The “vision” talk at Strings 2014 that I found most interesting was that of Greg Moore, whose topic was “Physical Mathematics and the Future”. He has a very extensive written version of the talk here, which includes both what he said, as well as a lot of detail about current topics at the interface of mathematics and physics.
I think what Moore has to say is quite fascinating, he’s giving a wonderful survey of where this intellectual subject is, how it has gotten there, and where it might be going. I’m very much in sympathy with his historical discussion of how math and physics have evolved, at times closely linked, at others finding themselves far apart. He’s concerned about an issue that I’ve commented on elsewhere, the fact that physics and math seem to be growing apart again, with no mathematicians speaking at the conference, instead attending their own conference (“Strings-Math 2014”). Physics departments increasingly want nothing to do with mathematics, which is a shame. One reason that Moore gave for this I found surprising, the idea that
most mathematicians are not as fully blessed with the opportunities for pursuing their research that many theoretical physicists enjoy.
It seems there’s a perception among many physicists that research mathematicians labor under some sort of difficult conditions of low pay and high teaching loads, but I think this is a misconception. Moore may be generalizing too much from the situation at Rutgers, where very unusual positions were created for string theorists at the height of that subject’s influence. From what I’ve seen, the salaries of top research mathematicians and theoretical physicists are quite comparable (if you don’t believe me, do some searches in the on-line data of salaries of faculty employed by public universities). Senior mathematicians do sometimes have slightly higher teaching loads, although often with a freedom to teach what they want. At the postdoc level, it is true that theoretical physics postdocs typically have no teaching, while similar positions in math often do require teaching. On the other hand, the job situation in theoretical physics is much more difficult than in mathematics. I’d say that working in an environment where you know you’re likely to find a permanent job is much preferable to one where you know this is unlikely, with doing some teaching not at all a significant problem.
On the question “What is String Theory?”, Moore’s take was that the “What is M-theory?” question is no longer getting much attention, with people kind of giving up. There was a very odd exchange at the end of the talk, when Witten asked him if he thought that maybe people should be emphasizing the string question, not the M-theory question, and Moore responded that the emphasis on M-theory was something he had learned from Witten himself.
His main point about this though was one I very much agree with, that the more interesting question now is “What is QFT?”. The standard way of thinking about QFTs in terms of an action principle doesn’t capture much of the interesting things about QFT we have learned over the years. Moore emphasizes certain examples, such as the (2,0) 6d superconformal theories, but discusses in his written version the relation of QFT to representation theory of some infinite dimensional groups, which I think provides even better examples of a different and more powerful way of thinking about QFT.
The written version contains a wealth of information surveying current topics in this area, is highly recommended to anyone who wants to try and understand what people working on “string theory” and mathematics have been up to. It appears that this document is a work in progress, with more material possibly to come (for instance, there’s a section 4.4 on Geometric Representation Theory still to be filled in). I look forward to future versions.