8.5 Nf = 3

The curve for SU(2) theory with Nf = 3 flavors was guessed in (6.4.11):

Σ : (x μ̃1)(x μ̃2) z + 2Λ(x μ̃3)z = x2 u. (8.5.1)

We see that λ diverges at z = 0, 1, independent of u, and there are four branch points which move as u changes, see Fig. 8.10. The reason we put tildes above the mass parameters will become clear soon.

Figure 8.10: The curve of Nf = 3 theory.

Let us check the behavior when |u||μi|2,|Λ|2. Two branch points are at z O(1) and another branch point is at uΛ. We now put the A-cycle around |z| = c, where 1 c uΛ. Then we see that the integral is given as before by

a = 1 2πi Axdz z u. (8.5.2)

The B-cycle integral can be approximated by

aD 2 2πiuΛ1adz z 2 2πia log a Λ. (8.5.3)

From this we find

τ(a) = aD a = 2 2πia log a Λ, (8.5.4)

thus reproducing the field-theoretical one-loop computation (8.1.6).

Figure 8.11: The u-plane of Nf = 3 theory, for equal masses

When μ1 = μ2 = μ3 = μ and |μ||Λ|, the coupling at the scale μ is still small, and the classical analysis using the superpotential (8.4.1) is almost valid. We expect that around a μ, i.e. when u μ2, the gauge group SU(2) is broken to U(1) with three charge-1 hypermultiplets. This point on the u-plane counts as three singularities, since when μ1,2,3 are slightly different, they should be at three slightly different points u μi2. When |u||μ|2, the theory can be effectively described by pure SU(2) gauge theory, which have the monopole point and the dyon point. In total we expect five singularities on the u-plane, see Fig. 8.11.

We would like to study the massless case, μ = 0. Here, we cannot just set μ̃i = 0 in the curve (8.5.1).11 We already saw that, when Nf = 2, the vev u can mix with the one-instanton factor Λ2. Here, with Nf = 3, the one-instanton factor is Λ and it can mix with any neutral chiral dimension-1 operator. The curve makes only U(3) flavor symmetry manifest. The mass parameter corresponding to the U(1) flavor symmetry is neutral, chiral, and of dimension 1. Therefore there can be a mixing of the form

μ̃i = μi + cΛ (8.5.5)

where c is a constant. Here, we fix the untilded mass parameter μi to transform linearly under the Weyl symmetry μi ±μi of the SO(2Nf) = SO(6) flavor symmetry.

To determine c, we set

(μ̃1,μ̃2,μ̃3) = (μ + cΛ,μ + cΛ,μ + cΛ) (8.5.6)

and study the singularities in the u-plane. This is just the SO(6) flavor Weyl transform of the SU(3) flavor symmetric choice of masses, therefore three out of five singularities on the u-plane should still collide as in (8.11). By an explicit computation, one finds that this happens only when c = 1.

Figure 8.12: The u-plane of massless Nf = 3 theory

Finally we can set μ = 0. The curve is now

(x Λ)2 z + 2Λ(x Λ)z = x2 u. (8.5.7)

There is an u-independent branch point of x(z) at z = 1. Two other branch points move with z, and are at the solutions of

Λ2z2 Λ2z + u Λ2 = 0. (8.5.8)

The branch points collide when u = Λ2 or u = (54)Λ2:

The monodromy at infinity is

M = 1 1 0 1 . (8.5.9)

Denoting the monodromies around u = Λ2, u = (54)Λ2 by M+ and M, we have

M = M+M,M+ = 1 04 1 ,M = 11 4 3 11 0 1 . (8.5.10)

By going to the S-dual frame at u = Λ2, we find that the running of the dual coupling is

τ(E) = + 4 2πi log E (8.5.11)

where the scale is set by E (u Λ2). Comparing with (8.3.13), the low energy physics can be guessed to be a U(1) gauge theory, coupled either (i) to just one charge-2 hypermultiplets or (ii) to four charge-1 hypermultiplets.

Recall that the classical theory has a Higgs branch. The choice (i) does not have a Higgs branch at u = Λ2. It does not have one at u = (54)Λ2 either. The Higgs branch should be preserved by the quantum correction, and thus this choice is ruled out.

The choice (ii) does have a Higgs branch at u = Λ2. We have four charge-1 hypermultiplets coupled to the U(1) gauge multiplet. Then the complex dimension of the Higgs branch is 2 4 2 1 = 6. This is acted on by the SU(4) flavor symmetry rotating four hypermultiplets.

Classically, we have three hypermultiplets in the doublet of SU(2). Then the complex dimension of the Higgs branch is 4 3 2 3 = 6. This agrees with the computation above. Recall that three hypermultiplets in the doublet of SU(2) count as six half-hypermultiplets of SU(2) doublet, with SO(6) flavor symmetry. As SO(6) SU(4), we see that the symmetry of the Higgs branch also agrees. We should recall that the monopole in this theory transforms as the spinor representation of the SO(2Nf) flavor symmetry. In our case the spinor of SO(6) is the fundamental four-dimensional representation of SU(4). This is also consistent with our choice that at u = Λ2 there are four charged hypermultiplets electrically coupled to the dual U(1). By a more detailed analysis we can check that the Higgs branches agree as hyperkähler manifolds.