Constructive field theory & N=4 super Yang-Mills

Question

Hello everyone! Is there any work on N=4 D=4 super Yang-Mills theory on the constructive field theory side? How is the  state of the art on that issue? Also, how is it's status on the integrability issues?

Thanks for any eventual answer. Paper references and a little of context on the subject will be most than welcomed.

Comments

Isn't it a "Hydrogen atom of QFT", about which David Gross is talking from time to time?

Answer 1

Here is my armchair answer:

It's true that in SUSY theories one can often compute many (perhaps even all, in this situation) quantities of interest but these calculations often rest either up a well-defined path integral formulation (which then localizes (pdf) to a sum of finite dimensional integrals thanks to SUSY) or operator-based formulation (as in the superconformal minimal models).

I think that if you have a formulation for quantum field theory like Martin Hairer's that takes the path integral seriously and works for traditional simple theories like scalar $\phi^4$ theory, then what you would want to do is first construct the path integral measure of N = 4 SYM and *then* show it localizes to a finite dimensional calculation.

My guess is that this will be a lot harder than constructing $\phi^4$ theory unless you can leverage SUSY from the beginning, considering it has so many more fields and exists at a critical dimension for the RG.