# Seiberg duality and IR fixed point

This question is related with Seiberg duality for $SU(N)$ gauge theory which states a duality between electric theory, $SU(N_c)$ gauge theory with $N_f$ flavors is dual to its magnetic theory, $SU(N_f-N_c)$ theory with $N_f$ flavors with an additional gauge invaraint massless field.

This duality valid for conformal window $\frac{3N_c}{2} < N_f < 3N_c$ where both electric and magnetic theory have interacting IR-fixed point.

Here i have question about $IR$ fixed points.

The beta function of electric theory(SQCD : $SU(N_c)$ gauge theory) is written as \begin{align} \beta(g) = -\frac{g^3}{16\pi^2} \frac{3N_c - N_f + N_f \gamma(g^2)}{1- N_c \frac{g^2}{8\pi^2}} \end{align} Thus we see $\beta>0$ for $N_f > 3 N_c$ for electric theory. And using above results for magnetic theory, $\beta>0$ for $\frac{3N_c}{2} > N_f$.

1. I want to know how we choose proper conformal window, and how we know that these window have interacting IR-fixed point rather than UV fixed point.

I know $\beta<0$ theory is asymptotically free theory which is a key properties of describing non-abelain gauge theories. And UV is related with for High energy, and IR is related with for Low energy dynamics.

1. (These might be part of question 1) And can anyone explain the difference of UV fixed point and IR fixed point?

1) If the theory is asymptotically free, that means the coupling gets weak at high energies, in the ultraviolet (UV), but it also means that the coupling gets strong at low energies, in the infrared (IR). You have pretty much answered your own question (1) above, by showing that the theory flows to strong coupling in the IR in the window ${3N_c \over 2} < N_f < 3N_c$ and correspondingly weak coupling in the UV, in both the magnetic and electric descriptions of the theory. There is much more to discuss here. The lecture notes seem to be a standard reference.