### This Week’s Finds in Mathematical Physics (Week 253)

#### Posted by John Baez

In week253, read about
mysterious relations between the Standard Model, the SU(5)
and SO(10) grand unified theories, the exceptional group
E_{6}, the complexified octonionic projective plane…
and maybe even E_{8}!

Here’s a lightning review of the Standard Model:

Alas, the fact that this chart looks like a square matrix seems to have no relation to any interesting physics whatsoever, except insofar as it shows that leptons and quarks come in 3 generations and the gauge bosons are something else. I know of no sense in which the $\gamma$, $g$, $W$ and $Z$ are like a “fourth generation”.

Also, this chart omits the Higgs.

But, it’s the best chart I could find when it came to simplicity, visual impact and actual information. It’s available at various places on the Fermi National Accelerator Laboratory website.

It’s possible to make a very nice chart of fermions in the SO(10) grand unified theory, and I believe such a chart can be found in Zee’s book *Quantum Field Theory in a Nutshell*. But, I couldn’t find a nice chart like this online. Maybe I’ll have to make one someday.

## Re: This Week’s Finds in Mathematical Physics (Week 253)

Very nice summary of standard model structure and beyond!

For those reading this who don’t know, one should maybe add, to avoid confusion here, that supersymmetry was never meant to make those bosons and fermions which have already been detected to be superpartners of each other. Rather, all their superpartners are hypothesized – in supersymmetric extensions of the standard model – to exist on top of that, but to have evaded detection so far due to their relatively high mass, which they are thought of as having acquired due to a “supersymmetry breaking mechanism” of one sort or another.

I wonder how all this $E_6$ magic which you mention is related to Alain Connes’ observation: that the fermions of the standard model naturally arise as

the direct sum of all inequivalent irreduciblebimodules of the algebra$\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C}) \,.$That’s another rather cute way to summarize all that information in one sentence!