A few things that may be of interest:
- Survey articles prepared for the 2018 ICM proceedings are starting to appear on the arXiv, and Peter Scholze (who will be getting a Fields Medal in Rio) has put his on his web-site. His title is p-adic Geometry, and it gives an overview of the ground-breaking work he has been doing over the last few years. The last section tells us that
Currently, the author is trying to understand to what extent it might be true that the “universal” cohomology theory is given by a shtuka relative to Spec Z. It seems that this is a very fruitful philosophy.
For some background about that section, I’d recommend his talk at the 2015 Clay Math conference.
- The New Yorker has a very detailed and interesting profile of Jim Simons and what he is up to with the Flatiron Institute he is now funding here in New York. This new Institute is costing him \$80 million a year, characterized as “a lark” for someone with his assets. David Spergel is running the Center for Computational Astrophysics there, and doing a lot of hiring. When I wrote here about his characterization of multiverse research, his final comment about being able to speak freely because he had tenure left me wondering “wait, what about grants, jobs, etc.?”. From the New Yorker article, I realized that while having tenure may give you some ability to speak freely, having a guy with \$18.5 billion willing to write large checks for you gives you a lot more…
- I’ve just finished teaching a course this semester which concentrated on the formalism for describing geometry in terms of connections and curvature. From the point of view of physicists, this formalism should be of interest because it applies equally well to gauge theory and general relativity. I’d been starting to think again about what light this formalism might shed on how to think about these two subjects together, when last night I noticed a wonderful new article on the arXiv, Gravity and Unification: A review by Krasnov and Percacci.
This article is an extremely lucid and comprehensive survey of the sort of thing I was thinking about, which can be re-expressed as the question of trying to find, at the classical level, a formalism uniting the vector potentials/field strengths of the SM and the different possible fields used to describe geometry in GR. Some of this has a very long history, going back to the things Einstein was trying in his later years. There have been many different ideas that people have tried since then, and the survey article does a great job of both explaining these ideas, as well as indicating why they haven’t worked out.
A couple of the general ideas that have always fascinated me make an appearance in the article. One of these is that of what mathematicians call a “Cartan connection”, the idea that you should think of a geometry as locally looking like a quotient space G/H of two Lie groups. A version of this is known to physicists as the MacDowell-Mansouri formulation, which gets a detailed treatment in the article. Another is the idea of using the fact that the complexified orthogonal group in 4 dimensions breaks up into two pieces, sometimes thought of at the Lie algebra level as self-dual vs. anti-self-dual pieces under the Hodge star operation. A version of this idea is known as the Plebanski formulation, and this decomposition is behind the story of Ashtekar variables. These variable have played a crucial role in modern treatments of GR by Hamiltonian methods, as well as the quantization program of loop quantum gravity.
The focus of the article is on Lagrangian and classical field theory methods for studying these ideas. There’s relatively little about the Hamiltonian story, and also relatively little about the geometry of spinors, two topics that I suspect might provide additional needed insights. For anyone interested in thinking about non-string theory-based ideas about unification of the SM and gravity, there’s a wealth of ideas, references and history here to think about. Perhaps future progress on unification will come from some new breakthrough in this field that shows how to get around the problems identified clearly in this article.
- For surveys of recent work on quantization of gravity and discussion by experts, a good place to look is videos of talks at a recent conference held at the IHES. Videos available here include my fellow Princeton student Costas Bachas surveying the approach growing out of string theory in Holographic Dualities and Quantum Gravity, Carlo Rovelli the opposition in Current Quantum Gravity Theories, Experimental Evidence, Philosophical Implications, and an even-handed overview from Steven Carlip with Why We Need Quantum Gravity and Why We Don’t Have It. Also well-worth watching, both for the talk and the discussion, is Alain Connes on Why Four Dimensions and the Standard Model Coupled to Gravity.
Finally, for fans of Lenny Susskind’s introductory level books on theoretical physics, Andre Cabannes writes to tell me that the most recent volume (which I wrote about here) is being translated by him into French, to appear next year. He also has notes from the lectures on General Relativity, Cosmology, and Statistical Mechanics, for which no book form has yet appeared.