10.1 $N_f=1$ theory and the simplest Argyres-Douglas CFT

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

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 $z+1\propto \delta u^{1∕3}$. This determines how the cycles are mapped, resulting in the monodromy. In either method, we find

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

Note that $\tau =e^{\pi i∕3}$ is a fixed point of this transformation; by an explicit computation, we can check that $\tau -e^{\pi i∕3}\propto \delta u^{1∕3}$. 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

The differential is

As the integral $\int \; <!--nolimits-->\lambda$ 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

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 $u=tr{\Phi }^2∕2$ 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:

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

As the left hand side is of scaling dimension $2$, we see that $\left[z\right]=-2∕5$, and we conclude

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:

and the equality is only attained when $O$ describes a free decoupled scalar boson. Then the operator $u$ with $\left[u\right]=6∕5$ 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 $\mathcal{𝒩}=2$ theory, we can consider a deformation of the prepotential

Here, $d^4𝜃$ is the chiral $\mathcal{𝒩}=2$ 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 $\mathcal{𝒩}=2$ superconformal theory, the combination $mO$ therefore needs to have a scaling dimension $2$. Then we should have

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 $u$.