4.3 The Seiberg-Witten solution

Let us construct holomorphic functions a and aD satisfying these monodromies explicitly. Note that a holomorphic function is uniquely determined by its singularities. Therefore, if we find a candidate with the correct properties around the singularities and at infinity of its domain of definition, it is necessarily the correct answer itself, assuming that we identified the singularities correctly. Therefore, it suffices to construct a candidate and then check that it satisfies the conditions.

4.3.1 The curve

We first introduce two auxiliary complex variables x and z, and then we consider an equation

Σ : Λ2z + Λ2 z = x2 u. (4.3.1)

We consider this equation as defining a complex one-dimensional subspace of a complex two-dimensional space of x and z.7 As the equation changes as we change u, the shape of this subspace also changes. This complex one-dimensional object is called the Seiberg-Witten curve.8 A differential

λ = xdz z , (4.3.2)

called the Seiberg-Witten differential, plays an important role later.9

The space parametrized by z is important in itself. We add the point at z = to the complex plane of z, or equivalently, we regard z to be the complex coordinate of a sphere. We denote this sphere by C, and call it the ultraviolet curve of this system. The variable x as a function of z has four square-root branch points, see Fig. 4.6.

Figure 4.6: The ultraviolet curve C of pure SU(2) theory.

Figure 4.7: The sheets of the Seiberg-Witten curve Σ of pure SU(2) theory.

Then the curve Σ is a two-sheeted cover of C,

Σ 2 : 1C, (4.3.3)

see Fig. 4.7. We then draw two one-dimensional cycles A, B on the curves as shown in the figures, and we declare that

a = 1 2πi Aλ,aD = 1 2πi Bλ. (4.3.4)

Let us check that the functions a(u) and aD(u) thus defined satisfy physically expected properties. First, let us compute τ(a):

τ(a) = aD a = aDu au . (4.3.5)

The u derivatives can be computed in the following way:

a u =A uλ =Adz xz (4.3.6) aD u =B uλ =Bdz xz (4.3.7)

where the u derivative within the integral is taken at fixed z. The differential ω = dz(xz) is finite on Σ, even at apparently dangerous points z = 0, z = or at x = 0. For example, when x = 0, z c + cx2 for some constants c and c. Then dz(xz) (2cc)dx.

Figure 4.8: The Seiberg-Witten curve Σ of the pure SU(2) theory, when smoothed out, is a torus.

Given an open path on the curve Σ from a fixed point P0, we find a map from the endpoint of the path to another complex plane

t =P0Pω. (4.3.8)

As shown in Fig. 4.8, the curve Σ is mapped to a parallelogram in the complex plane, bounded by the lines which are the images of the cycles A and B. Now, any holomorphic mapping such as (4.3.8) preserves the angles. Therefore, the image of the cycle B is always to the left of the image of the cycle A. Then

τ(a) = aDu au = Bdz(xz) Adz(xz) (4.3.9)

takes the values to the left of the real axis, and therefore

Imτ(a) > 0, (4.3.10)

which guarantees that the coupling squared g2(a) is always positive. This complex number τ(a) is called the period or the complex structure of the torus.

4.3.2 The monodromy M

Let us check the curve (4.3.1) reproduces the monodromy we determined from physical considerations. Write the curve Σ as

z + 1 z = x2 Λ2 u Λ2. (4.3.11)

From this we see that when |u| Λ2, we find two branch points z± of the function x(z) around

z+ uΛ2,z Λ2u. (4.3.12)

We also have branch points at z = 0 and z = , and we take the branch cuts to run from z = 0 to z = z, and from z = z+ to z = .

We put the A-cycle at |z| = 1. Then the integral over it is very easy: x u around |z| = 1, and therefore

a = 1 2πi xdz z u. (4.3.13)

As for the B-cycle integral, the dominant contribution comes when the variable z is not very close to the branch points. The variable x can be again approximated by u a, and therefore

aD = 2 2πiz+zxdz z 2 2 2πi uΛ21adz z 8a 2πi log a Λ. (4.3.14)

From these two equations we find that a and aD defined via the curve Σ have the correct monodromy around u ,

M = 1 4 0 1 . (4.3.15)

By a more careful computation, we can explicitly find corrections to (4.3.14), or to its derivative τ(a). From the form of the curve (4.3.1), it is clear that the corrections can be expanded in powers of Λ2, but in fact they are given by powers of Λ4. We find

τ(a) = 8 2πi log a Λ + k=0ck Λ a4k (4.3.16)

where ck are dimensionless rational numbers. We now know the terms hidden as in (4.1.8). This expansion can be understood for example by introducing z̃ = Λ2z. Then the curve is z̃ + Λ4z̃ = x2 u, and we can compute a, ãD by considering Λ4z̃ as a perturbation to the leading-order form of the curve z̃ = x2 u. An efficient method to compute ck from the contour integral can be found e.g. in [43].

Let us interpret these corrections in the powers of Λ4. From (4.1.9), we know that the term Λ4k carries the phase eik𝜃UV where 𝜃UV is the theta angle. It corresponds to a configuration with instanton number k, as we learned in (3.2.6). This expansion explicitly demonstrates that the only perturbative correction to the low-energy coupling τ(a) is from the one-loop level, and there are non-perturbative corrections from the instantons. An honest path-integral computation in the instanton background should reproduce the coefficients ck. For the one-instanton contribution c1 this was done in [44]. It was later extended to all k in [745]. Summarizing, we see that various quantities are given by a combination of a one-loop logarithmic contribution plus instanton corrections. It is now known that they agree to all orders in the instanton expansion, thanks to the developments starting from [7].

4.3.3 The monodromies M±

Let us next study the monodromy around the strongly-coupled singularities. Taking a look at (4.3.11) again, it is clear that when z + 1z = ±2 we have a rather special situation. When u = 2Λ2, the two branch points collide at z± = 1, and when u = 2Λ2, they collide at z± = +1. These are the singularities u = ±u0 introduced in Fig. 4.4.

Let us study the behavior close to u = u0 = 2Λ2 as an example. We let u = 2Λ2 + δu. Then the branch points are at

z± 1 ±δu. (4.3.17)

The close up of the branch points z± and the cycles A, B are shown in Fig. 4.9.

Figure 4.9: Monodromy action on cycles around the monopole point.

When we slowly change the value of u around u = 2Λ2, two branch points z = z± are exchanged. This modifies the cycle A as shown in the figure, which is equivalent to the original cycle A minus the cycle B. The cycle B is clearly unchanged. Therefore we have

a a aD,aD aD (4.3.18)

or equivalently, the monodromy is

M+ = 1 01 1 , (4.3.19)

reproducing (4.2.9).

Let us study the physics at u = u0 = 2Λ2. We perform the S transformation (1.2.19)

a = aD,aD = a (4.3.20)

exchanging the electric and magnetic charges. These are given as functions of u by

a = c(u u0),aD = a 2πi log c(u u0) (4.3.21)

where c and c are two constants, from which we find

τD(a) = aD a + log a 2πi . (4.3.22)

Note that a sets the energy scale of the system. The result shows the same behavior as the running of the coupling of an 𝒩=2 supersymmetric U(1) gauge theory with one charged hypermultiplet, consisting of 𝒩=1 chiral multiplets (Q,Q̃). The superpotential coupling is then

d2𝜃QaQ̃. (4.3.23)


τD = 4πi gD2 + 𝜃D 2π, (4.3.24)

we find

gD 0 (4.3.25)

as we approach u u0. The mass of the quantum of Q is given by the BPS mass formula to be

mass of quantum of Q = |a| = |aD|. (4.3.26)

Therefore, we identify the charged chiral multiplet Q as the second quantized version of the monopole in the original theory. The monopoles, which were very heavy in the weakly coupled region, are now very light.

The behavior at u = u0 is easily given by applying the discrete R-symmetry (4.2.5). As we map by T2, we find that the very light particles now have electric charge n = 2 and magnetic charge m = 1, i.e. they are dyons. From these reasons, the point u = u0 = 2Λ is often called the monopole point, and the point u = u0 = 2Λ the dyon point.