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Standard set up and review:

Let us consider $SU(N)$ SQCD with $N_f$ flavors as our electric theory (just like in Seiberg's paper) and also let $N_f \geq N$. This theory is completely Higgsed in the IR and we end up describing the low energy degrees of freedom using mesons $M$ and baryons $B, \tilde{B}$ out of the Higgsed (massless in the UV) degrees of freedom.

The idea is that we can describe the theory using $M, B, \tilde{B}$ in the IR. I thought that this is correct also when $N_f < 3N$, i.e., when the theory becomes confining. It turns out that there is a conformal window and the theory flows to an IR fixed point when $3N/2 < N_f < 3N$.

The resulting SCFT is strongly coupled and after passing through the conformal window lowering further $N_f$ the theory should be confining. But when we calculate the scaling dimension of the meson, for example, we find that it must be a free operator, and even the whole theory to be free. Thus this is not the correct description of the theory at the confining phase after the conformal window and we need to use a dual magnetic theory theory that is indeed free in that region.

My point:

Now, I do not understand very well this very last bit. My question is the following:

Why would the dual magnetic theory correctly describe the physics of the original strongly coupled IR electric theory in the region $N_f < 3N/2$?

The duality holds only within the conformal window, i.e. the electric SCFT is dual to the magnetic SCFT only within the window. So when one SCFT is very strongly coupled I understand why we can use the dual one. But why would that be the case outside the conformal window?

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  • $\begingroup$ I thought the duality held for all $N$. It's just that in the conformal window, the IR limit in both descriptions is a conformal field theory. Outside that window, we trade a confining theory in one description for a free one in the other. $\endgroup$ – user2309840 Oct 17 '17 at 4:36
  • $\begingroup$ @user2309840 Once $N$ is fixed the duality holds for sure if $3N/2 < N_f M 3N$. Now let us assume we start with a theory whose $N_f $ is not in this regime. If larger, the UV theory flows to a IR free theory. If $N_f < 3N/2$ and according to the beta function, it should be a confining theory in the IR. In what sense we exchange this theory with a free magnetic one? Electric and magnetic theories should be dual only when they flow in the IR fixed point INSIDE the conformal window. Outside they are not the same. $\endgroup$ – Marion Oct 17 '17 at 8:56
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Though this isn't exactly the language I speak, I have heard susy folks describe the far-IR limit of confining theories (like SQCD with $N+2 \leq F \leq 3N/2$) as a "free pion gas". That is, in the IR they expect the confined theory to approach a system of non-interacting massless composite particles, which is also the case for the IR limit of the dual magnetic theory you describe. Although we know that strongly coupled quarks and gluons are involved, once those are confined they're no longer part of the physical spectrum and hence no longer the correct degrees of freedom to think about in the far IR. (I am most familiar with this style of argument in the context of hep-th/9901109.)

Poking around I was able to find relevant statements in Section 10.4 of Terning's Modern Supersymmetry ("the dual theory leaves the conformal regime to become IR free at exactly the point where the meson of the original theory becomes a free field"), as well as Section 5.4 of hep-th/9509066 (which may be the source of the standard setup you summarize).

I believe the consideration of far-IR limits like those described above is also the sense in which Seiberg duality can be "extended" beyond the conformal window. In all cases the duality strictly applies only to the asymptotic low-energy limits of the electric and magnetic theories. Within the conformal window, these limits are interacting IR fixed points, while outside of the conformal window they are free theories of either elementary or composite massless particles. (Fig. 2.2 of 1611.04883 comes to my mind, though that review doesn't provide many details in the corresponding text.)

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