Standard Model Floccinaucinihilipilification
Grand Unified Theories are kind of an old game. The aim is to combine all the forces of nature - sans gravity - into a single force and then explain why it all falls apart. Electricity and Magnetism were shown to be the same at the turn of the 20th century. It took well over 70 years to show that the weak nuclear force and electromagnetism were shown to descend from the same force. That basically leaves us with the strong nuclear force.
To prove that forces are unified typically means finding new particles. The photon is shared by electricity and magnetism. The photon itself mixes with the Z-boson at high energies. If the strong force and the weak force were to combine, we'd expect to find some of the left over particles - perhaps other matter particles too. In part, that's why some folks have falling out of their chairs lately about leptoquarks.
String theory has given us the theoretical technology to build arbitrarily complicated forces of nature, and so much work has been done to study the various possibilities. Most attempts have failed. Some have come closer. There have also been several rogue attempts that have also failed, some more interestingthan others.
A large part of the problem is the simplicity of the mathematical objects that form the basis for the Standard Model of Particle Physics. The Lie group SU(2) - responsible for the weak nuclear force - is by coincidence the same mathematical object that can explain rotations in three-dimensions. Two copies of it describe our 4-dimensional spacetime and are used routinely to model quark flavor and the existence of pions. SU(2) is commonplace precisely because it is so simple, not because of anything fundamental.
This simplicity, together with the plethora of possibilities from String theory, has brought about attempts to understand the Standard Model from a statistical perspective. Although it is not clear what priors we should be employing or even if this approach makes sense, some attempts have been compelling. Others, in this author's opinion, less so.
Nevertheless, it is true that only a finite number of grand unification possibilities exist, at least with a fixed set of rules: like including only the currently observed matter particles of nature. Recently, three physicists from DAMTP have devised a computational process to play the game of grand unification, exhaustively. Reducing all the high-minded particle philosophy to cold, hard, arguably boring category theory. No doubt at least one of them attended Eton, suggesting the name for this algorithm, Floccinaucinihilipilification.
For a single generation of fermions: including only up and down quarks, the electron and its neutrino, this approach yield the typical scenarios: Pati-Salam, su(5) and so(10). Amusingly, it rules out all others. With two generations of matter - including the muon, muon neutrino, strange and charm quarks, the number of possibilities jumps to 45. With three generations, the possibilities are tabulated in their paper.
Notably absent from these possibilities are the flipped-SU(5) type models, which while not fundamental, do include more string theory friendly properties.
