5.1 Structure of the u-plane

5.1.1 Schematic running of the coupling

In terms of 𝒩=1 chiral multiplets, the hypermultiplet consists of two SU(2) doublets Qa and Q̃a where a = 1, 2 is the SU(2) index. There is an 𝒩=1 superpotential

W = QΦQ̃ + μQQ̃ (5.1.1)

where μ is the bare mass of the hypermultiplet. Classically, Φ = diag(a,a) together with Q = Q̃ = 0 still gives supersymmetric vacua. With nonzero a, the gauge group is broken to U(1), and the physical mass of the fields Q and Q̃ can be found by explicitly expanding the superpotential above:

W = (Q1,Q2) a 0 0 a Q̃1 Q̃2 +μ(Q1,Q2) Q̃1 Q̃2 . (5.1.2)

We see that the masses are

|±a ± μ| (5.1.3)

where we allow all four choices of signs.

In general, the BPS mass formula is

mass |na + maD + fμ| (5.1.4)

where f is the charge under the U(1) flavor symmetry, under which Q has charge 1 and Q̃ has charge 1.

From the one-loop running of the coupling constant, we find

τ(a) = 2τUV 6 2πi log a ΛUV + (5.1.5) = 6 2πi log a Λ1 + (5.1.6)

in the ultraviolet region. Here we defined

Λ16 = ΛUV 6e4πiτUV (5.1.7)

where the subscript 1 is a reminder that we are dealing with the Nf = 1 theory. From this, we can determine the monodromy M at infinity acting on (a,aD):

M = 1 3 0 1 (5.1.8)

exactly as in the pure SU(2) case.

To study the strong coupling region of the system, let us first consider two extreme cases. When |μ| is very big, we expect the running of the coupling to be given roughly as in Fig. 5.1. Namely, at around the scale |μ|, the fields Q and Q̃ decouple, and the system effectively becomes the pure SU(2) gauge theory, which we studied in the last section. Correspondingly, the structure of the u-plane in the region |u||μ2| should be effectively the same with that of the pure SU(2) theory, with two singularities at u = ±2Λ02.

Figure 5.1: Schematic running of the coupling of Nf = 1 theory, when |Λ||a||μ|

A rough relation between Λ0 and Λ1 can be read off from the schematic graph of the running coupling shown in Fig. 5.1. The rightmost segment in the graph is given by

τ(E) = 6 2πi log E Λ1 (5.1.9)

and the middle segment in the graph, representing the effectively pure SU(2) theory, is

τ(E) = 8 2πi log E Λ0. (5.1.10)

Equating these two values at E = μ, we obtain

Λ04 = μΛ13. (5.1.11)

In addition, we know from (5.1.3) that the quanta of one component of Q and Q̃ become very light when ± a μ. This should produce a singularity in the u-plane at around u μ2. We therefore expect that the u-plane to have three singularities, as shown in Fig. 5.2. Note that local physics at the three singularities, at u μ2 and at u ±2Λ02, is always just U(1) gauge theory with one charged hypermultiplet.

Figure 5.2: Singularities on the u-plane when m Λ

In the other extreme when μ = 0, we can make use of the discrete R symmetry. The standard R-charge assignment is as follows:

R = 0 A1λλ 2 Φ , R = 1 ψQ 0Q Q̃ 1 ψQ̃ . (5.1.12)

The rotation

λ eiφλ,ψQ,Q̃ eiφψQ,Q̃ (5.1.13)

is anomalous, but can be compensated by

𝜃UV 𝜃UV + 6φ. (5.1.14)

Therefore φ = 2π6 is a genuine symmetry, which does

𝜃 𝜃 + 2π,Φ e2πi3Φ,u e4πi3u. (5.1.15)

This guarantees that singularities in the u-plane should appear in triples, related by 120 rotation. A minimal assumption is then to have exactly three singularities, as shown in Fig. 5.3. Having three singularities is consistent with our previous analysis when |μ| was very big. We expect that the situation in Fig. 5.2 will smoothly change into the one in Fig. 5.3 when μ is adiabatically changed.

Figure 5.3: Singularities on the u-plane when m = 0

5.1.2 Monodromies

Figure 5.4: Monodromy of Nf = 1

Let us denote the monodromies around each of the three singularities by M1,2,3, see Fig. 5.4. Clearly, we should have

M = M3M2M1 (5.1.16)

where M was given in (5.1.8). As the three singularities are related by discrete R-symmetry, they should be conjugate. For example, as shown in Fig. 5.5, we expect M2 = Y M1Y 1.

Figure 5.5: Relation of M1, M2

A solution is given by

M2 = T1M1T1,M3 = T2M1T2, (5.1.17)

together with

M1 = STS1 = 1 01 1 . (5.1.18)

As M1 found here is the same as M+ found in the pure case (4.2.9), the local physics close to the singularity is also the same, i.e. it is described by an 𝒩=2 U(1) gauge theory coupled to one charged hypermultiplet. The same can be said for M2 and M3.

For the pure case, we saw that the light charged hypermultiplet in this low energy U(1) description was a monopole in the original description. Is the same true in this case? It is easier to give a definitive answer when |μ| is very big. Then, the two singularities in the strong coupled region have the same physics as that of the pure SU(2) theory, and thus we should have light monopoles and dyons there. At the third singularity u μ2, one component of the doublet hypermultiplet (Q,Q̃) becomes very light. For all three singularities, the low-energy description is that of a U(1) gauge theory coupled to one charged hypermultiplet.

By gradually decreasing μ to be zero, these three singularities become the three singularities related by the discrete R symmetry. At this stage, it is not possible to say which of the three was originally the one whose light particle came from the doublet hypermultiplet and which two of the three were the ones with monopoles and dyons. This loss of the distinction between the hypermultiplets which were elementary fields and the hypermultiplets which came from solitons such as monopoles or dyons is somewhat surprising to an eye trained in the classical field theory. We will see this more explicitly below, in Fig. 5.10.