8.2 Nf = 2: the curve and the monodromies

Let us start with the SU(2) with Nf = 2 flavors. The Seiberg-Witten curve was guessed in (6.4.6), which we repeat here:

Σ : 2Λ(x μ1) z + 2Λ(x μ2)z = x2 u (8.2.1)

with the Seiberg-Witten differential λ = xdzz. The ultraviolet curve C is still just an S2, shown in Fig. 8.1.

Figure 8.1: The curve of Nf = 2 theory.

When |u||Λ2|,|μi|2, we can estimate the line integrals easily. First, we put the A-cycle at |z| = 1. Using x u around there, we have

a = 1 2πi Axdz z u. (8.2.2)

The positions of the branch points of x(z) on the curve C can also be easily estimated: there are two around z uΛ and two more around z Λu. Then we see

aD = 1 2πi Bxdz z 2 2 2πi uΛ1adz z 4 2πia log a Λ. (8.2.3)

From this we can compute τ(a) = aDa, which reproduces the one-loop running (8.1.6).

Let us next study the structure of the singularities on the u-plane. When μ1,2 Λ, the gauge coupling is rather small around the energy scale μ1,2. Then we expect that when

u μi2for i = 1, 2 (8.2.4)

one component of (Qi,Q̃i) become very light, producing a singularity. Below the scale of μ1,2, the theory is effectively equivalent to pure SU(2) theory, which should have two singularities where either monopoles or dyons are very light. In total we expect that there are four singularities on the u-plane, see Fig. 8.2.

Figure 8.2: The u-plane for Nf = 2.

This structure can be checked starting from the curve (8.2.1) by studying its discriminant, which is left as an exercise to the reader. Here we study the massless case μ1 = μ2 = 0 in detail. The Seiberg-Witten curve for the massless case is simply

x2 2Λ(z + 1 z)x u = 0. (8.2.5)

Then we see that four branch points of x(z) meet in pairs when u = 0 or u = 4Λ2 as depicted in Fig. 8.3.

Figure 8.3: The curve of Nf = 2 theory degenerates when u = 0 or u = 4Λ2.

Explicitly, when u = 0 they meet at z = ±i and when u = Λ2 they meet at z = ±1. There are no other singularities on the u-plane, so we see that when μ1 = μ2 = 0 the u-plane has the structure shown in Fig. 8.4.

Figure 8.4: The u-plane for massless Nf = 2.

At each of u = 0, u = Λ2, two pairs of branch points of x(z) collide. This means that each of u = 0, u = Λ2 should be considered as two singularities on the u-plane. This situation was shown in Fig. 8.4 by putting almost overlapping two blobs at u = 0, Λ2. In total there are four singularities, matching what we found above for μ1,2 Λ. Let us denote the monodromies around various closed paths as shown in Fig. 8.4.

The monodromy M at infinity can be found from the explicit form of a, aD found in (8.2.2), (8.2.3) to be

M = 1 2 0 1 . (8.2.6)

The monodromy M+ around u = 0 can be found by following the motion of the branch points when we make a slow change along the path u = 𝜖ei𝜃 for a very small 𝜖 from 𝜃 = 0 to 𝜃 = 2π.

Figure 8.5: Monodromy action on cycles for Nf = 2.

The pair of branch points exchanges positions as shown in Fig. 8.5. We see that the B cycle remains the same, while A is sent to A 2B, thus generating

M+ = 1 02 1 = ST2S1. (8.2.7)

We see that

M = M+M (8.2.8)


M = TM+T1. (8.2.9)

Before proceeding, it is instructive to use another description of the curve to find the same u-plane structure. The curve was given in (6.4.7). When massless, this just becomes

x2 z + Λ2z = x2 u. (8.2.10)

The branch points collide when u = 0 or u = Λ2 as before, but it looks rather different on the ultraviolet curve, as shown in Fig. 8.6.

Figure 8.6: The curve of Nf = 2 theory degenerates when u = 0 or u = Λ2, the second description.

Note that λ diverges at z = 0 and z = 1 independent of u. One branch point of x(z) on the ultraviolet curve moves as u changes, and this point hits either z = 0 or z = 1 at u = 0 or u = Λ2 respectively. It is left to the reader to recover the monodromies M± from this latter view point.