Consider the curve of the ${N}_{f}=2$ theory,

$$\frac{2\Lambda \left(x-\mu \right)}{z}+2\Lambda \left(x-\mu \right)z={x}^{2}-u$$ | (10.2.1) |

where we set the masses of the two ﬂavors the same. We can also use the curve of the alternative form

$$\frac{{\left(x-\mu \right)}^{2}}{z}+4{\Lambda}^{2}z={x}^{2}-u.$$ | (10.2.2) |

Its moduli space for generic $\mu $ was shown in Fig. 8.9.

When $\mu =0$, two singularities without the Higgs branch attached collide. Instead, let us tune the parameter $\mu $ so that the singularity with the Higgs branch collides with a singularity without, see Fig. 10.3. For deﬁniteness let us use the ﬁrst form of the curve. Then this collision happens when $\mu =2\Lambda $, at $u=4{\Lambda}^{2}$. The four branch points then collide at $z=1$.

We ﬁnd that the monodromy around the resulting singularity is

$${M}_{AD2}\sim \left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right),$$ | (10.2.3) |

acting on the coupling as

$$\tau \to {\tau}^{\prime}=-\frac{1}{\tau}.$$ | (10.2.4) |

The strong coupling value $\tau =i$ is the ﬁxed point of this transformation, and the low-energy coupling approaches this value when we let $u\to 4{\Lambda}^{2}$.

Expanding the variables as before,

we ﬁnd that the curve in the limit is

$${\left(\delta x\right)}^{2}+\delta u={\left(\delta z\right)}^{4}+\delta \mu \delta {z}^{2}+\Delta \mu \delta z$$ | (10.2.6) |

with the diﬀerential $\lambda \sim \delta xd\delta z$. Here we reinstated a small diﬀerence $\Delta \mu ={\mu}_{1}-{\mu}_{2}$ between the bare masses ${\mu}_{1}$, ${\mu}_{2}$ of two doublet hypermultiplets. Demanding $\lambda $ to have scaling dimension 1, we see that

$$\left[\delta x\right]=\frac{2}{3},\phantom{\rule{1em}{0ex}}\left[\delta z\right]=\frac{1}{3}.$$ | (10.2.7) |

Then we ﬁnd

$$\left[\delta u\right]=\frac{4}{3},\phantom{\rule{1em}{0ex}}\left[\delta \mu \right]=\frac{2}{3},\phantom{\rule{1em}{0ex}}\left[\Delta \mu \right]=1.$$ | (10.2.8) |

We see again that $\left[\delta u\right]+\left[\delta \mu \right]=2$, and therefore $\delta \mu $ is a deformation parameter corresponding to the operator $\delta u$. $\Delta \mu $ is a mass parameter for the non-Abelian ﬂavor symmetry $SU{\left(2\right)}_{F}$. In general, in a conformal theory, a non-Abelian ﬂavor symmetry current ${J}^{a}$ should have scaling dimension 3. The $\mathcal{\mathcal{N}}=2$ supersymmetry relates it to the mass term, which is given for a Lagrangian theory by the familiar term $Q\stackrel{\u0303}{Q}$ and has scaling dimension 2. Therefore, the non-Abelian mass parameter of $\mathcal{\mathcal{N}}=2$ superconformal theory should always have scaling dimension 1. Our computation of $\left[\Delta \mu \right]$ is consistent with this general argument. We call this resulting theory $A{D}_{{N}_{f}=2}\left(SU\left(2\right)\right)$.

Let us study the limiting procedure of the ${N}_{f}=2$ theory from the 6d point of view. Before taking the limit, the curve in the ﬁrst form (10.2.1) was ${\lambda}^{2}=\varphi \left(z\right)$ with two order-4 poles of $\varphi \left(z\right)$. We collide them, and we end up with a singularity of order 8. Just as in the analysis before, we conclude that the curve in the limit is given by

$${\lambda}^{2}=\frac{1+\mu {z}^{2}+\Delta \mu {z}^{3}+u{z}^{4}}{{z}^{8}}d{z}^{2}.$$ | (10.2.9) |

We easily see that $\left[z\right]=-1\u22153$. Then we ﬁnd the same scaling dimensions as in (10.2.8).

The curve in the second form (10.2.2), when written as ${\lambda}^{2}=\varphi \left(z\right)$, had two poles of order 2, and another of order 3. At the two order-two poles, the residues of $xdz\u2215z$ are $\pm \left({\mu}_{1}+{\mu}_{2}\right)\u22152$ and $\pm \left({\mu}_{1}-{\mu}_{2}\right)\u22152$, respectively. Let us collide an order-2 pole with the residue $\pm \left({\mu}_{1}+{\mu}_{2}\right)\u22152$ and an order-3 pole to form a pole of order 5. We end up having a $\varphi \left(z\right)$ with one pole of order 5, say at $z=0$, and another pole of order 2, with the residue $\pm \left({\mu}_{1}-{\mu}_{2}\right)\u22152$, see Fig. 10.4. The curve in the limit can also be easily found:

$${\lambda}^{2}=\frac{1+\delta \mu z+\delta u{z}^{2}+{\left(\frac{{\mu}_{1}-{\mu}_{2}}{2}\right)}^{2}{z}^{3}}{{z}^{5}}d{z}^{2}$$ | (10.2.10) |

The last coeﬃcient was ﬁxed by the condition at $z=\infty $. Demanding $\lambda $ to have scaling dimension 1, we see that $\left[z\right]=-2\u22153$, and

$$\left[\delta \mu \right]=\frac{2}{3},\phantom{\rule{1em}{0ex}}\left[\delta u\right]=\frac{4}{3}.$$ | (10.2.11) |

It is reassuring to ﬁnd the same answer.