8.4 $N_f=2$: the moduli space

We studied in Sec. 7.3 that $U\left(1\right)$ gauge theory with two charge-1 hypermultiplets has a Higgs branch of the form ${ℂ}^2∕{\mathbb{Z}}_2$. Together with the $u$-plane describing the Coulomb branch, we can visualize the totality of the supersymmetric vacuum moduli space as shown in Fig. 8.7. Note that two singular points on the $u$-plane where Higgs branches meet are exchanged by the discrete R-symmetry and the flavor parity.

Compare this with the classical moduli space of $SU\left(2\right)$ theory with $N_f=2$ flavors. The Coulomb branch is still described by $u=tr{\Phi }^2∕2$. When $\Phi =0$, we can go to the Higgs branch; we studied this system in Sec. 7.3 too, where we saw that it is given by ${ℂ}^2∕{\mathbb{Z}}_2\wedge {ℂ}^2∕{\mathbb{Z}}_2$. We can visualize them as in Fig. 8.8.

In Sec. 7.1, we argued that the local metric of the Higgs branch cannot be corrected by the gauge dynamics. We see here that the quantum dynamics can still split the point where two copies of ${ℂ}^2∕{\mathbb{Z}}_2$ meet the $u$-plane; the argument in that section is not applicable at the points where the metric is singular.

Recall that there is a flavor symmetry $SO\left(4\right)\simeq SU{\left(2\right)}_A\times SU{\left(2\right)}_B$, so that $SU{\left(2\right)}_{A,B}$ acts separately on the two copies of ${ℂ}^2∕{\mathbb{Z}}_2$. Then, after non-perturbative correction, $SU{\left(2\right)}_A$ acts on the hypermultiplets at $u^{\prime }={\Lambda }^2∕2$ and $SU{\left(2\right)}_B$ at $u^{\prime }=-{\Lambda }^2∕2$. This is consistent with the action of the flavor parity exchanging $u^{\prime }=\pm {\Lambda }^2∕2$, recall (8.3.7).

We learned in Sec. 1.3 that the monopole in this type of theories transforms as the spinor representation of the $SO\left(2N_f\right)$ flavor symmetry. Here the spinor of $SO\left(4\right)$ is the fundamental doublet of $SU{\left(2\right)}_A$ or $SU{\left(2\right)}_B$, and they are indeed interchanged by the flavor parity. This is consistent with what we have found so far.

Before closing this section, let us discuss what happens when we turn on a small but nonzero $\mu ={\mu }_1={\mu }_2$. This breaks the $SO\left(4\right)=SU{\left(2\right)}_A\times SU{\left(2\right)}_B$ flavor symmetry to $SU{\left(2\right)}_A$, say. Correspondingly, we can check that the two singularities sitting at the same point $u=-{\Lambda }^2$ splits into two, by directly performing the analysis of the discriminant of the curve. We are still left with one point on the $u$-plane where two singularities still collide, and the local monodromy around it is unchanged from $M_{+}$. There, we have a Higgs branch of the form ${ℂ}^2∕{\mathbb{Z}}_2$. The resulting structure is shown in Fig. 8.9. When $\mu$ is continuously made large, eventually the situation is better described as a special case of Fig. 8.2 with $\mu ={\mu }_1={\mu }_2$. Namely, the gauge coupling at the scale $\mu$ is still very weak, and the classical Lagrangian analysis is valid. The superpotential is

and therefore when $a=\mu$, the components $\left(Q_i^2,{\tilde{Q}}_2^i\right)$ for $i=1,2$ remain massless. The gauge group is broken from $SU\left(2\right)$ to $U\left(1\right)$, and we have two charge-1 hypermultiplets, producing a Higgs branch of the form ${ℂ}^2∕{\mathbb{Z}}_2$.