4.1 One-loop running and the monodromy at infinity

The pure SU(2) theory contains only an 𝒩=2 vector multiplet for the SU(2) gauge group, with its Lagrangian given by (2.1.11). For reference we reproduce it here:

L = Imτ 4π d4𝜃trΦe[V,]Φ +d2𝜃 i 8π τtrWαWα + cc. (4.1.1)

A supersymmetric vacuum is classically characterized by the solution to the D-term constraint

[Φ, Φ] = 0. (4.1.2)

This means that Φ can be diagonalized by a gauge rotation. Let

Φ = diag(a,a). (4.1.3)

Roughly speaking, the gauge coupling τ runs from a very high energy scale down to the energy scale a according to the one-loop renormalization of the SU(2) theory. Then the vev a breaks the gauge group SU(2) to U(1). There are massive excitations charged under the unbroken U(1), but they will soon decouple, and the coupling remains almost constant below the energy scale a. This evolution is shown in Fig. 4.1.

Figure 4.1: Schematic drawing of the running of the coupling.

Let us describe it slightly more quantitatively. Our normalization of the U(1) Lagrangian and the gauge coupling was given in (1.2.7) and (1.2.15), which we reproduce here:

1 2e2FμνU(1)FμνU(1) + 𝜃 16π2FμνU(1)F̃μνU(1),andτU(1) = 4πi e2 + 𝜃 2π. (4.1.4)

In the broken vacuum, the low-energy U(1) and the high-energy SU(2) are related as in (1.3.3), which we also reproduce here

FμνSU(2) = diag(FμνU(1),FμνU(1)). (4.1.5)

Plugging this in to the high-energy Lagrangian (4.1.1) and comparing the definitions of τs, we find

τU(1) = 2τSU(2). (4.1.6)

This relation gets modified by the quantum corrections.

Let us denote by τ(a) the low-energy coupling of the U(1) gauge field when the vev is given by (4.1.3), and by τUV the high-energy coupling of the SU(2) gauge field at the high-energy renormalization point ΛUV . The one-loop running (3.1.6) then gives

τ(a) = 2τUV 8 2πi log a ΛUV + (4.1.7) = 8 2πi log a Λ + (4.1.8)

where we defined

Λ4 = ΛUV 4e2πiτUV . (4.1.9)

The dual variable aD can be obtained by integrating (4.1.8) once, and we find

aD = 8a 2πi log a Λ + . (4.1.10)

As long as we keep |a||Λ|, the coupling τ(a) remains weak, and the computation above gives a reliable approximation.

A gauge-invariant way to label the supersymmetric vacua is to use

u = 1 2trϕ2 = a2 + (4.1.11)

where are quantum corrections. Let us consider adiabatically rotating the phase of u by 2π:

u = ei𝜃|u|,𝜃 = 0 2π (4.1.12)

We have a a. From the explicit form of aD we find aD aD + 4a. We denote it as

(a,aD) (a,aD) 1 4 0 1 . (4.1.13)

The mass formula of BPS particles is

M = |na+maD| = (a,aD) nm . (4.1.14)

Therefore, the transformation (4.1.16) can also be ascribed to the transformation of the charges:

nm 1 4 0 1 nm . (4.1.15)

We call this matrix

M = 1 4 0 1 (4.1.16)

the monodromy at infinity. The situation is schematically shown in Fig 4.2. The space of the supersymmetric vacua, parametrized by u, is often called the u-plane.

Figure 4.2: Monodromy at infinity.

In our argument, the matrix (4.1.13) could have had non-integral entries, as we read the matrix elements off from an approximate formula of a and aD. However, the transformation (4.1.15) should necessarily map integral vectors to integral vectors, which guarantees that the matrix (4.1.15) is integral. Not only that, this transformation is just a relabeling of the charges and should not change the Dirac pairing

nmmn = det nnm m (4.1.17)

which measure the angular momentum carried in the space when we have two particles with charges (n,m) and (n,m), respectively. A transformation given by

nm M nm (4.1.18)

affects the Dirac pairing as

det nnm m det M det nnm m . (4.1.19)

Therefore, M should necessarily has unit determinant. A 2 × 2 integral matrix with unit determinant is called an element of SL(2, ). It is reassuring that the matrix (4.1.16) satisfies this condition.