Section 8.9 The Geometry of \(E_8\)
We've come to the end of the line. The Lie group \(E_8\) is the largest member of its class, both as the largest of the exceptional Lie groups, and as the largest element in the magic square.
Our previous analyses of Lie groups and Lie algebras have all used the notion that a symmetry group acts on a representation, such as vectors or spinors. Yes, groups also act on themselves; that's the adjoint representation. But all of the groups we have examined so far admit other representations that are smaller and easier to work with.
Not so for \(E_8\text{.}\) Alone among the simple Lie groups, its smallest representation is itself; there is no smaller representation. However, analyzing an adjoint representation over the octonions is not easy It's one thing to use nesting to implement octonionic symmetries as iterated actions, but quite another for the things being acted on to be nested!
It is indeed possible to introduce such a notion of “nesting”; see [7]. But there is another way to interpret \(E_8\text{.}\)
In Section 7.2, we attempted to combine spinors and vectors into some sort of \(3\times3\) representation. We can instead combine the spinors with the \(2\times2\) adjoint, and this process yields the corresponding element of the \(3\times3\) magic square. Thus, we consider anti-Hermitian matrices of the form
where \(M\) is an alement of the corresponding orthogonal group, \(\theta\) a spinor, and \(\alpha\) must now be pure imaginary. Remarkably, because of some special properties of the octonions, the “extra” degrees of freedom in \(\alpha\) are already present in \(M\text{!}\) This additional symmetry, known as the triality of \(\SO(8)\text{,}\) means that, we can decompose \(E_8\) precisely into the corresponding orthogonal group together with its spinors—and nothing more.
In the half-split magic square, this decomposition takes the form
where \(\mathbf{128}\) refers to the Majorana–Weyl spinor representation of \(\so(12,4)\text{,}\) which has dimension \(2^7=128\text{.}\) Since \(\so(12,4)\) has dimension \({16\choose2}=120\text{,}\) we conclude that \(\ee_8\) has dimension \(120+128=248\text{.}\)
This natural decomposition may have applications in the context of particle physics, since \(\SO(12,4)\) is just large enough to contain both the Standard Model symmetry group \(\SU(3)\times\SU(2)\times\UU(1)\) and the Lorentz group \(\SO(3,1)\text{,}\) and the \(\mathbf{128}\) is just large enough to contain the fermions (quarks and leptons) that the Standard Model acts on. A Grand Unified Theory (GUT) based on \(E_8\text{,}\) in a certain sense the “largest” possible symmetry group, appears to provide a remarkably good description of nature! 1