d^4\theta
M_j^\dagger M_j$, rescaled by powers of $s_0$, $\mu$. The magnetic superpotential is: The auxiliary scale $\mu$ (mentioned in the introduction) is needed for dimensional reasons . In the next sections we will see that it is very natural to normalize $Y$ such that $\bar s_0=-s_0$; this choice leads to all the coefficients in the sum over $j$ in  being $1$, as indicated.

The transformation properties of the magnetic matter fields, $q$, $\tilde q$, $Y$ and $M_j$ under the global symmetries are:

The auxiliary scale $\mu$ in  is actually not an independent parameter. The scale of the electric theory $\Lambda$, that of the magnetic theory $\bar \Lambda$, and $\mu$ satisfy the scale matching relation described in the introduction: It is easy to check that  is invariant under all global symmetries including those under which $s_0$, $\mu$, $\Lambda$ transform. It is in fact uniquely fixed by these symmetries. As in , the scale matching condition  implies that when the electric theory is weakly coupled the magnetic one is strongly coupled, and vice versa. Differentiating the actions with respect to $\Lambda$ holding $s_0$ and $\mu$ fixed, the electric gauge field strength $W_\alpha^2$ is related to the magnetic one $\bar W_\alpha^2$ by the 