This question is extension of Electro magnetic duality, Strong weak duality and N=4 super Yangmils which i asked before.
Here what i want to know is compare of QED and N=4 SYM in terms of electro-magnetic duality.
As i heard, In QED, there is no such electro-magnetic duality but N=4 SYM theory has such duality. I want to know why this is true, and what is the distinction of two theories in the viewpoint of electro-magnetic duality.
I found proper(?) information for this question.
QED has a running coupling constant, but in N=4 SYM is conformal invaraint, β=0.
Answer
relevant answer monopole might give the answer. i.e,
In the sense of electro-magnetic duality, we require magnetic monopoles (We assume B=g2πrr3 like E.) And see the duality g and e, B and E.
In QED we have no magnetic monopole. Considering Gerogi-Glashow model coupled with Higgs transforming adjoint representation for given gauge group. i.e SO(3), or SU(2). In the process of BPS saturation, we obtain 'thooft polyakov monopole which is a kind of magnetic monopole but without any singular behaviour.
By computing N=2 SYM and N=4 SYM theories, computing its Hamiltonian and BPS saturation we obtain exactly same as 'thooft polyakov monopole, (i.e, Georgi-Glashow model coupled with Hiigs), thus magnetic monopole and associated W boson exsits.
N=4 SYM, the magnetic monopole and W boson is in same supersymmetry short multiplet. Thus in N=4 SYM we see exact Monoten-Olive duality (em duality). (Note N=2 case they are in different multiplet... spin ...)
I noticed from original papers of H. Osborn "TOPOLOGICAL CHARGES FOR N = 4 SUPERSYMMETRIC GAUGE THEORIES AND MONOPOLES OF SPIN 1"
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