8.4 Nf = 2: the moduli space

We studied in Sec. 7.3 that U(1) gauge theory with two charge-1 hypermultiplets has a Higgs branch of the form 22. 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.

Figure 8.7: Quantum moduli space of the Nf = 2 theory.

Compare this with the classical moduli space of SU(2) theory with Nf = 2 flavors. The Coulomb branch is still described by u = trΦ22. When Φ = 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 22 22. We can visualize them as in Fig. 8.8.

Figure 8.8: Classical moduli space of the Nf = 2 theory.

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 22 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(4) SU(2)A ×SU(2)B, so that SU(2)A,B acts separately on the two copies of 22. Then, after non-perturbative correction, SU(2)A acts on the hypermultiplets at u = Λ22 and SU(2)B at u = Λ22. This is consistent with the action of the flavor parity exchanging u = ±Λ22, 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(2Nf) flavor symmetry. Here the spinor of SO(4) is the fundamental doublet of SU(2)A or SU(2)B, and they are indeed interchanged by the flavor parity. This is consistent with what we have found so far.

Figure 8.9: The moduli space of Nf = 2 theory when μ1 = μ2.

Before closing this section, let us discuss what happens when we turn on a small but nonzero μ = μ1 = μ2. This breaks the SO(4) = SU(2)A ×SU(2)B flavor symmetry to SU(2)A, say. Correspondingly, we can check that the two singularities sitting at the same point u = Λ2 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 22. The resulting structure is shown in Fig. 8.9. When μ is continuously made large, eventually the situation is better described as a special case of Fig. 8.2 with μ = μ1 = μ2. Namely, the gauge coupling at the scale μ is still very weak, and the classical Lagrangian analysis is valid. The superpotential is

(Qi1,Qi2) μ + a 0 0 a + μ Q̃1i,Q̃2i (8.4.1)

and therefore when a = μ, the components (Qi2,Q̃2i) for i = 1, 2 remain massless. The gauge group is broken from SU(2) to U(1), and we have two charge-1 hypermultiplets, producing a Higgs branch of the form 22.