Let us come back to the curve of $SU\left(2\right)$ gauge theory with ${N}_{f}=1$ ﬂavor again:

$$\Sigma :\phantom{\rule{1em}{0ex}}\frac{2\Lambda \left(x-\mu \right)}{z}+{\Lambda}^{2}z={x}^{2}-u.$$ | (10.1.1) |

With a generic choice of $\Lambda $ and $\mu $, there are three singularities on the $u$-plane. As we saw at the end of Sec. 5.2, two singularities collide at $u=3{\Lambda}^{2}$ when we set $\mu =-\frac{3}{2}\Lambda $, see Fig. 10.1.

When $u\sim 3{\Lambda}^{2}$, three branch points of $x\left(z\right)$ collide at $z=-1$. Then, both the A-cycle and the B-cycle deﬁning $a$ and ${a}_{D}$ can be taken to be small loops around $z=0$. This guarantees that both $a$ and ${a}_{D}$ are small. Therefore we simultaneously have very light electric and magnetic particles. Such a point on the Coulomb branch is called the Argyres-Douglas point. This was ﬁrst identiﬁed in the case of pure $SU\left(3\right)$ theory in [57], and extended to $SU\left(2\right)$ theories with ﬂavors in [58].

The monodromy ${M}_{AD1}$ around $u=3{\Lambda}^{2}$ can be found in various ways. One is to multiply the monodromies of the two colliding singularities of the ${N}_{f}=1$ theory. Another is to follow how the three branch points move. Setting $u=3{\Lambda}^{2}+\delta u$, we ﬁnd that the three branch points are at $z+1\propto \delta {u}^{1\u22153}$. This determines how the cycles are mapped, resulting in the monodromy. In either method, we ﬁnd

$${M}_{AD1}\sim \left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right).$$ | (10.1.2) |

The transformation on the low energy coupling by ${M}_{AD1}$ is

$$\tau \mapsto {\tau}^{\prime}=\frac{1}{1-\tau}.$$ | (10.1.3) |

Note that $\tau ={e}^{\pi i\u22153}$ is a ﬁxed point of this transformation; by an explicit computation, we can check that $\tau -{e}^{\pi i\u22153}\propto \delta {u}^{1\u22153}$. We ﬁnd that the coupling is pinned at this strongly-coupled value.

The low energy limit is believed to be conformal. To isolate the physics in this limit, let us take

and make all variables with $\delta $ to be very small. The curve in terms of the new variables is approximately given by

$${\left(\delta x\right)}^{2}+\delta u={\left(\delta z\right)}^{3}+\delta \mu \delta z.$$ | (10.1.5) |

The diﬀerential is

$$\lambda =x\frac{dz}{z}\sim \delta xd\delta z.$$ | (10.1.6) |

As the integral $\int \lambda $ gives the mass of BPS particles, $\lambda $ itself should have the scaling dimension 1. The relation (10.1.5) means that the scaling dimensions $\left[\delta x\right]$ and $\left[\delta z\right]$ should satisfy

$$\left[\delta x\right]:\left[\delta z\right]=3:2.$$ | (10.1.7) |

This ﬁxes the scaling dimensions of all the variables involved:

Note that the mass dimension, or equivalently the scaling dimension at the ultraviolet of the operator $u=tr\phantom{\rule{0.3em}{0ex}}{\Phi}^{2}\u22152$ was 2. We ﬁnd that the anomalous dimension is of order one, reducing $\left[\delta u\right]$ signiﬁcantly.

As we are taking the limit $\delta u\to 0$,
we are zooming into the neighborhood of the u-plane around
$u=3{\Lambda}^{2}$. In the
limit, we can think of the low energy theory to be described by a theory with only a singularity at
$\delta u=0$, as
shown on the left hand side of Fig. 10.2. We call the resulting theory the Argyres-Douglas CFT
$A{D}_{{N}_{f}=1}\left(SU\left(2\right)\right)$.^{14}

Let us revisit this limiting procedure from the 6d point of view. We ﬁrst write the original ${N}_{f}=1$ curve in the form ${\lambda}^{2}-\varphi \left(z\right)=0$. Recall that $\varphi \left(z\right)$ has one order-3 pole and one order-4 pole, as was studied in Sec. 9.6 and shown in Fig. 9.17. We also have three branch points of $\varphi \left(z\right)$ on generic points. Suppose that we tune the parameters carefully so that two poles of $\varphi \left(z\right)$ collide:

where ${P}_{d}$, ${Q}_{d}$ are generic polynomials of degree $d$ at this stage. We end up having just one singularity with an order-7 pole, as shown on the right hand side of Fig. 10.2. To have no singularity at $z=\infty $, we see that ${Q}_{7}\left(z\right)$ should be in fact of degree 3:

$${\lambda}^{2}=\frac{c+{c}^{\prime}z+\mu {z}^{2}+u{z}^{3}}{{z}^{7}}d{z}^{2}.$$ | (10.1.10) |

By the coordinate transformation $z\to z\u2215\left(az-b\right)$, we can set $c=1$ and ${c}^{\prime}=0$. We then have

$${\lambda}^{2}=\frac{1+\mu {z}^{2}+u{z}^{3}}{{z}^{7}}d{z}^{2}.$$ | (10.1.11) |

As the left hand side is of scaling dimension $2$, we see that $\left[z\right]=-2\u22155$, and we conclude

$$\left[\mu \right]=\frac{4}{5},\phantom{\rule{2em}{0ex}}\left[u\right]=\frac{6}{5}$$ | (10.1.12) |

which agree with what we found above. Note that the variable $z$ is auxiliary, and therefore there is no reason for its dimension to match.

In general for any conformal ﬁeld theory, any dynamical scalar operator $O$ should have scaling dimension larger than or equal to one:

$$\left[O\right]\ge 1,$$ | (10.1.13) |

and the equality is only attained when $O$ describes a free decoupled scalar boson. Then the operator $u$ with $\left[u\right]=6\u22155$ is a genuine operator in the theory $A{D}_{{N}_{f}=1}\left(SU\left(2\right)\right)$. The object $\mu $ is regarded as a parameter conjugate to $u$ in the following sense. In an $\mathcal{\mathcal{N}}=2$ theory, we can consider a deformation of the prepotential

$$\int {d}^{4}\mathit{\theta}F\to \int {d}^{4}\mathit{\theta}\left(F+mO\right).$$ | (10.1.14) |

Here, ${d}^{4}\mathit{\theta}$ is the chiral $\mathcal{\mathcal{N}}=2$ superspace integral we brieﬂy mentioned at the end of Sec. 2.4, $O$ is an operator and $m$ is a parameter multiplying it. In an $\mathcal{\mathcal{N}}=2$ superconformal theory, the combination $mO$ therefore needs to have a scaling dimension $2$. Then we should have

$$\left[m\right]+\left[O\right]=2.$$ | (10.1.15) |

We see that the pair $\mu $ and $u$ satisﬁes this condition, see (10.1.12). We therefore regard $\mu $ as the deformation parameter corresponding to the operator $u$.