John Baez has a very interesting new paper on the arXiv this evening entitled Calabi-Yau Manifolds and the Standard Model. In it he points out that the standard model gauge group (which he carefully defines as SU(3)xSU(2)xU(1)/N, where N is a six-element subgroup that acts trivially on the standard model particles) is the subgroup of SU(5) that preserves a splitting of C5 into orthogonal 2 and 3 dimensional complex subspaces. Furthermore, if you think of SU(5) as a subgroup of SO(10), then the spinor representation of SO(10) on restriction to the standard model group has exactly the properties of a single generation of the standard model.
Baez would like to think of SO(10) as the frame rotations in the Riemannian geometry of a 10d manifold X. The SU(5) is then the holonomy subgroup picked out by a choice of Calabi-Yau complex structure on the manifold. One way to get such an X is as the product of R4 and a compact 6-manifold M6, picking Calabi-Yau structures on both manifolds in the product. What is happening here is related to an old idea I wrote a paper about a very long time ago (see Nuclear Physics B, vol. 303, pgs. 329-342, from 1988). By picking an orthogonal complex structure on R4, one picks out a U(2) in SO(4) (the Euclideanized Lorentz group), and it is tempting to identify this with the electroweak U(2). This is one part of what is happening in Baez’s construction. It’s very hard though to see what to do with this within the standard gauge theory framework; this is true both for my old idea and for Baez’s newer one. Maybe string theorists can come up with some way of implementing this idea of thinking of the standard model gauge group in terms of the Riemannian geometry of the target space of a string. If so I might even get interested in string theory…..
I don’t immediately see from Baez’s paper why the hypercharge assignments come out right. I need to sit down and work that out, but it’s getting late this evening. There are some other issues his paper raises that I’d like to think about, and maybe I’ll finally get around to doing some work to see whether what I’ve learned about spin geometry in recent years has any use in this context.
I also noticed today that Baez is advertising for students to come to UC Riverside to study Quantum Mathematics. I like the term, and for many students who really care about mathematics and fundamental physics, this would be worth thinking about.
Please, commenters who want to write about their favorite ideas about standard model geometry, try and stick to any aspects of this directly related to Baez’s paper.