9.1 The curve as λ2 = ϕ2(z)

Let us consider SU(2) gauge theory with four doublet hypermultiplets with masses μ1,2,3,4. In the very high energy region, the one-loop running is given by (8.1.6), which is just

τ(a) = τUV . (9.1.1)

From this we learn a distinguishing feature of the Nf = 4 theory compared to the theories with less flavors: it has a dimensionless parameter τUV . When Nf < 4, the bare coupling τUV was combined with the scale ΛUV to form the dynamical scale Λ, which just set the overall scale of the theory.

Now suppose the gauge coupling is small at the ultraviolet. Equivalently, suppose τUV has a large positive imaginary part. Further suppose μ1,2,3,4 are all of the same order, μ. Then the coupling at the energy scale μ is small, and the semiclassical analysis is OK. We see that when a ±μi, or equivalently when u μi2, the low-energy limit is described by U(1) gauge theory with one charged hypermultiplet. Far below this scale, the theory is effectively the pure SU(2) theory, which has the monopole point and the dyon point. Then the u-plane schematically has the structure shown in Fig. 9.1.

Figure 9.1: The u-plane of Nf = 4 theory

When μ1,2,3,4 = 0, we can consider the R-symmetry with the charge assignment

R = 0 A1λλ 2 Φ , R = 1 ψI 0qI . (9.1.2)

This is not anomalous. Then the only sensible point to have a singularity in the u-plane is at the origin, where six singularities in the generic case collide, see Fig. 9.2.

Figure 9.2: The u-plane of massless Nf = 4 theory

The coupling is given by τUV everywhere,

a = u,aD = τUV aD. (9.1.3)

Therefore the monodromy M at infinity is just

M = 1 0 0 1 . (9.1.4)

It looks relatively uninteresting. We will see however that there is a lot of interesting physics going on when we study the dependence on τUV .

Figure 9.3: The curve of Nf = 4 theory

The curve of the Nf = 4 theory is given by

Σ : f (x̃ μ̃1)(x̃ μ̃2) z̃ + f (x̃ μ̃3)(x̃ μ̃4)z̃ = x̃2 u (9.1.5)

where f and f are complex numbers, whose ratio will eventually be related to τUV . The differential is λ̃ = x̃dz̃z̃. The reason for additional tildes will become clear later.

The structure of the function x̃(z̃) over the ultraviolet curve C which is a sphere with coordinate z is shown in Fig. 9.3. The differential λ̃ always diverges at z̃ = 0, , and at the two solutions z̃ = c1,2(f) of fz̃ + fz̃ = 1. These points do not move when u is changed, and shown in red blobs in the figure. There are four additional branch points where x̃(z) is finite, shown in black blobs.

Let us rewrite the curve in a more illuminating way. We first rescale the coordinate z to set f = 1. We then collect terms with the same power of x̃:

(1 z̃ f z̃)x̃2 x̃ = 0 (9.1.6)

where , are some complicated expressions, which readers should fill in. We divide the whole expression by (1 z̃ fz̃), and find

x̃2 x̃ = 0. (9.1.7)

We note that and have poles at the two solutions c1,2(f) of 1 z fz = 0. Here it is instructive to spell out , which is given by

= f (μ̃1 + μ̃2)z̃ + (μ̃3 + μ̃4)z̃ 1 z̃ fz̃ . (9.1.8)

Defining x = x̃ 2, we have

x2 = 0 (9.1.9)

where now has double poles at c1,2(f) due to the completion of the square. Instead of λ̃ = x̃dz̃z̃ we will use λ = xdz̃z̃ henceforth. Note that

λ̃ λ = 2 dz̃ z̃ = 1 2 f (μ̃1 + μ̃2)z̃ + (μ̃3 + μ̃4)z̃ 1 z̃ fz̃ dz̃ z̃ . (9.1.10)

This is independent of the Coulomb branch modulus u, and its residues are all linear combinations of μ̃i. We encountered in (6.4.8) a similar shift of λ by a one-form which is independent of u and whose residues are given by the mass terms only. Such shift only amounts to a re-definition of the flavor charge and the mass terms, and does not affect the physics, as discussed there.

Figure 9.4: A step in the derivation of the curve in the Gaiotto form

Now we define the coordinate z = z̃c1(f) so that the double poles are at z = q and 1 for |q| < 1, see Fig. 9.4. The final form of the curve is then:

λ2 ϕ2(z) = 0,ϕ2(z) = P(z) (z 1)2(z q)2 dz2 z2 (9.1.11)

where P(z) is a quartic polynomial, as can be seen by re-following the change of variables starting from (9.1.5). The explicit expression of P(z) in terms of u, f and μ̃i is not very important, however.

The quadratic differential ϕ2(z) has double poles at z = 0,q, 1,. To see this for z = , set w = 1z. Then dz2z2 = dw2w2. This has poles of order two when w = 0, i.e. when z = . We identify this dimensionless parameter q as a function of the UV coupling τUV .