8.3 Nf = 2: the discrete R-symmetry

Let us now study the discrete R-symmetry. We assign the charges under continuous R-symmetry to be given by

R = 0 A1λλ 2 Φ , R = 1 ψI 0qI . (8.3.1)

The rotation

λ eiφλ,ψI eiφψI (8.3.2)

is anomalous, but can be compensated by

𝜃UV 𝜃UV + 4φ. (8.3.3)

Equivalently, the dynamical scale Λ transforms as

Λ2 e4iφΛ2. (8.3.4)

Therefore φ = π2 is a genuine symmetry, which does

𝜃UV 𝜃UV + 2π,Φ Φ,u u (8.3.5)

where u = trΦ22. The reason why we put a prime to the symbol u here will be explained shortly. Unfortunately this does not tell us much about the structure on the u-plane, as it acts trivially on it.

We can perform a slightly subtler operation. Consider the action on the hypermultiplets given by

(qI=1,q2,q3,q4) (qI=1,q2,q3,q4), (8.3.6) (ψI=1,ψ2,ψ3,ψ4) (ψI=1,ψ2,ψ3,ψ4). (8.3.7)

So far we always said that the flavor symmetry is SO(2Nf) = SO(4). This operation is a flavor parity action

diag(1, +1, +1, +1) O(4) SO(4). (8.3.8)

Recall that in an SU(2) k-instanton background, the number of zero-modes of ψI=1 is just k. Then the operation (8.3.7) multiplies the path integral measure by (1)k. This means that the parity part of the classical flavor symmetry O(4) is anomalous. That said, as we have a term ei𝜃k in the integrand of the path integral, we can compensate it by the shift 𝜃 𝜃 + π.

Then, we can combine phase rotations (8.3.2), (8.3.3) with φ = π4 and the flavor parity (8.3.7) to have a genuine unbroken symmetry. Summarizing, this is a combination of two actions: the first one is

𝜃 𝜃 + π,Φ iΦ,uu (8.3.9)

and the second one is

𝜃 + π 𝜃 + 2π,qI=1 qI=1,ψI=1 ψI=1. (8.3.10)

In total this is a 4 symmetry acting on the u-plane by 2.

At the first sight this looks contradictory with the structure of the u-plane found in Fig. 8.4: the two singularities are at u = 0 and u = Λ2. The way out is to set

u = uΛ2 2 . (8.3.11)

This illustrates a subtlety which is often there in the non-perturbative analysis of field theories. Naively, u is defined to be trΦ22. But a composite operator needs to be defined with care, by carefully performing the regularization and the renormalization. As there are almost no divergence between two chiral operators in a supersymmetric theory, it is relatively safe to do this for chiral composite operators, although one still needs to take care of the point splitting between two gauge-dependent chiral operators, which is known as a source of Konishi’s anomaly [55], for example. At least perturbatively, we can take the holomorphic scheme and that uniquely fixes the regularization and the renormalization of chiral composite operators to all orders in perturbation theory. There still is, however, non-perturbative ambiguity in the definition of the scheme. In our present case, u and Λ2 both have mass dimension two and has charge 2 under the continuous broken R-symmetry, therefore they tend to mix. When we guessed the curve in Sec. 6.4.4, we did not take the discrete unbroken R-symmetry into account, thus there was a discrepancy between the u appearing in the curve and the u which was constructed by definition to transform nicely under the discrete R-symmetry.

We learned that the low energy behavior at u = 0 and u = Λ2, or equivalently at u = ±Λ22 is related by the discrete R-symmetry combined with the flavor parity. Let us study them in more detail. We know that the monodromy at u = 0 is given by (8.2.7). Let us say aD cu close to u = 0, where c is a constant. Applying the S transformation once, we see that the running of the dual coupling is

τD(E) + 2 2πi log E (8.3.12)

where E cu sets the energy scale. Compare this with the running of the dual coupling (4.3.22) at the monopole point of the pure SU(2) theory. The factor 2 in the numerator comes from the lower-left entry of M+ in (8.2.7), or more physically from the fact that two pairs of the branch points simultaneously collide as shown in Fig. 8.3. In general, when a U(1) gauge theory is coupled to several hypermultiplets with charges given by qi, the running is given by

τ + iqi2 2πi log E (8.3.13)

Then we can conclude uniquely that there are two hypermultiplets with charge 1. This can be seen from the higher-dimensional perspective: there are disk-shaped membranes as in Fig. 6.4 for each pair of colliding branch points. They become massless when the branch points do collide, thus providing two charged hypermultiplets.