"What i have question above is how we know N=4 SYM has such kind of duality."

The basic idea is that you make a change of variables in the Feynman integral, by mapping the field strength 2-form $F$ to its dual $*F$. Then you observe that, with respect to these new variables, the Feynman integral still describes a gauge theory, except with the coupling constant $g$ interchanged with $\kappa / g$ and the electric and magnetic observables interchanged.

To really make this argument work, you'd need to regularize and renormalize the two Feynman integrals. This amounts to introducing renormalization flows into the coupling constants and appropriately dressing the observables. In N=4 SYM, Seiberg's holomorphy arguments tell you that the renormalization flows are trivial; likewise, that renormalization doesn't significantly alter the observables. Which indicates that the naive argument actually works for N=4 SYM. I don't think the fine details on this argument have been written down anywhere, however. N=4 duality is still, strictly speaking, a conjecture.

The strongest argument that it's a true conjecture is due to Vafa, who observed that N=4 SYM arises as the low energy limit of both IIA and IIB string theory, compactified on the product $K \times T^2$ of an ALE space $K$ of type ADE and a 2-torus. The strength of the gauge coupling is proportional to the inverse volume of the 2-torus. String theory tells us there is an exact duality which relates IIA and IIB while switch $T^2$ with its dual. In the low energy gauge theory descriptions, this duality flips the electric and magnetic observables. Likewise, it switches the volume of $T^2$ with its inverse, so flips the gauge coupling.

I've also heard speculations the naive argument for duality can be carried out more or less exactly in certain lattice regularizaitons of N=4 SYM. Haven't seen a paper yet though.

This post imported from StackExchange Physics at 2015-12-14 21:40 (UTC), posted by SE-user user1504