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

The rotation

$$\lambda \to {e}^{i\phi}\lambda ,\phantom{\rule{2em}{0ex}}{\psi}_{I}\to {e}^{-i\phi}{\psi}_{I}$$ | (8.3.2) |

is anomalous, but can be compensated by

$${\mathit{\theta}}_{UV}\to {\mathit{\theta}}_{UV}+4\phi .$$ | (8.3.3) |

Equivalently, the dynamical scale $\Lambda $ transforms as

$${\Lambda}^{2}\to {e}^{4i\phi}{\Lambda}^{2}.$$ | (8.3.4) |

Therefore $\phi =\pi \u22152$ is a genuine symmetry, which does

$${\mathit{\theta}}_{UV}\to {\mathit{\theta}}_{UV}+2\pi ,\phantom{\rule{1em}{0ex}}\Phi \to -\Phi ,\phantom{\rule{1em}{0ex}}{u}^{\prime}\to {u}^{\prime}$$ | (8.3.5) |

where ${u}^{\prime}=\u27e8tr\phantom{\rule{0.3em}{0ex}}{\Phi}^{2}\u22152\u27e9$. 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 ${u}^{\prime}$-plane, as it acts trivially on it.

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

$$\begin{array}{lll}\hfill \left({q}_{I=1},{q}_{2},{q}_{3},{q}_{4}\right)& \mapsto \left(-{q}_{I=1},{q}_{2},{q}_{3},{q}_{4}\right),\phantom{\rule{2em}{0ex}}& \hfill \text{(8.3.6)}\\ \hfill \left({\psi}_{I=1},{\psi}_{2},{\psi}_{3},{\psi}_{4}\right)& \mapsto \left(-{\psi}_{I=1},{\psi}_{2},{\psi}_{3},{\psi}_{4}\right).\phantom{\rule{2em}{0ex}}& \hfill \text{(8.3.7)}\end{array}$$So far we always said that the ﬂavor symmetry is $SO\left(2{N}_{f}\right)=SO\left(4\right)$. This operation is a ﬂavor parity action

$$diag\left(-1,+1,+1,+1\right)\in O\left(4\right)\supset SO\left(4\right).$$ | (8.3.8) |

Recall that in an $SU\left(2\right)$ $k$-instanton background, the number of zero-modes of ${\psi}_{I=1}$ is just $k$. Then the operation (8.3.7) multiplies the path integral measure by ${\left(-1\right)}^{k}$. This means that the parity part of the classical ﬂavor symmetry $O\left(4\right)$ is anomalous. That said, as we have a term ${e}^{i\mathit{\theta}k}$ in the integrand of the path integral, we can compensate it by the shift $\mathit{\theta}\to \mathit{\theta}+\pi $.

Then, we can combine phase rotations (8.3.2), (8.3.3) with $\phi =\pi \u22154$ 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

$$\mathit{\theta}\to \mathit{\theta}+\pi ,\phantom{\rule{1em}{0ex}}\Phi \to i\Phi ,\phantom{\rule{1em}{0ex}}{u}^{\prime}\to -{u}^{\prime}$$ | (8.3.9) |

and the second one is

$$\mathit{\theta}+\pi \to \mathit{\theta}+2\pi ,\phantom{\rule{1em}{0ex}}{q}_{I=1}\to -{q}_{I=1},\phantom{\rule{1em}{0ex}}{\psi}_{I=1}\to -{\psi}_{I=1}.$$ | (8.3.10) |

In total this is a ${\mathbb{Z}}_{4}$ symmetry acting on the $u$-plane by ${\mathbb{Z}}_{2}$.

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 $u=-{\Lambda}^{2}$. The way out is to set

$$u={u}^{\prime}-\frac{{\Lambda}^{2}}{2}.$$ | (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 $\u27e8tr\phantom{\rule{0.3em}{0ex}}{\Phi}^{2}\u22152\u27e9$. 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 ${\Lambda}^{2}$ 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 ${u}^{\prime}$ which was constructed by deﬁnition to transform nicely under the discrete R-symmetry.

We learned that the low energy behavior at $u=0$ and $u=-{\Lambda}^{2}$, or equivalently at ${u}^{\prime}=\pm {\Lambda}^{2}\u22152$ 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 ${a}_{D}\sim cu$ close to $u=0$, where $c$ is a constant. Applying the $S$ transformation once, we see that the running of the dual coupling is

$${\tau}_{D}\left(E\right)\simeq +\frac{2}{2\pi i}logE$$ | (8.3.12) |

where $E\sim cu$ sets the energy scale. Compare this with the running of the dual coupling (4.3.22) at the monopole point of the pure $SU\left(2\right)$ 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 $U\left(1\right)$ gauge theory is coupled to several hypermultiplets with charges given by ${q}_{i}$, the running is given by

$$\tau \simeq +\frac{\sum _{i}{q}_{i}^{2}}{2\pi i}logE$$ | (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.