On self-duality of N=4 super Yang Mills theories

Question

I am looking at S-duality a bit, and was wondering if anyone had the answer to the following question. It is known that the supersymmetric Yang-Mills theory in 4 space dimensions is self-dual with gauge group or U(N) or SU(2). But is it known whether there are more Yang-Mills theories that are self-dual? On the other hand, is it known that some are not?

Any help much appreciated!

This post imported from StackExchange Physics at 2014-08-05 15:04 (UCT), posted by SE-user user30564

Comments

Isn't the Langlands dual of $SU(2)$ $SO(3)$?

Answer 1

In general, S-duality of N=4 super Yang-Mills in 4 dimensions exchanges a theory of gauge group $G$ with a theory of gauge group $G^L$ where $G^L$ denotes the Langlands dual group of $G$ (first introduced in a physical context by Goddard, Nuyts, Olive). In particular, S-duality is a self-duality if and only if $G = G^L$. This is the case for $G=U(N)$ but not for $G=SU(2)$ because in this case $G^L=SO(3)$ (as suggested by  Ryan Thorngren in its comment to the question).

Comments

Could you expand your answer a little bit more please? Or give some reference maybe?

Thanks a lot.