With so many 11's around today, it seems fitting to mention some M-theory related material. A few days ago, Hisham Sati updated his
On the geometry of the supermultiplet in M-theory paper which argues that the massless supermultiplet of D=11 supergravity can be generated from the decomposition of reps of the
exceptional Lie group F4 and its maximal compact subgroup Spin(9). The dynamical origin of this is proposed to result from Cayley plane bundles over eleven-dimensional spacetime.
The
Cayley plane, OP^2, is a projective plane over the
octonions and its isometries form the group F4. Lines in OP^2 are 8-spheres and given any two points in OP^2 there is a unique 8-sphere passing through them. Given any three distinct points, if we apply an F4 transformation that fixes one of the points, we get a Spin(9) transformation.
In matrix parlance, F4 is the automorphism group of the algebra of 3x3 Hermitian matrices over the octonions, the
exceptional Jordan algebra J(3,O). We can construct OP^2 using the rank one projectors of J(3,O). It can actually be defined as the space of all such rank one projectors. Normalizing the rank one projectors turns them into primitive idempotents, that is, matrices P that satisfy P^2=P which cannot be decomposed as an orthogonal sum of other idempotents. As the identity matrix of J(3,O) is just a 3x3 matrix with ones on the diagonal, its straightforward to see that it decomposes into an orthogonal sum of three primitive idempotents. This is called the capacity and is why J(3,O) is an algebra of degree three.
Going back to the geometry of OP^2, the three distinct points mentioned earlier can be interpreted as three orthogonal primitive idempotents of J(3,O) with orthogonality being a result of these matrices satisfying P1.P2=0 under regular matrix multiplication. To simplify the picture, let's just imagine applying an F4 transformation on the identity matrix where we want to keep one of the diagonal ones fixed. This can be done with a Spin(9) transformation. Since we can fix any of the three diagonal ones of the identity matrix, there are three copies of Spin(9) inside F4 we can use. This freedom of choice we have is what some people refer to as triality.