I was going to just provide the following links with a some comments, but decided it would be a good idea to put them into what seems to me the larger context of where we are in fundamental physics and its relationship to mathematics.
For the latest on the conventional physics approach to unification (GUTS, SUSY, strings, M-theory), there’s:
- The Lex Fridman podcast has an interview with Cumrun Vafa. Going to the section (1:19:48) – Skepticism regarding string theory) where Vafa answers the skeptics, he has just one argument for string theory as a predictive theory: it predicts that the number of spacetime dimensions is between 1 and 11.
- A second edition of Gordon Kane’s String Theory and the Real World has just appeared. One learns there (page 1-19) that
There is good reason, based on theory, to think discovery of the superpartners of Standard Model particles should occur at the CERN LHC in the next few years.
For the latest in mathematics and the interface of math and physics, there’s
- Quanta magazine has a good article about the dramatic Fargues-Scholze result linking geometry and number theory.
- A long paper by Gaiotto and Witten on their new ideas about quantization, giving a lot of background and details, in particular exploring the link to the currently best-developed set of ideas relating geometry and quantization (so-called “geometric quantization”).
About the first two links, I’m at a loss for words.
The second two are extremely interesting topics indicating a deep unity of number theory, geometry and physics. They’re also not topics easy to say much about in a blog posting. In the Fargues-Scholze case that’s partly because the new ideas they have come up with relating arithmetic and geometry are ones I don’t understand very well at all (although I hope to learn more about them in the future). The connections they have found between representation theory, arithmetic geometry, and geometric Langlands are very new and it will likely be quite a few years before they are well understood and their implications well-developed.
In the Gaiotto-Witten case, some of what they discuss is very familiar to me: geometric quantization has been a topic of fascination since my student days, and one major goal of my QM book was to work out in detail (for the case of $\mathbf R^{2d}$) some of the subtleties about quantization that they discuss. For co-adjoint orbits in Lie algebras, geometric quantization has a long history, and “brane quantization” may or may not have anything new to say about this. For moduli spaces of vector bundles on Riemann surfaces, and Hitchin moduli spaces of Higgs bundles on Riemann surfaces, “brane quantization” might come into its own.
There is a fairly short path now potentially connecting fundamental unifying ideas in number theory and geometry to our best fundamental theories in physics (and seminars on arithmetic geometry and QFT are now a thing). The Fargues-Scholze work relates arithmetic and the central objects in geometric Langlands involving categories of bundles over curves. These categories in turn are related (in work of Witten and collaborators) to 4d TQFTs based on twistings of N=4 super Yang-Mills. This sort of 4d QFT involves much the same ingredients as 4d QFTs describing the Standard Model and gravity. For some better indication of the relation of number theory to this sort of QFT, a good source is David Ben-Zvi’s lectures this past semester (see here and here). I’m hopeful that the ideas about twistors and QFT in Euclidean signature discussed here will provide a close connection of such 4d QFTs to the Standard Model and gravity (more to come on this topic in the near future).
