Stony Brook Dialogues in Mathematics and Physics

Last week I spent a day out at Stony Brook, attending the second day of a two-day symposium devoted to mathematics and physics, held in honor of C. N. Yang and Jim Simons. Peter Steinberg was there for the first day, and has a report about this on his blog. The symposium was in many ways also a celebration of the new Simons Center for Geometry and Physics, which is just getting off the ground with a $60 million donation from Simons. A new building will be constructed over the next couple years, and already one permanent member (Michael Douglas, who will not be the Director, as mistakenly reported in some press accounts and here) has been hired. In an era when string theory has caused a backlash against mathematical and formal research at many physics departments, the Simons Center may be one of very few places where a physicist working at the boundary of mathematics and physics will be able to find employment. To get some idea of how dramatic the situation is this year, with only “phenomenologists” and “cosmologists” getting hired into tenure-track positions, take a look at the Theoretical Particle Physics Rumor Mill.

Stony Brook played a very important role in the interaction of mathematicians and physicists around the topic of gauge theory, and many of the speakers at the symposium discussed this. Since his early work on Yang-Mills, Yang had been intrigued by the similarities between gauge theory and the Riemannian geometry of GR. He built up the ITP at Stony Brook, in the same building and at the same time as a great mathematics department focused on geometry was being built up by Simons. He discussed these similarities with Simons, who told him that gauge theory must be related to connections on fiber bundles and pointed him to Steenrod’s The Topology of Fibre Bundles. Yang didn’t get much out of that (not surprising, since Steenrod is purely topological, with nothing about connections and curvature), leading him to make the statement:

There exist only two kinds of modern mathematics books: ones which you cannot read beyond the first page and ones which you cannot read beyond the first sentence.

In early 1975 Simons gave a series of lectures to the physicists on differential forms, geometry and bundles, and some real communication between the two camps began. This led to Yang writing a paper that year with Wu, Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields, Phys. Rev. D12, 3845, that included the famous “Wu-Yang dictionary” explaining how to translate back and forth between mathematician’s and physicist’s language. The crucial example was the Dirac monopole, where the bundle (for a sphere enclosing the monopole) is what a mathematician would call the Hopf fibration. This was already becoming a hot topic among physicists, with the ‘t Hooft-Polyakov monopole having been discovered in 1974.

Is Singer visited Stony Brook in the summer of 1976 and talked to the physicists about gauge theories and geometry. In early 1977 he traveled to Oxford, where he, Atiyah and Hitchin began working on instantons, i.e. solutions of the self-dual Yang-Mills equations. Again the physicists had started this, with the discovery of the BPST (Belavin, Schwarz, Polyakov and Tyupkin) solution in 1975, followed by its use by ‘t Hooft, Polyakov, Jackiw, Rebbi, Callan, Dashen, Gross and others soon thereafter. In his 1977 Erice lectures on The Uses of Instantons, Sidney Coleman refers to the “classic part of the theory”:

“Classic”, in this context, means work done more than six months ago.

Atiyah and collaborators were to devote much of the next decade to work on gauge theories. In 1977 he also met Witten, and this began a long and fruitful collaboration. During the years after 1977 Witten would become by far the dominant figure in the subject.

At Stony Brook last Friday, I arrived just in time to catch a morning talk by Dennis Sullivan about the classification of 3-manifolds (he was speaking in place of Iz Singer, who wasn’t able to come due to a respiratory infection, but an e-mail from him was read to the audience). Yang made some short comments about a problem in condensed-matter physics. The afternoon featured three hour-long talks. The first, by Dijkgraaf on The Unreasonable Effectiveness of Physics in Mathematics was a talk for a general audience advertising some of the high points of how ideas originating in string theory have had influence in mathematics. The main example was the computation of the number of rational curves on a quintic. The talk was extremely polished, featuring very impressive graphics. He described the current situation of string theorists as being very enthusiastic about “emergent geometry”, but struggling for the right mathematical language to express these ideas. This is an interesting research program, but as far as I can tell it is one that has to some extent ground to a halt, with little progress in recent years, and Dijkgraaf’s talk being essentially the same one he has given on other occasions during the last 5-6 years.

Michael Douglas, who is joining Stony Brook next year as the first permanent member of the Simons Center, spoke on Physics and Geometry: past, present and future. He emphasized ideas about “branes” and non-commutative geometry that have been popular since the late 1990s, like Dijkgraaf giving a take on the question of what new sort of geometry string theory might be pointing to. He ended his talk with a sentiment that I would heartily agree with, that he thought the time had come for a deeper investigation into quantum field theory. Unfortunately it was in a context I find not so promising: he has been thinking about how to count quantum field theories as part of anthropic landscape research, and has realized that this is a pretty ill-defined question. His final remarks seemed to be designed to answer skeptics who have noticed that string theory has stopped making progress, noting that physicists like himself are always moving on to something new, and this something new might soon not be string theory. In answer to a question from the audience about what the LHC might tell us about string theory, he gave a defensive set of remarks about the testability of string theory.

The last talk of the symposium was Witten on the topic of Electric-magnetic duality on a half-space, and this was a breath of fresh air and an extremely impressive performance. He was discussing joint work with Davide Gaiotto that I wrote a bit about here, based on his recent series of lectures at the IAS, which you can follow in lecture notes from David Ben-Zvi and Sergei Gukov here. The Stony Brook talk was an extended version of one he gave recently at the Linde-fest, available here.

The talk began with a motivational example from d=2, with a 1-d boundary, of a duality in the QFT of real scalars, taking Dirichlet boundary conditions on one side of the duality to Neumann boundary conditions on the other. The next example was 4d U(1) gauge theory, with its electro-magnetic duality, again relating by duality Dirichlet boundary conditions and Neumann boundary conditions for the field-strength F at the boundary. Most of the talk was about his new work on the surprising ways in which duality is reflected in choices of boundary conditions in N=4 super Yang-Mills. He claimed this to be a physicist’s way of understanding geometric Langlands and its duality between D-modules and coherent sheaves, but ended after an hour without having much to say about these mathematical implications, (although he jokingly threatened to go on for another hour on this topic if people were willing to stay).

In response to a question, he noted that unfortunately there seemed to be no useful relation between this S-duality and the AdS/CFT duality of the theory that is the reason for its central importance in modern string theory and particle theory.

Witten’s talk ended the symposium on a high note. This summer he’ll be temporarily moving to CERN as a visitor for the next academic year, so he may be on-site there as, if all goes well, the first results come in from experiments at the LHC.