Witten and Langlands

Edward Witten was speaking about the Langlands duality. The huge lecture hall "Science Center D" was completely full. Full of mathematicians and theoretical physicists. And probably many others.

It may be fair to say that Edward Witten is probably not only the holy father of theoretical high energy physics - using the term invented by Friedwardt Winterberg - but quite possibly also the most respected mathematician in the world as of 2005. This fact makes such a talk a special event; but that's not the only reason.

Of course, I apologize to all who feel that I am not fair if I put their name below Witten's name in the list of mathematicians. If you have any doubts that Edward Witten, a leading figure of string theory, has figured out important insights that have been experimentally verified, open the "fast comments" and see examples from a reader whose name happens to be Peter Woit.

The talk was addressed both to the mathematical as well as physical audiences. Recently I discussed the Langlands duality a bit and included a link to a written version of Witten's talk in Stony Brook that was similar to the talk today which is why I decided not to describe the talk in detail right now.

Update:

On Friday, Edward Witten gave another talk in "Science Center C" about various issues related to many holomorphic properties of the Bogomolnyi equations in Yang-Mills theory (which included correlators of 't Hooft loops) and about topological twists of the N=4 Yang-Mills theory.




There are three types of topological twists of this maximally supersymmetric Yang-Mills theory. Some of the postmodern "physics critics" like to admire topological field theories but they often say that they have no connections with string theory. Witten's talk has shown many things - and one of them is the fact that these "lit crits" have not only an insufficient understanding of string theory but also an outdated idea about topological field theory. In fact, the only way how you may easily remember what are the three topological twists of the N=4 Yang-Mills theory is to realize that they can be constructed as descriptions of D3-branes on supersymmetric (calibrated) 4-cycles of manifolds of three types of special holonomy:

• spin(7) - special holonomy - spin bundles are relevant
  
• G2 - Omega2+ is relevant
  
• SU(4) - Calabi-Yau four-folds

Each kind of manifold comes with its natural calibration, preserved supersymmetry, and be sure that you can always see that the theories "know" that they come from string theory and you may always find the total of 10 dimensions if you count it properly. In the last case that is relevant for the Langlands duality, you find a CP1-worth of theories. The previous two twists are analogous to the Donaldson theory.

String theory is not only the only natural organizational scheme for quantum gravity and for the explanation of the origin of fields in particle physics and relations between gauge theories, gravitational theories, and other types of physics; it is also the natural framework that gives you this portion of mathematics connected with topological field theories. Those who deny that string theory has become necessary as unifying framework for a full understanding of physics of gauge theories, gravity, topologically twisted field theories, and their relationships are analogous to those who deny that there are reasons to believe the evolutionary framework in biology.

There have been many interesting things in the talk but a more accurate report would require me to use some formulae and maybe pictures and it just seems too difficult at this moment. Neither of these things can stop Jacques Distler whose nice description I recommend you all of those who followed the abstract above.