The past few weeks I’ve often been going down to the IAS in Princeton on Thursdays to hear talks given as part of the special program there this semester in mathematics. These talks included a series of five talks by Witten; notes from David Ben-Zvi and Sergei Gukov are available here.
The first three talks concentrated on the existence of a very special superconformal six-dimensional QFT, and information that could be derived from what is known of its properties. Such a theory is an inherently quantum object, lacking a usual sort of classical limit or Lagrangian formulation. Witten compares it to the holomorphic conformal field theory that appear as “square roots” of the WZW model. These field theories are closely related to the representation theory of loop groups and at the core of a several important mathematical developments of the last couple of decades. The mathematical significance of the six-dimensional theory remains much more mysterious, and Witten argues that understanding this mystery is a very worth goal for both mathematicians and physicists. . For more about this, see the article Conformal Field Theory in Four and Six Dimensions, based on his lecture at the Oxford conference in honor of Graeme Segal’s 60th birthday back in 2002. Taking the six dimensions to be the product of a torus and a four dimensional space, the existence of such a superconformal six dimensional theory implies an SL(2,Z) symmetry of N=4 Super-Yang-Mills on the four dimensional space. This includes the famous Olive-Montonen non-abelian electric-magnetic symmetry that is responsible for Langlands duality in Witten’s 4d QFT approach to Geometric Langlands.
The last two talks of the series dealt with a different topic, boundary conditions in N=4 SYM. Taking this theory on the half-space with boundary conditions, one can ask about the implications of non-Abelian electric-magnetic duality for these boundary conditions. Witten has recently been working on this subject with Davide Gaiotto, he’ll be talking about it later this month at a Stony Brook symposium in honor of C. N. Yang and Jim Simons, and I assume a paper will appear sooner or later. In his IAS lectures Witten was talking to mathematicians and arguing that “universal” operations (ones that can be done uniformly for all Riemann surfaces) in geometric Langlands should all come from the properties of these boundary conditions. Note that in this work what appears is the full N=4 SYM theory, not just the topological twisted version. This theory plays a central role in AdS/CFT, so if new information about its physics arises from this study, this should be directly interesting for physics, although Witten did not discuss this in his talks.
The two sorts of boundary conditions that get related by duality are analogs of Neumann and Dirichlet boundary conditions. The Neumann boundary conditions involve superconformal 3d QFTs, examples of which were studied by Intriligator and Seiberg in their 1996 paper Mirror Symmetry in Three Dimensions. Witten has previously worked on this kind of thing in the Abelian case, see here.
During these visits to the IAS I got the chance to meet Meng-Chwan Tan, who is there in the Physics group this year. He has been working on a different QFT approach to geometric Langlands, one that is purely two-dimensional and based in conformal field theory, using (0,2) sigma models on flag manifolds, and has just posted a the revised for publication version of his paper on the subject here. This is much closer to the approach to geometric Langlands via conformal field theory that Edward Frenkel has described here.
In other geometric Langlands news, there was a workshop on Homological Mirror Symmetry recently in Miami, with notes from many of the talks available here (and a blog posting by Joel Kamnitzer here). And there’s another one (notes here from David Ben-Zvi) going on this week at the IAS. I better stop now, go and get some sleep so I can head down there tomorrow morning to catch the last day of it.