diagram.

But note that there cannot be three or more 1's in the marked Dynkin diagram for $h$.  Indeed, for every connected subset $S$ of vertices of the Dynkin diagram of $\E_8$, $\sum_{i \in S} \alpha_i$ is a root \cite[\S{VI.1.6}, Cor.~3b]{Bou:g4}.  If the number of 1's in the marked diagram of $h$ is at least three, then one can pick $S$ so that it meets exactly three of the $\alpha_i$'s with $\alpha_i(h) = 1$, in which case $\sum_{i \in S} \alpha_i(h)$ is odd and at least 3, violating the hypothesis of (1).

For sake of contradiction, suppose that there are two 1's in the marked diagram for $h$.  If $\alpha_8(h)$ and $\alpha_i(h)$ are the two 1's, then one can find a root $\alpha$ of $\E_8$ where $\qform{\omega_i + \omega_8, \alpha}$ is odd and at least 3, i.e., the coefficients of $\alpha_i$ and $\alpha_8$ in $\alpha$ have opposite parities and sum at least 3.  This is a contradiction, so we may assume that both 1's in the marked Dynkin diagram lie in the $\E_7$ subdiagram.  Repeating this argument with smaller and smaller subsystems shows the contradiction.

We are left with the case the there is exactly one 1 in the marked diagram for $h$.  If $\alpha_i(h) = 1$ for some $i \ne 1, 8$, then we find a contradiction because there is a root $\alpha$ of $\E_8$ with $\alpha_i$-coordinate 3.  Therefore $\alpha_i(h) = 1$ only for $i = 1$ or 8 and not for both.  By the fact used two paragraphs above, $\beta := \sum_i \alpha_i$ is a root of $\E_8$, so $\beta(h) = \sum \alpha_i(h)$ is odd and must be 1.  It follows that $h = \omega_1$ or $\omega_8$.
\end{proof}


\begin{borel}{Centralizer for index 1} \label{eg1.cent}
The $\sl_{2,\C}$ of index 1 in $\e_8$ has centralizer the regular subalgebra $\e_7$ of type $\E_7$.  Indeed, it is clear that $\e_7$ centralizes this $\sl_{2,\C}$, and conversely $\e_7 \times \sl_{2,\C}$ is a maximal proper subalgebra of $\e_8$, so the centralizer cannot be any larger.
\end{borel}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Index 2 copies of $\sl_{2,\C}$ in the complex $\E_8$} \label{ind2.sec}

\begin{lem} \label{eg2.cent}
The centralizer of the index 2 $\sl_{2,\C}$ in $\e_8$ from Example \ref{sl2.eg2}  is a copy of $\so_{13}$ contained in the regular subalgebra $\so_{14}$ of $\e_8$.
\end{lem}

\begin{proof}
The centralizer of the $\sl_{2,\C}$ of index 2 in $\e_8$ is contained in the centralizer of the defining vector $h$; this centralizer is reductive with semisimple part the regular subalgebra $\so_{14}$ of type $\D_7$.  The centralizer of $\sl_{2,\C}$ contains the centralizer of the $\sl_{2,\C} \times \sl_{2,\C}$ from Example \ref{sl2.eg2}, which is the regular subalgebra $\so_{12}$ of type $\D_6$.  Furthermore, using the standard representation theory of $\sl_{2,\C}$, it is easy to see that that the the centralizer of $\sl_{2,\C}$ has dimension 78 (as is implicitly claimed in the statement of the lemma).

The regular $\so_{14}$ does not centralize this $\sl_{2,\C}$.  On the other hand, the centralizer is strictly bigger than $\so_{12}$ because it includes a copy of $\sl_{2,\C}$ obtained by embedding diagonally in the $\sl_{2,\C} \times \sl_{2,\C}$ generated by nilpotents corresponding to the highest root of the $\D_7$ subsystem and $-\alpha_8$.  (To check that this $\sl_{2,\C}$ centralizes the index 2 copy we started with, it is helpful to have a table of structure constants as in \cite{Vavilov:const}.)  Together with $\so_{12}$, this generates an $\so_{13}$ subalgebra of $\so_{14}$.  Dimension count gives the claim.
\end{proof}

\begin{borel*} \label{so13}
Implicit in the proof above is an inclusion of $\so_{13}$ in $\e_8$ and a comparison of the pinnings of the two algebras, and in particular an inclusion of coroot lattices in terms of those pinnings.  Number the simple roots of $\so_{13}$ according to the diagram
\[
\begin{picture}(7,0.5)
    % put in the horizontal lines
    \put(0.25,0.10){\line(1,0){4}} %multiput(1,1)(1,0){2}{\line(1,0){1}}
    \put(4.25,0.13){\line(1,0){1}}
    \put(4.25,0.05){\line(1,0){1}}
 
    % label horizontal circles
    \put(0.25,0.27){\makebox(0,0.4)[b]{$1$}}
    \put(1.25,0.27){\makebox(0,0.4)[b]{$2$}}
    \put(2.25,0.27){\makebox(0,0.4)[b]{$3$}}
    \put(3.25,0.27){\makebox(0,0.4)[b]{$4$}}
    \put(4.25,0.27){\makebox(0,0.4)[b]{$5$}}
    \put(5.25, 0.27){\makebox(0,0.4)[b]{$6$}}
    
    \put(4.75,0){\makebox(0,0.4)[b]{$>$}}

    % put in the horizontal circles
    \multiput(0.25,0.1)(1,0){6}{\circle*{\darkrad}}

%    % put in the diagonal circles
%    \put(5.7,0.3){\circle{0.2}}
%    \put(5.7,1.7){\circle{0.2}}
%    \put(5.7,-0.1){\makebox(0,0.4)[b]{$\ell^{-}$}}
%    \put(5.7,1.9){\makebox(0,0.4)[b]{$\ell^{+}$}}
\end{picture}
\]
We write $\bch_i$ for the simple coroot corresponding to the simple root $i$.  Here is a translation table between the coroots of $\so_{13}$ and the (co)roots of $\e_8$:
\[
\begin{array}{c|cccccc}
\so_{13}&\bch_1&\bch_2&\bch_3&\bch_4&\bch_5&\bch_6 \\ \hline
\e_8&\alpha_3&\alpha_4&\alpha_5&\alpha_6&\alpha_7&\alpha_2 + \alpha_3 + 2\alpha_4 + 2\alpha_5 + 2\alpha_6 + 2\alpha_7
\end{array}
\]

The index 2 $\sl_{2,\C}$ and the copy of $\so_{13,\C}$ give an $\sl_{2,\C} \times \so_{13,\C}$ subalgebra of $\e_{8,\C}$, and as a representation, $\e_{8,\C}$ decomposes  as a direct sum of irreducibles:
\[
1 \ot \so_{13,\C} \quad \oplus \quad 2 \ot (\text{spin}) \quad \oplus \quad 3 \ot 1 \quad \oplus \quad 3 \ot (\text{vector})
\]
\end{borel*}

\begin{eg} \label{sl2.eg2p}
By Dynkin's game of adding the highest root to the Dynkin diagram and deleting a vertex as in \cite[\S5]{Dynk:ssub}, $\so_{13,\C}$ contains a maximal subalgebra $\so_{8,\C} \times \sp_{4,\C}$.  In turn, $\so_8$ contains a maximal subalgebra $\sl_2 \times \sp_4$ \cite[Th.~1.4]{Dynk:max}.  This gives an $\sl_2 \times \sp_4 \times \sp_4$ subalgebra of $\so_{13}$.

We remark that this $\sl_2$ has index 2 in $\e_8$.  As the inclusions $\so_8 \subset \so_{13} \subset \e_8$ have index 1, it suffices to check that $\sl_2$ has index 2 in $\so_8$.  This follows from the fact that the adjoint representation of $\so_8$ has index 12, whereas its restriction to $\sl_2$ decomposes as six copies of the 3-dimensional irreducible representation and a 10-dimensional trivial representation \cite[p.~260]{McKP}, so has index $6 \cdot 4 + 10 \cdot 0 = 24$.
\end{eg}

The main result of this section is the following:
\begin{prop} \label{ind22}
Up to conjugacy, there is a unique copy of $\SL_{2,\C} \times \SL_{2,\C}$ in $\E_{8,\C}$ so that each inclusion of $\SL_{2,\C}$ in $\E_{8,\C}$ has index 2.  The Lie algebra of the centralizer of this $\SL_{2,\C} \times \SL_{2,\C}$ is an $\sp_{4,\C} \times \sp_{4,\C}$ subalgebra of $\so_{13,\C}$ in $\e_8$.
\end{prop}

\begin{proof}
An analysis similar to the one in the proof of Proposition \ref{sl2}---but more complicated because there is more than one root length and $\Spin_{13}$ has a nontrivial center---shows that there are two conjugacy classes of copies of $\sl_2$ in $\so_{13}$ of index 2, corresponding to marked Dynkin diagrams
\[
\text{(a)}
\quad
\begin{tabular}{cccccc}
2&0&0&0&0&0
\end{tabular}
\quad
\text{and}
\quad
\text{(b)}
\quad
\begin{tabular}{cccccc}
0&0&0&1&0&0
\end{tabular}
\]
We can pair each of (a) and (b) with the copy of $\sl_2$ from Example \ref{sl2.eg2} to get an $\sl_2 \times \sl_2$ subalgebra of $\e_8$ where both $\sl_2$'s have index 2.  Clearly, these represent the only two $\E_8$-conjugacy classes of such subalgebras.  With the marked Dynkin diagram in hand, it is not difficult to calculate the decomposition of $\e_8$ into a direct sum of irreducible representations of $\sl_2 \times \sl_2$.

In case (a), every irreducible summand $m \ot n$ has $m + n$ even.  Therefore, this copy of $\sl_2 \times \sl_2$ is the Lie algebra of a subgroup of $\E_8$ isomorphic to $(\SL_2 \times \SL_2)/(-1, -1)$.

In case (b), we have the following table of multiplicities for $m \ot n$:
\begin{equation} \label{22.table}
\begin{array}{cc|rrrc} 
&&1&2&3&m\\ \hline
&1&20&20&6 \\
n&2&20&16&4 \\
&3&6&4&0 
\end{array}
\end{equation}
In particular, it is the Lie algebra of a copy of $\SL_2 \times \SL_2$ in $\E_8$.  Combining the $\sl_2$'s from Examples \ref{sl2.eg2} and \ref{sl2.eg2p} gives an $\sl_2 \times \sl_2$ subalgebra with both $\sl_2$'s of index 2 and the same decomposition of $\e_8$ into irreducible representations, so it is in the same conjugacy class.  Clearly, its centralizer contains the copy of $\sp_4 \times \sp_4$ from Example \ref{sl2.eg2p}, and the number 20 in the upper-left corner of \eqref{22.table} shows that $\sp_4 \times \sp_4$ is the whole centralizer.
\end{proof}

\begin{borel*} \label{decomp.22}
We can decompose $\e_8$ into a direct sum of irreducible representations of the $\sl_2 \times \sl_2 \times \sp_4 \times \sp_4$ subalgebra from the proposition by combining 
the decomposition of $\e_8$ into irreducible representations of $\sl_2 \times \so_{13}$ from \ref{so13} with the tables in \cite{McKP}.  Recall that $\sp_4$ has two fundamental irreducible representations: one that is 4-dimensional symplectic and another that is 5-dimensional orthogonal; we denote them by their dimensions.  With this notation and \ref{decomp}, we find:
\[
V_{2,1} = 5 \ot 4, \quad V_{1,2} = 4 \ot 5, \quad V_{2,3} = 1 \ot 4, \eand V_{3,2} = 4 \ot 1.
\]
\end{borel*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Copies of $\SL(2, \C)$ in a real form of $\E_8$}

Suppose now that we have a copy of $\SL(2, \C)$ inside a real Lie group $\cE$ of type $\E_8$.  Over the complex numbers, we decompose $\Lie(\cE) \ot \C$ into a direct sum of irreducible representations of $\SL(2, \C) \times \C \cong \SL_{2,\C} \times \SL_{2,\C}$; each irreducible representation can be written as $m \ot n$ where $m$ and $n$ denote the dimension of an irreducible representation of the first or second $\SL_{2,\C}$ respectively.  The goal of this section is to prove:

\begin{prop} \label{real.z}
Maintain the notation of the previous paragraph.  If $\Lie(\cE)$ contains no irreducible summands $m \ot n$ with $m+n > 4$, then the identity component of the centralizer of $\SL(2, \C)$ in $\E$ is a
\begin{enumerate}
\item a regular subgroup $\Spin(7,5)$ if $\cE$ is split; or 
\item a regular subgroup $\Spin(9,3)$ or $\Spin(11, 1)$ if the Killing form of $\Lie(\cE)$ has signature $-24$.
\end{enumerate}
\end{prop}

\begin{proof}
Complexifying the inclusion of $\SL(2, \C)$ in $\cE$ and going to Lie algebras gives an inclusion of $\sl_{2,\C} \times \sl_{2,\C}$ in the complex Lie algebra $\e_8$ from \S\ref{sl2.e8.sec}.  The hypothesis on the irreducible summands $m \ot n$ amounts to the statement that each of the two $\sl_{2,\C}$'s has index 1 or 2 by Proposition \ref{sl2}.  As complex conjugation interchanges the two components, they must have the same index.  

Suppose first that both $\sl_2$'s have index 1.  Lemma \ref{conj1} (twice) gives that this $\sl_2 \times \sl_2$ is conjugate to the one generated by the highest root of $\E_8$ from Example \ref{sl2.eg1} (so the second $\sl_2$ belongs to the centralizer of type $\E_7$) and by the highest root of the $\E_7$ subsystem and makes up the first two summands of an $\sl_2 \times \sl_2 \times \so_{12}$ subalgebra as in \cite[pp.~147, 148]{Dynk:ssub}.  As this subalgebra has rank 8, it follows that the Lie algebra of the centralizer $\z$ of $\SL(2, \C)$ in $\E$ is a real form of $\spin_{12}$.

We can decompose $\Lie(\cE) \ot \C$ into irreducible representations of $(\sl_2 \times \sl_2 \times \spin_{12}) \ot \C$ using the tables in \cite{McKP} to find the adjoint representation plus
\[
2 \ot 1 \ot  S_+,\quad 1 \ot 2 \ot S_-, \quad \text{and} \quad 2 \ot 2 \ot V,
\]
where $S_\pm$ denotes the half-spin representations of $\spin_{12}$ and $V$ is the vector representation.  But $\Lie(\cE)$ is a real representation of $\z$, so we deduce that $V$ is also a real representation of $\z$ but $S_+$ and $S_-$ are not; they are interchanged by the Galois action.  The first observation shows that $\z$ is $\spin(12-a, a)$ for some $0 \le a \le 6$.  The second shows that $a$ must be 1, 3, or 5, as claimed.

It remains to prove the correspondence between $a$ and the real forms of $\E_8$.  For $a = 5$, this is clear: the subgroup generated by $\SL(2, \C)$ and $\Spin(7,5)$ has real rank 6, so it can only be contained in the split real form.  Now suppose that $a = 3$ or 1 and that $\SL(2, \C)$ is in the split $\E_8$; we will obtain a contradiction.  Over $\C$, $\SL(2, \C)$ is conjugate to the copy of $\SL_{2,\C} \times \SL_{2,\C}$ in $\E_{8,\C}$ generated by the highest root of $\E_8$ and the highest root of the natural subsystem of type $\E_7$.  One calculates using the tables in \cite{Bou:g4}, e.g., that the element $-1 \in \SL(2, \C)$, equivalently, $(-1, -1) \in \SL_2 \times \SL_2$ is $h_{\alpha_2}(-1)\,h_{\alpha_3}(-1)$ where $h_{\alpha_i}$ is the cocharacter corresponding to the coroot $\alpha_i^\vee$ as in \cite{St}.  That is, the subgroup fixed by conjugation by this $-1$ has root system consisting of roots $\alpha$ such that $\qform{\omega_1, \alpha}$ is even.  These roots form the natural $\D_8$ subsystem of $\E_8$ and our $\SL(2, \C) \cdot \Spin(12-a, a)$ is a standard subgroup.  The vector representation of this $\D_8$ restricts to a sum of the vector representation of $\Spin(12-a,a)$ and the 4-dimensional vector representation of $\SL(2, \C)$ (which factors through $\SO(3,1)$).  In particular, this representation is real, so this $\D_8$ subgroup is, according to \cite[p.~161]{Berger:symm}, isogenous to $\SO(8,8)$.   But then the invariant symmetric bilinear form on the vector representation has signature 0 but restricts to have signature $\pm(12-2a) \in \{ \pm 6, \pm 10 \}$ and $\pm 2$ on each summand, which cannot add up to get 0.  This is a contradiction, so for $a = 3$ or 1, the real form of $\E_8$ is neither split nor compact.

\smallskip

Now suppose that both $\sl_2$'s have index 2.  When we decompose $\e_8$ as in \ref{decomp}, we find the representation $2 \ot 3$ with positive multiplicity 4 by \eqref{22.table}, which violates our hypothesis on the $\SL(2, \C)$ subgroup of $\cE$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{No Theory of Everything in a real form of $\E_8$} \label{real.phys.sec}

We now prove the second claim in Theorem \ref{MT}, namely that each real form $\cE$ of $\E_8$ contains no ToE subgroups.  Suppose $\cE$ contains a copy of $\SL(2, \C)$ (so $\cE$ is non-compact) and a subgroup $G$ satisfying (ToE1) and (ToE3).  We will show that (ToE2) fails.

The $-1$-eigenspace in $\Lie(\cE)$ is a real representation of $\SL(2, \C)\cdot G$.  By Proposition \ref{real.z}, $G$ is contained in a copy of $\Spin(12-a,a)$ for $a = 1$, 3, or 5.  As in the proof of Proposition \ref{real.z}, there is a representation $W$ of $\SL(2, \C) \times \Spin(12-a,a)$ defined over $\R$ that is isomorphic to 
\[
(2 \ot 1 \ot  S_+) \quad \oplus \quad (1 \ot 2 \ot S_-)
\]
over $\C$.  Now $G$ is contained in the maximal compact subgroup of $\Spin(12-a,a)$, i.e., $\Lie(G)$ is a subalgebra of $\so(11)$, $\so(9) \times \so(3)$, or $\so(7) \times \so(5)$.  The restriction of the two half-spin representations of $\Spin(12-a, a)$ to the compact subalgebra are equivalent \cite[p.~264]{McKP}, and we see that in each case the restriction is \emph{quaternionic}.  (To see this, one uses the standard fact that the spin representation of $\so(2\ell + 1)$ is real for $\ell \equiv 0, 3 \pmod{4}$ and quaternionic for $\ell \equiv 1, 2 \pmod{4}$.)  That is, the restrictions of $S_+$, $S_-$, and their complex conjugates to the maximal compact subgroup are all equivalent (over $\C$), hence the same is true for their further restrictions to $G$, and (ToE2) fails.$\hfill\qed$

\begin{rmk}
It is worthwhile noting that, in each of the three cases, it is possible to embed $G_\SM$ in the centralizer, thus showing that (ToE1) is satisfied. Given such an embedding, a simple computation verifies explicitly that $S_+$ has a self-conjugate structure as a representation of $G_\SM$.

First consider $\Spin(11,1)$. There is an obvious embedding of $G_\GUT := \Spin(10)$. Under this embedding, $S_+$ decomposes as the direct sum of the two half-spinor representations, \emph{i.e.}~as a generation and an anti-generation.

For $\Spin(7,5)$, there is an obvious embedding of the Pati-Salam group, $G_\GUT := (\Spin(6)\times \Spin(4))/ (\Zm2)$. Again, $S_+$ decomposes as the direct sum of a generation and an anti-generation.

Finally, $\Spin(3,9)$ contains $(\SU(3)\times \SU(2)\times \SU(2)\times \U(1))/(\Zm6)$ as a subgroup. Under this subgroup,
\[
  S_+ = (3,2,2)_{1/6} \oplus (\overline{3},2,2)_{-1/6} + (1,2,2)_{-1/2} + (1,2,2)_{1/2}
\]
where the subscript indicates the $\U(1)$ weights, and the overall normalization is chosen to agree with the physicists' convention for the weights of the Standard Model's $\U(1)_Y$. Embedding the $\SU(2)$ of the Standard model in one of the two $\SU(2)$s, we obtain an embedding of $G_\SM\subset \Spin(3,9)$ where, again $S_+$ has a self-conjugate structure as a representation of $G_\SM$.
\end{rmk}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{No Theory of Everything in complex $\E_8$} \label{c.phys.sec}

We now complete the proof of Theorem \ref{MT} by showing that there are no ToE subgroups in the transfer $\cE := R(\E_{8,\C})$ of the complex Lie group of type $\E_8$.

\begin{borel*}
First, recall the transfer $R(G_\C)$ of a complex group $G_\C$ as described, e.g., in \cite[\S2.1.2]{PlatRap}.  Its complexification can be viewed as $G_\C \times G_\C$, where complex conjugation acts via
\begin{equation} \label{conj}
\overline{(g, g')} = (\overline{g'}, \overline{g}).
\end{equation}
One can view $R(G_\C)$ as the subgroup of the complexification consisting of elements fixed by \eqref{conj}.

Now consider an inclusion $\phi \!: \SL(2, \C) = R(\SL_{2,\C}) \injects R(\E_{8,\C})$.  Complexifying, we identify $R(\SL_{2,\C})\times \C$ with $\SL_{2,\C} \times \SL_{2,\C}$ and similarly for $R(E_{8,\C})$ and write out $\phi$ as:
\begin{equation} \label{phi.img}
\phi(g_1, g_2) = (\phi_1(g_1) \phi_2(g_2), \psi_1(g_1) \psi_2(g_2))
\end{equation}
for some homomorphisms $\phi_1, \phi_2, \psi_1, \psi_2 \!: \SL_{2,\C} \ra \E_{8,\C}$.  As $\phi$ is defined over $\R$, we have:
\[
\phi(g_1, g_2) 
= \overline{\phi(\overline{g_2}, \overline{g_1})}  =
(\overline{\psi_1(\overline{g_2}) \psi_2(\overline{g_1})}, \overline{\phi_1(\overline{g_2}) \phi_2(\overline{g_1})}),
\]
and it follows that $\psi_1(g_1) = \overline{\phi_2(\overline{g_1})}$ and $\psi_2(g_2) = \overline{\phi_1(\overline{g_2})}$.  Conversely, given any two homomorphisms $\phi_1, \phi_2 \!: \SL_{2,\C} \ra \E_{8,\C}$ (over $\C$) with commuting images, the same equations define a homomorphism $\phi \!: \SL(2, \C) \ra R(\E_{8,\C})$ defined over $\R$.
\end{borel*}

\begin{borel}{Plan of the proof}
Now suppose that we have a subgroup $\SL(2,\C) \cdot G$ of $R(\E_{8,\C})$ satisfying (ToE1) and (ToE3).  
Write $C$ for the identity component of the centralizer of the image of the map $(\phi_1, \phi_2) \!: \SL_{2,\C} \times \SL_{2,\C} \ra \E_{8,\C}$ from \eqref{phi.img}.  Clearly, $G$ is contained in the transfer $R(C)$ of $C$.  In each of the cases below, we verify that
\begin{equation} \label{complex.ss}
\parbox{4in}{$C$ is semisimple,}
\end{equation}
so the maximal compact subgroup of $R(C)$ is the compact real form $C_\R$ of $C$.  Furthermore, in each of the cases below, we will observe that \begin{equation} \label{complex.sc}
\parbox{4in}{$-1$ is in the Weyl group of $C$.}
\end{equation}
It follows that $C_\R$ is an inner form, hence every irreducible representation of $C_\R$ is real or quaternionic, hence every representation of $C_\R$ is self-conjugate.  That is, (ToE2) fails, which is the desired contradiction.
\end{borel}

\begin{borel}{Case 1: $\phi_1$ or $\phi_2$ is trivial}
Consider the easiest-to-understand case where $\phi_1$ or $\phi_2$ is the zero map, say $\phi_2$.  That is, $\phi$ is the transfer of a homomorphism $\phi_1 \!: \SL_{2,\C} \ra \E_{8,\C}$, which by Proposition \ref{sl2} has index 1 or 2.

If $\phi_1$ has index 1, then $C$ is simply connected of type $\E_7$, hence \eqref{complex.ss} and \eqref{complex.sc} hold.  If $\phi_1$ has index 2, then $C$ is isogenous to $\Spin_{13,\C}$ by Lemma \ref{eg2.cent}, and again \eqref{complex.ss} and \eqref{complex.sc} hold.
\end{borel}

\begin{borel}{Case 2: $\phi_1$ and $\phi_2$ are injections}
Finally, we consider the case where $\phi_1$ and $\phi_2$ are both injections.  Again, (ToE3) and Proposition \ref{sl2} implies that $\phi_1$ and $\phi_2$ have Dynkin index 1 or 2.

If $\phi_1$ and $\phi_2$ both have index 1, then over $\C$ this is the same embedding of $\SL_{2,\C} \times \SL_{2,\C}$ in $\E_{8,\C}$ as the one in the proof of Proposition \ref{real.z}.  The centralizer 
$C$ is the standard $\D_6$ subgroup of $\E_{8,\C}$.  

If $\phi_1$ and $\phi_2$ both have index 2, then $\phi_1 \times \phi_2$ gives an embedding as in Proposition \ref{ind22}, and $C$ has Lie algebra $\sp_{4,\C} \times \sp_{4,\C}$ of type $\B_2 \times \B_2$.  Note that (ToE3) fails in this case by \eqref{22.table}.

\smallskip
Suppose finally that $\phi_1$ has index 1 and $\phi_2$ has index 2.
We conjugate so that $\phi_2(\sl_2)$ is the copy of $\sl_2$ from Example \ref{sl2.eg2}, and (by Lemma \ref{conj1} for the centralizer $\so_{13}$ of $\phi_2(\sl_2)$) we can take $\phi_1(\sl_2)$ to be a copy of $\sl_2$ generated by the highest root of $\Spin_{13}$.  Calculating the weights of the representation of this $\sl_2 \times \sl_2$ on $\e_8$ gives the following decomposition into irreducibles:
\begin{equation} \label{12.table}
\begin{array}{cc|rrrc} 
&&1&2&3&m\\ \hline
&1&39&18&1 \\
n&2&32&16&0 \\
&3&10&2&0 
\end{array}
\end{equation}
with the same notation as \eqref{22.table}.  In particular, the $\mathrm{A}_1 \times \B_4$ subgroup of $\Spin_{13}$ that centralizes the image of $\phi_1 \times \phi_2$ is all of the identity component $C$ of the centralizer in $\E_8$.
Again \eqref{complex.ss} and \eqref{complex.sc} hold.  (Of course, \eqref{12.table} shows that (ToE3) fails anyway.)
\end{borel}

This completes the proof of Theorem \ref{MT}. $\hfill\qed$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Relaxing (ToE3)} \label{supergravity}

Technically, $(m,n)=(2,3)\, \text{and}\, (3,2)$ are \emph{possible} in an interacting theory, but only in the presence of local supersymmetry (i.e., in supergravity theories) \cite{Grisaru:1977kk}. Lisi's framework is not compatible with local supersymmetry, so we excluded this possibility above.
If we relax (ToE3) by replacing it with 
\begin{equation}
\text{$V_{1,4} = V_{4,1} = 0$ and $V_{m,n} = 0$ if $m + n > 6$} \tag{ToE3'}
\end{equation}
then we still don't find anything further.  More precisely, we have the following strengthening of Theorem \ref{MT}.

\begin{thm} \label{MTp}
There are no subgroups $\SL(2, \C) \cdot G$ satisfying (ToE1), (ToE2), and (ToE3') in the (transfer of the) complex $\E_8$ or any real form of $\E_8$.
\end{thm}

\begin{proof}
Note that (ToE3') still forces that in the decomposition of $\Lie(\cE)$, the representation $V_{m,n} = 0$ if $m > 3$ or $n > 3$, so each of the two $\sl_{2,\C}$ summands in the complexification of $\SL(2, \C)$ have index 1 or 2 by Proposition \ref{sl2}.  Looking back, we see that we already proved the theorem for the transfer of the complex $\E_8$ in \S\ref{c.phys.sec}.

Suppose we have an $\SL(2, \C) \cdot G$ subgroup of a real form $\cE$ of $\E_8$.  Imitating the proof in \S\ref{real.phys.sec}, we appeal to Proposition \ref{real.z}.  We now have the additional possibility that the complexification $\SL_{2,\C} \times \SL_{2,\C}$ of the $\SL(2, \C)$ subgroup is such that both $\SL_{2,\C}$'s have index 2 as in \S\ref{ind2.sec}.  The centralizer of such an $\SL(2, \C)$ is a real form of $\sp_{4,\C} \times \sp_{4,\C}$ by Proposition \ref{ind22}.  When we decompose $\e_8$ as in \ref{decomp}, we find $V_{2,1}$ and $V_{1,2}$ as in \ref{decomp.22}.  As complex conjugation interchanges these two representations, it follows that complex conjugation interchanges the two $\sp_{4,\C}$ factors, i.e., the centralizer of $\SL(2, \C)$ has identity component the transfer $R(\Sp_{4,\C})$ of $\Sp_{4,\C}$.  Its maximal compact subgroup is the compact form of $\Sp_{4,\C}$ (also known as $\Spin(5)$), all of whose irreducible representations are self-conjugate.  Therefore, (ToE2) fails.
\end{proof}

\begin{rmk}
As we have already mentioned, weakening (ToE3) to (ToE3') is only consistent in supergravity theories. In the case at hand, with $G_{\text{max}}=\Spin(5)$, we find
\begin{equation}\label{nonsugradecomp}
V_{3,2} \simeq V_{2,3} = 4, \qquad V_{2,1}\simeq V_{1,2} = 4 \oplus 16
\end{equation}
where we have indicated the irreducible representations of $\Spin(5)$ by their dimensions. Since the gravitinos transform nontrivially under $G_{\text{max}}$ and, given their multiplicity, the only consistent possibility would be a gauged $\mathcal{N}=4$ supergravity theory (for a recent review of such theories, see \cite{SchonWeidner}). Unfortunately, the rest of the matter content (it suffices to look at $V_{2,1}$) is not compatible with $\mathcal{N}=4$ supersymmetry. Even if it were, $\mathcal{N}=4$ supersymmetry would, of course, necessitate that the theory be non-chiral, making it unsuitable as a candidate Theory of Everything.
\end{rmk}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}

In paragraph \ref{dim} above, we observed by an easy dimension count that no proposed Theory of Everything constructed using subgroups of a real form $\cE$ of $\E_8$ has a sufficient number of weight vectors in the $-1$-eigenspace to identify with all known fermions.  The proof of our Theorem  \ref{MT} was quite a bit more complicated, but it also gives much more.  It shows that you cannot obtain a \emph{chiral} gauge theory for \emph{any ToE subgroup} of $\cE$, whether $\cE$ is a real form or the complex form of $\E_8$. In particular, it is impossible to obtain even the one-generation Standard Model in this fashion.

\bigskip
\noindent{\small{\textbf{Acknowledgments.}} The second author thanks Fred Helenius for pointing out a reference and Patrick Brosnan for suggesting this project. The authors' research was supported by  the National Science Foundation under grant nos.\ PHY-0455649 (Distler) and DMS-0653502 (Garibaldi).}

\bibliographystyle{utphys}
%\bibliography{skip_master,physics}

\providecommand{\href}[2]{#2}\begingroup\raggedright\begin{thebibliography}{10}

\bibitem{Lisi}
A.~G. Lisi, ``An exceptionally simple theory of everything,''
\href{http://arxiv.org/abs/0711.0770}{{\tt arXiv:0711.0770 [hep-th]}}.
%%CITATION = 0711.0770;%%.

\bibitem{Ti:R}
J.~Tits, ``Repr\'esentations lin\'eaires irr\'eductibles d'un groupe r\'eductif
  sur un corps quelconque,'' {\em J. Reine Angew. Math.} {\bf 247} (1971)
  196--220.

\bibitem{GeorgiGlashow}
H.~Georgi and S.~Glashow, ``Unity of all elementary particle forces,''
\href{http://dx.doi.org/10.1103/PhysRevLett.32.438}{{\em Phys. Rev. Lett.} {\bf
  32} (1974)  438--441}.
%%CITATION = PRLTA,32,438;%%.

\bibitem{PatiSalam}
J.~Pati and A.~Salam, ``Lepton number as the fourth color,''
\href{http://dx.doi.org/10.1103/PhysRevD.10.275}{{\em Phys. Rev.} {\bf D10}
  (1974)  275--289}.
%%CITATION = PHRVA,D10,275;%%.

\bibitem{Distler:SuperConnections}
J.~Distler, ``Superconnections for dummies.'' Weblog entry, May, 2008.
\newblock \url{http://golem.ph.utexas.edu/~distler/blog/archives/001680.html}.

\bibitem{StreaterWightman:PCT}
R.~F. Streater and A.~S. Wightman, {\em PCT, Spin and Statistics, and All
  That}.
\newblock Princeton Landmarks in Mathematics and Physics. Princeton University
  Press, 2000.

\bibitem{Weinberg:QFT}
S.~Weinberg, {\em The Quantum Theory of Fields. 3 volumes}.
\newblock Cambridge University Press, 1995-2000.

\bibitem{Helgason}
S.~Helgason, {\em Differential geometry, {L}ie groups, and symmetric spaces},
  vol.~34 of {\em Graduate Studies in Mathematics}.
\newblock Amer. Math. Soc., 2001.

\bibitem{Se:trace}
J.-P. Serre, ``On the values of the characters of compact {L}ie groups,'' {\em
  Oberwolfach Reports} {\bf 1} (2004) no.~1, 666--667.

\bibitem{Dynk:ssub}
E.~Dynkin, ``Semisimple subalgebras of semisimple {L}ie algebras,'' {\em Amer.
  Math. Soc. Transl. (2)} {\bf 6} (1957)  111--244. [Russian original: Mat.\
  Sbornik N.S.\ \textbf{30(72)} (1952), 349--462].

\bibitem{Bou:g7}
N.~Bourbaki, {\em Lie groups and {L}ie algebras: Chapters 7--9}.
\newblock Springer-Verlag, Berlin, 2005.

\bibitem{GrossNebe}
B.~Gross and G.~Nebe, ``Globally maximal arithmetic groups,'' {\em J. Algebra}
  {\bf 272} (2004) no.~2, 625--642.

\bibitem{Hum:LA}
J.~Humphreys, {\em Introduction to {L}ie algebras and representation theory},
  vol.~9 of {\em Graduate Texts in Mathematics}.
\newblock Springer-Verlag, 1980.
\newblock Third printing, revised.

\bibitem{OV3}
A.~Onischik and E.~Vinberg, {\em Lie groups and {L}ie algebras {III}}, vol.~41
  of {\em Encyclopaedia Math. Sci.}
\newblock Springer, 1994.

\bibitem{Bou:g4}
N.~Bourbaki, {\em Lie groups and {L}ie algebras: Chapters 4--6}.
\newblock Springer-Verlag, Berlin, 2002.

\bibitem{Vavilov:const}
N.~Vavilov, ``Do it yourself structure constants for {L}ie algebras of type
  ${E}_{\ell}$,'' {\em J. Math. Sci. (N.Y.)} {\bf 120} (2004) no.~4,
  1513--1548.

\bibitem{Dynk:max}
E.~Dynkin, ``Maximal subgroups of the classical groups,'' {\em Amer. Math. Soc.
  Transl. (2)} {\bf 6} (1957)  245--378. [Russian original: Trudy Moskov.\
  Mat.\ Ob\v s\v c. \textbf{1} (1952), 39--166].

\bibitem{McKP}
W.~McKay and J.~Patera, {\em Tables of dimensions, indices, and branching rules
  for representations of simple {L}ie algebras}, vol.~69 of {\em Lecture Notes
  in Pure and Applied Mathematics}.
\newblock Marcel Dekker Inc., New York, 1981.

\bibitem{St}
R.~Steinberg, {\em Lectures on {C}hevalley groups}.
\newblock Yale University, New Haven, Conn., 1968.

\bibitem{Berger:symm}
M.~Berger, ``Les espaces sym{\'e}triques noncompacts,'' {\em Ann. Sci. \'Ecole
  Norm. Sup. (3)} {\bf 74} (1957)  85--177.

\bibitem{PlatRap}
V.~Platonov and A.~Rapinchuk, {\em Algebraic groups and number theory}.
\newblock Academic Press, Boston, 1994.

\bibitem{Grisaru:1977kk}
M.~T. Grisaru and H.~N. Pendleton, ``Soft spin 3/2 fermions require gravity and
  supersymmetry,''
\href{http://dx.doi.org/10.1016/0370-2693(77)90383-5}{{\em Phys. Lett.} {\bf
  B67} (1977)  323}.
%%CITATION = PHLTA,B67,323;%%.

\bibitem{SchonWeidner}
J.~Schon and M.~Weidner, ``Gauged {N}=4 supergravities,'' {\em JHEP} {\bf 05}
  (2006)  034,
\href{http://arxiv.org/abs/hep-th/0602024}{{\tt arXiv:hep-th/0602024}}.
%%CITATION = HEP-TH/0602024;%%.

\end{thebibliography}\endgroup



\end{document}
