John Baez has just put out a new issue of his This Week’s Finds in Mathematical Physics, dealing partly in more detail with the material about Clifford modules mentioned here a couple weeks ago. I’ve added as the first comment here something he had some trouble submitting as a comment to the older posting on this topic.
John briefly mentions a relation of all this to Bott periodicity in topology, using a very abstract homotopy construction involving spectra. A more concrete version of this can be found in Milnor’s book on Morse theory. For the relation of Clifford algebras and K-theory, the standard refererence is the 1964 paper “Clifford Modules” by Atiyah, Bott and Shapiro published in the journal “Topology”. The crucial fact they describe is how the Thom isomorphism in K-theory (which is essentially the same fact as Bott periodicity) is related to the structure of Clifford modules. Greg Landweber has recently worked out an interesting equivariant version of this story.
Greg also has a nice new paper with Megumi Harada about the K-theory of a symplectic quotient, that looks like it should imminently appear on the arXiv.
John also mentions some recent work of Dror Bar-Natan, Thang Le and Dylan Thurston on the Duflo isomorphism. This is a beautiful story, and also has a relation to Clifford algebras that John doesn’t mention. For this, see Eckhard Meinrenken’s talk at the 2002 ICM in Beijing.
Thanks, should be fixed now.
Your link to the Landweber-Harada paper is broken.
Dear Peter –
I’m glad you liked week211 of This Week’s Finds. I just put up week212, which digs deeper into the relation Clifford algebras and supersymmetry. A bunch of it comes from Deligne’s talks at the Institute for Advanced Studies – the ones that became part of that book Quantum Fields and Strings: a Course for Mathematicians.
I think there are still hopes for understanding the details of the Standard Model with the help of a deeper understanding of Clifford algebras, division algebras and related algebraic structures. Garrett Lisi mentioned Greg Trayling’s work. I’ve never taken the time to properly understand that, in part because of some nonstandard terminology that it uses, but I’m encouraged by the fact that Garrett thinks it’s good. I’ve spent a lot more time on Geoffrey Dixon’s work. The SU(5) and SO(10) grand unified theories, and Jogesh Pati’s work on left-right symmetric theories, also have something very beautiful about them. I suspect that they’re all grasping various parts of some truth that we’re not yet able to fathom, perhaps because we don’t have the right language.
If none of these theories are currently fashionable, well, that’s in part because none of them quite hit the nail on the head – but also because the most influential particle theorists seem to have given up hope on the idea of staring at the Standard Model until something clicks and a beautiful theory takes form which explains all its baroque peculiarities. Most of the physicists who are really good at math seem willing to let the internal logic of string theory guide them where it will: either to a triumphant victory in physics, or to a journey through beautiful mathematics increasingly distant from the physical world. It’s a pity that pondering the Standard Model is being left to mere "phenomenologists".
Best,
jb