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I am brand new to this subject, so this will probably be a very stupid question, but I would appreciate any patient explanations.

S-duality is typically described as a relationship between two QFTs (I am ignoring string theory here) where one theory describing objects interacting with some characteristic strength or coupling constant $g$ is equivalent to a different theory containing different objects (which might be solitonic objects in the dual description, like monopoles) with characteristic interaction strength $1/g$. We can then use the $g \to 0$ limit in one theory to infer information about the strong-coupling dynamics of the dual.

However, according to the renormalization group ideas, any garden-variety QFT will undergo dimensional transmutation and the perturbation theory really ends up being in an effective coupling $g(\mu)$ which flows according to the RG. Even the most simplistic loop calculations in dimensional regularization reflect this, and from a more highbrow point of view, the theory is defined by its UV fixed point. From this point of view, when we look at asymptotically free theories like QCD or gauge theories, the correct question to ask is not "What happens when $g \to \infty$?" but "What does the theory flow to in the infrared?". While it is evident that the IR theory will have some kind of strong-coupling dynamics, how does S duality incorporate this idea?

I know that some examples like Seiberg duality are phrased in terms of the infrared behavior according to the RG, but it seems that the general paradigm is still weak/strong coupling. But even then, exactly what coupling are we referring to? In the classical theory there is just "the coupling", but as soon as you go to the quantum theory dimensional transmutation destroys this, right? Any help would be appreciated.

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In QFT, the notion of couplings is only reliable in the perturbation setting. So it doesn't make much sense to have a strong/weak duality, unless the theory is actually free or integrable. For example, in N=4 SYM, you have S duality which sends the coupling $\tau$ to $-1/\tau$.

However, as you said, in the interacting QFT, dualities are normally meant to be infrared dualities with Seiberg duality being one of the examples, which refers to the phenomenon that two distinctly looking theories in the UV flow to the same infrared fix point, in other words, they belong to the same universality class. And it doesn't make much sense to talk about couplings here.

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  • $\begingroup$ What do you mean, that couplings are only "reliable" in perturbation theory? They are also employed in lattice computations. $\endgroup$ – Mitchell Porter Aug 17 '18 at 23:02
  • $\begingroup$ You are right. I was only talking about continuous quantum field theories, usually described by Lagrangians. And there are many dualities in lattice system and statistical mechanics as well, which I don't know well enough to say anything smart. $\endgroup$ – Jing Aug 18 '18 at 2:24
  • $\begingroup$ Thanks for the answer. After doing a bit more reading, I can see this point more clearly. I guess the best way to think about this is that the theory exists abstractly by itself, and the only technical tools we really have are Lagrangians and perturbation theory/RG. The theory and its dual, both in their respective expansions, are like “local coordinates” on the moduli space of the theory, as Dijkgraaf puts it. $\endgroup$ – Spencer Tamagni Aug 18 '18 at 6:41

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