Yesterday I sneaked on to campus for a few hours. I’m on family leave (as I mentioned earlier) and so I’ve not been going to campus unless I more or less have to. Yesterday was one of those days that I decided was a visit day and so visit I did. I went to say hi to a visitor to the Mathematics Department, Sylvester James Gates Jr., an old friend who I’ve known for many years. He was giving the CAMS (Center for Applied Mathematical Sciences) distinguished lecture with the title “How Attempting To Answer A Physics Question Led Me to Graph Theory, Error-Correcting Codes, Coxeter Algebras, and Algebraic Geometry”. You can see him in action in the picture above.
I was able to visit with Jim for a while (lunch with him and CAMS director Susan Friedlander), and then hear the talk, which was very interesting. I wish he’d had time to say more on all the connections he mentioned in the title, but what he did explain sounded rather interesting. It is all about the long unsolved problem of finding certain kinds of (unconstrained, off-shell) representations of extended supersymmetry. (Supersymmetry is, you may know, a symmetry that relates matter-tpye particles, like electrons and quarks, to force-type particles, like photons and gluons.) The supersymmetry veterans like Jim have been puzzling over this for some time (since the 1970s!), and his explorations over the years have let to what looks to be a rich playground of new connections.
The problem is interesting and important for physics, since we know that theories of extended supersymmmetry are highly instructive in understanding how various physical systems work (even if not directly experimentally relevant) – most of what we’ve learned in the remarkable revolution in our understanding of quantum field theory, string theory, and gravity, since the 1990s have as their best understood examples models which have extended supersymmetry. Writing and fully formulating the quantum versions of these theories requires* coming to grips with the “off shell” (and unconstrained) representations – they remain elusive. This is why supersymmetry experts want to find ways of handling them.
Jim’s new work starts by turning the details of a representation into a decorated graph (which he and his collaborators call an “adinkra” – the example shown is from this paper), and the properties of the graphs turn out to be mathematically very interesting, and lead to connections with all sorts of other fields of mathematical investigation. Who knows where it might lead?
*At least if we want to quantise things the way we usually do.