Savdeep Sethi from University of Chicago was the duality seminar speaker yesterday, despite the holidays.
His title was:
A curious truncation of N=4 supersymmetric Yang-Mills
He chose to present a blackboard talk, even though PowerPoint was another option. The talk focused on his recent paper with Anirban Basu and Michael Green. Savdeep is, of course, a very interesting speaker. OK, what's the problem?
Choose a correlator of some operators in N=4 super Yang-Mills, and try to calculate the derivative with respect to tau, the complexified coupling constant. You will obtain a term that encodes the explicit dependence of the operators on tau, but you will also obtain a "dynamical" term arising from the fact that the action is tau-dependent. The latter can be written as a correlator of the previous operators with one more insertion added. This extra insertion is essentially the integral of the Lagrangian.
A funny feature is that one can take the operator called the Lagrangian and construct all possible supersymmetric variations of it. One obtains a short multiplet that contains a lot of other "special" operators such as the R-symmetry currents. Savdeep played with these special operators in many ways and he expressed various correlators as Eisenstein series (modular forms). Also, he studied how these correlators transform under SL(2,Z). His most favorite correlator was a correlator of 16 fields. He also considered a "gauge connection" on the space of complex coupling constants which was interesting on its own right.
As far as I understand, his curious truncation is the gauge theoretical counterpart of the truncation of the string theory to supergravity (operators whose dimensions remain finite in the large N limit). Yes, I probably misunderstand something because this does not seem too curious - in fact, this is how AdS/CFT started. Moreover, we (with Mina and perhaps others) were a bit confused whether you can really and consistently truncate string theory into supergravity in this way. All of us understand that classically - or equivalently, with non-planar diagrams neglected - one has a beautiful theory of supergravity, but we thought that it's not possible to define any quantum theory, or equivalently a theory with a finite 't Hooft's coupling, that would contain supergravity only. Correspondingly, such a truncation should be invalid in the gauge theory, too.
Because I don't understand this basic point too well, I must redirect all interested readers to their paper. If you find this stuff interesting, reading this efficient paper seems as a good investment of your time.