The Algebra of Grand Unified Theories
Posted by John Baez
John Huerta is a student of mine who’s really interested in particle physics. Pretty soon he’ll plunge into his thesis work on exceptional algebraic structures and their role in physics — especially superYang–Mills theory, superstring theory and supergravity, but maybe also grand unified theories. But first he needs to pass his oral. Here are the slides for his talk:
 John Huerta, The Algebra of Grand Unified Theories. Also available in printerfriendly form.
Since most of the oral examiners will be from the math department, Huerta’s talk will start with a presentation of facts that particle physicists already know… but reformulated in a way that mathematicians can enjoy. So, he’ll start by explaining the group representation theory underlying the Standard Model. Then he’ll explain — in deliberately very brief and sketchy terms! — the idea of a Grand Unified Theory, or GUT. Then he’ll review three famous GUTs:

The SU(5) theory proposed by Georgi and Glashow. This unifies the three colors of quark (red, green, and blue) with the two weak isospin states (up and down) by stuffing the color symmetry group SU(3) and the weak isospin symmetry group SU(2) into SU(5) in an obvious way. The Standard Model symmetry group is SU(3) $\times$ SU(2) $\times$ U(1), so the theory also needs to stuff U(1) inside SU(5) as a subgroup that commutes with SU(3) $\times$ SU(2). This can be done, at least mod a discrete subgroup — and amazingly, the result naturally ‘explains’ the bizarre pattern of hypercharges (that is, U(1) representations) seen in the Standard Model! Instead of a complicated looking representation of the complicatedlooking group SU(3) $\times$ SU(2) $\times$ U(1), quarks and leptons are now described by the obvious representation of SU(5) on the exterior algebra $\Lambda \mathbb{C}^5$.

The Spin(10) theory proposed by Georgi. Using the fact that a 5dimensional complex vector space can be seen as a 10dimensional real vector space, we can see SU(5) as a subgroup of Spin(10). The representation of SU(5) on $\Lambda \mathbb{C}^5$ is reducible, but it extends to a representation of Spin(10) that’s almost irreducible: it splits into just two parts, one for particles and the other for antiparticles.
 The Spin(4) $\times$ Spin(6) theory proposed by Pati and Salam. By what I’ve said, the Standard Model symmetry group is a subgroup of Spin(10). But in fact, it’s a subgroup of the smaller group Spin(4) $\times$ Spin(6). Since Spin(4) $\cong$ SU(2) $\times$ SU(2) and Spin(6) $\cong$ SU(4), we can also think of the symmetry group of the Pati–Salam model as SU(2) $\times$ SU(2) $\times$ SU(4). One copy of SU(2) corresponds to weak isospin — and it acts nontrivially only on lefthanded particles. The other corresponds to a righthanded version of weak isospin. Since we don’t see this, it must be spontaneously broken in this model. Finally, the SU(4) treats the leptons as a fourth color of quark — say, ‘white’.
Then comes the fun part: all three of these models, together with the Standard Model, fit into a unified framework. John Huerta explains this in terms of a commutative cube of groups.
Of course all three of these models have their problems! They predict proton decay at too fast a rate, and so on. So at best, they are just steppingstones towards a better theory. Still, they’re intriguing attempts to fit the messy pack of particles we know into an elegant picture. And, it’s possible for mathematicians who know some group theory but not much physics to have a lot of fun learning this stuff. So, John and I are writing a gentle introduction to the algebra of Grand Unified Theories, suitable for mathematicians, which covers the material in his talk. I’ll announce it on this blog soon. But where should we publish it? Suggestions, anyone?
Re: The Algebra of Grand Unified Theories
Super! This will be a great resource as I’ve finally started reading through Peter Woit’s book. Is there an easy way to put the slides “together” so it is more printable? (If the only way is to go and delete all the partial slides, don’t bother–I can do that myself before printing.)