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!

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    $\begingroup$ Generally, S-duality of SYM4 exchanges a theory with gauge group G with a theory of the Langlands dual of G, as described in this discussion. $\endgroup$ – Dilaton Aug 5 '14 at 20:18

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