5.2 The curve

Let us now construct the holomorphic functions a(u), aD(u) satisfying the monodromies determined above. It is again done by using the Seiberg-Witten curve, which is given in this case by

Σ : 2Λ(x μ) z + Λ2z = x2 u (5.2.1)

with auxiliary complex variables z and x, together with the Seiberg-Witten differential

λ = xdz z . (5.2.2)

We dropped the subscript 1 from Λ to lighten the notation.

Figure 5.6: The ultraviolet curve of SU(2) Nf = 1 theory.

Again, we add a point z = and regard z as a complex coordinate on the sphere C. This is the ultraviolet curve. The variable x is now a function on it, see Fig. 5.6. Note that z = 0 is no longer a branch point; indeed, the local behavior of x there is now

x+ 2Λ z μ + O(z), (5.2.3) x + μ + O(z). (5.2.4)

Note also that λ has a residue μ at z = 0. The curve Σ is a two-sheeted cover of C shown in Fig. 5.7.

Figure 5.7: The sheets of the Seiberg-Witten curve of SU(2) Nf = 1 theory.

We define cycles A and B as shown, and then the functions a(u) and aD(u) are given by

a = 1 2πi Aλ,aD = 1 2πi Bλ. (5.2.5)

Figure 5.8: The smoothed-out torus of the curve of the SU(2) Nf = 1 theory.

The proof Imτ(a) > 0 goes exactly as in the pure case. The curve Σ can be mapped to a parallelogram within a complex t plane by P0Pλu =P0Pdz(xz), see Fig. 5.8. The poles with residues ± μ of λ are denoted explicitly in the figure. When a closed cycle L on the torus winds the A cycles n times, B cycles m times, and the poles f times, the integral of λ is then

1 2πi Lλ = na + maD + fμ, (5.2.6)

just as in the BPS mass formula (5.1.4).

Let us check that the curve correctly reproduces the running of the coupling in the weakly-coupled region. For simplicity, set μ = 0, and assume |u||Λ|. We put the A cycle at |z| = 1. We easily find

1 2πi Axdz z u (5.2.7)

as before. As for the B integral, two branch points are around z Λu and one branch point is around z uΛ2. The dominant contribution to the integral is then

1 2πi Bxdz z 2 2πiuΛ2Λuadz z = 6 2πia log a Λ. (5.2.8)

Then we find

τ(a) = aD a = 6 2πi log a Λ, (5.2.9)

reproducing the running (5.1.6).

Let us next check that the curve correctly reproduces the singularity structure on the u-plane. The branch points of the function x(z) can be determined by studying when the equation of Σ, given in (5.2.1), has double roots. The equation for the branch points is given by

z3 + uz2 Λ2 2μz Λ + 1 = 0. (5.2.10)

The singularity in the u-plane is caused by two of the branch points of x(z) colliding in the ultraviolet curve C with the coordinate z. This condition can be found by taking the discriminant of the equation of z above, giving

u3 μ2u2 + 9Λ3μu + 27 4 Λ6 8Λ3μ3 = 0. (5.2.11)

When μ = 0, this equation simplifies to u3 + 27 4 Λ6 = 0, giving the solutions

u = cΛ2,e2πi3cΛ2,e4πi3cΛ2 (5.2.12)

for a constant c, reproducing Fig. 5.3.

When |μ||Λ|, the equation (5.2.11) can be solved by making two separate approximations. Assuming u is rather big, we can truncate the equation to just u3 μ2u2 0, finding a singularity at

u μ2. (5.2.13)

Next, assuming u is rather small, we find μ2u2 8Λ3μ3 0 giving

u ±8Λ 3 μ. (5.2.14)

Together, they reproduce Fig. 5.2. From this, we find that the effective pure SU(2) theory in the region |u||μ| has the dynamical scale

Λ02 Λ 3 μ. (5.2.15)

This agrees with what we saw in (5.1.11).

It is instructive to study another way to derive the singularity at u μ2 from the curve. We would like to take the approximation |Λ||μ|. To facilitate to take the limit, we introduce z̃ = zΛ in (5.2.1) and find

2(x μ) z̃ + Λ3z̃ = x2 u. (5.2.16)

Now the limit is easy to take: we just find

2(x μ) z̃ = x2 u. (5.2.17)

Then it is clear that when u = μ2, the equation can be factorized to

(x μ)(x + μ 2 z̃) = 0, (5.2.18)

therefore it represents two sheets intersecting at a point. When uμ2, two sheets are connected smoothly. The change is schematically shown in Fig. 5.9. We learned that the singularity at u μ2 arises essentially from the structure 2Λ(x μ)z in the curve.

uμ2 u = μ2

Figure 5.9: The schematic change in the Seiberg-Witten curve when u μ2.