M theory: life, the universe and everything
Short story: we know how to do M theory rigorously. This is a fairly compelling case that our ideas about quantum gravity are, er, let's say, on the right track. Personally, I do not claim to understand much of it at all. I don't see how anyone could claim to, except maybe Ed Witten and the mysterious kneemo and a few other wizards.
Physically, M theory needs to formulate a Machian duality (Witten's favourite word) that looks like supersymmetry, but not in the sense of ordinary algebras or superpartners. Years ago, when I was trying to picture this, and stumbling clumsily with varying hbar or speed of light (while Louise was working it all out), I asked myself: if horizons are like boundaries, but the holographic mirror has to turn everything inside out, then what is a boundary? How do primordial black holes and the cosmological horizon fit together? Many of the most fundamental ideas in physics are about understanding boundaries. Consider Stokes' theorem, for instance.
After playing with a little mathematics I fell wildly in love with Algebraic Topology, because Stokes' theorem could be written pretty simply! The Machian principle also suggested an interconnectedness-of-all-things idea. It slowly became clear that this was impossible without category theory, because category theory is the mathematics of relationships (and people had already tried pretty well everything else).
But I won't bore you with diagrams, which I can't draw here anyway. It turns out that to understand M theory we need several pictures at once in our mind: twistor String theory, spin foam QG and matrix models.
It is a powerful mathematical theorem that the complex moduli of Riemann surfaces with punctures is closely related to a moduli of labelled (metric) ribbon graphs. A ribbon graph is a closed diagram (graph) of flat ribbons, which are allowed to cross over and under one another. By allowing twists, one can also study matrix models for the quaternionic ensemble. The expert on this is Mulase.
Anyway, since the Bilson-Thompson preons were made of ribbons, it made sense to think about these ribbon moduli. These spaces turn out to have cell decompositions that look a lot like the special polytopes that turn up in higher category theory. This is no accident, because it is all really about operads. Now, I knew that there was a first class maverick amateur physicist named Carl Brannen, who had already calculated the neutrino and charged lepton masses based entirely on ideas from the Geometric Algebra of Hestenes. What Carl did was associate preons (not quite the same as the Bilson-Thompson ones) with idempotent eigenmatrices in Clifford algebras.
A few days ago, people started talking about the Bilson-Thompson preons, yet again, and Carl briefly outlined his vastly superior version, which could explain the number of generations. I showed up, and so did Michael Rios (kneemo), who happens to be a young wizard expert in Jordan algrebras, and just about everything else it seems. The three of us got talking, and while I was struggling to understand all the algebra it dawned on me that, geometrically, the model for the idempotents were the special points in projective space from Mulase's ribbon graph theory. As soon as we realised that the orbifold Euler characteristic of the moduli space of the 6 punctured sphere was -6 we knew that was a derivation of the number of generations of String theoretic power. The gallant kneemo got working, and I tried hard too but mostly I just went into panic!
This is all about doing not only higher categorical cohomology (boundary theory), but also Poincare duality - the mother of all Poincare dualities. Mulase had already shown how to get T duality out of the real-symplectic Penner model. This used twisty single ribbons. The triple ribbons come in because the idempotents are 3x3. Carl's preons can be built out of a basis of 2 flat ribbons (particle, antiparticle) and two twisted ribbons (1 left, 1 right), each of which effectively has 4 labels. A left handed electron must pair with a right handed electron to generate mass.
Don't worry: spin foam models are in there, too. Louis Crane's geometrization of matter proposal was about keeping the spin foam topological and relaxing the restraint on using manifolds. Matter should somehow be related to the singularities, where the nature of a point (vertex of the spin foam) is given by the surrounding 3-space. It was a lot of fun to play around with 3 dimensional hyperbolic geometry and knots, but it wasn't clear how to make the higher genus surfaces (boundaries of the 3-spaces) look exactly like black holes (or 'dark matter'). Now we can do it. The trick is to think not of an actual surface, but the whole moduli space being modelled by projective geometry (twistors).
More on this later. If anyone happens to be in Sydney, I'll be talking about ribbon graphs to some category theory guys this coming week, at 2pm on Wednesday August 23 in the Maths department at Macquarie.