11.5 Argyres-Douglas CFTs

Let us study the most singular point in the Coulomb branches of the theories we analyzed in this section.

11.5.1 Pure $SU\left(N\right)$ theory

First, take the curve of the pure $SU\left(N\right)$ theory:

with the differential $\lambda =xdz∕z$. We set $z=1+\delta z$, $u_N=2{\Lambda }^N+\delta u_N$ and take the limit where both $\delta z$ and $\delta u_N$ are very small. We find

where $c$ is an unimportant constant. The differential is now given by $\lambda =xd\delta z\sim \delta zdx$. Introducing $\tilde{z}=1∕x$, we find that the curve in this limit can be written as

Note that it has the same form as the curve we saw in Sec. 10.5, which arose from considering the curve

where $\varphi \left(\tilde{z}\right)$ is a quadratic differential with one pole of order $N+4$, see Fig. 11.7. This is the same as the theory $Y_{N+4}$ introduced in Fig. 10.8. We have

Demanding that $\lambda$ has scaling dimension 1, we find that

Note that we have $\left[u_k\right]+\left[u_{N+2-k}\right]=2$. At this point it is instructive to recall our discussions around (10.1.13). We consider the prepotential deformation

where $d^4𝜃$ is the chiral $\mathcal{𝒩}=2$ superspace integral. As $\left[u_k\right]\le 1\le \left[u_{N+2-k}\right]$ when $k\le N+2-k$, we consider $u_k$ is the deformation parameter for the physical operator $u_{N+2-k}$.

Take the simplest case $N=3$. We have the theory with one operator with $\left[u_3\right]=6∕5$ and a corresponding parameter with $\left[u_2\right]=4∕5$. These are the same as those of the Argyres-Douglas CFT which arose from $SU\left(2\right)$ with one flavor; in fact the curve and the differential are completely the same:

11.5.2 $SU\left(N\right)$ theory with two flavors

Next, consider $SU\left(N\right)$ theory with two flavors. The curve is

We already studied the case $N=2$, so let us set $N>3$. Then we expand as

and take the limit where $\delta z$, $\delta u_{N-1}$ and ${\mu }_{1,2}$ are all small. The curve is

with the differential $\lambda =xd\delta z\sim \delta zdx$. Here $c$ and $c^{\prime }$ are unimportant constants.

We now define $x^{\prime }$ by $x=x^{\prime }-\left({\mu }_1+{\mu }_2\right)∕2$, shift $\delta z$ by $\delta z\to \delta z-\left(c^{\prime }∕c\right)\left({\mu }_1-{\mu }_2\right)∕\left(2x^{\prime }\right)$, and introduce $\tilde{z}=1∕x^{\prime }$. The curve is now

Here we absorbed various unimportant numerical constants into the definition of variables with tildes.

This is the curve ${\lambda }^2=\varphi \left(z\right)$ with $\varphi$ having one pole of order $N+3$ and another of order 2, see Fig. 11.8. This is the theory $X_{N+3}$ introduced in Fig. 10.8. We have

The most singular point of $SU\left(N\right)$ theory with odd number of flavors gives an $\mathcal{𝒩}=2$ CFT, analyzed in [65, 19]. The most singular point of $SU\left(N\right)$ theory with even number of flavors $N_f\ge 4$ does not give an $\mathcal{𝒩}=2$ CFT, as we will see in Sec. 12.4.4.

11.5.3 Pure $SO\left(2N\right)$ theory

Next, take the pure $SO\left(2N\right)$ theory

Take

and go to the limit where $\delta u_{2N-2}$, $\delta z$ are both small. The curve is

where $c$ is an unimportant constant. The differential is given by $\lambda =xd\delta z\sim \delta zdz$. In terms of $\tilde{z}=1∕x^2$, the curve is

This is again the curve ${\lambda }^2=\varphi \left(z\right)$ with $\varphi$ having one pole of rather high order $N+2$ and another of order 2, see Fig. 11.9. Therefore we find

Now, $SU\left(4\right)$ and $SO\left(6\right)$ have the same Lie algebra. Using (11.5.5) and (11.5.18), we find

Using (11.5.13), we find that these are also equivalent to $A{D}_{N_f=2}\left(SU\left(2\right)\right)$. This set of equivalences explains what we saw in (10.5.1).

11.5.4 Argyres-Douglas CFTs and the Higgs branch

The $SU\left(2\right)$ theory with $N_f=2$ flavors has a Higgs branch of the form ${ℂ}^2∕{\mathbb{Z}}_2$, but the pure $SU\left(4\right)$ theory does not have it in the ultraviolet. We just claimed

How is this compatible? The discussion below summarizes the content of [66].

Note that the limiting Argyres-Douglas theory has an operator $u$ of scaling dimension 4/3, a corresponding parameter $m$ of scaling dimension $2∕3$ and an additional mass parameter ${\mu }_1-{\mu }_2$ of scaling dimension 1. When we realize it as a limit of the $SU\left(2\right)$ theory with $N_f=2$ flavors, clearly the low energy theory has just one $U\left(1\right)$ multiplet and ${\mu }_1-{\mu }_2$ is an external parameter.

When we realize the same theory as a limit of the pure $SU\left(4\right)$ theory, however, originally the low energy theory has $U{\left(1\right)}^3$ vector multiplet, and three Coulomb branch parameters $u_2$, $u_3$ and $u_4$. We saw that $\delta u_2$ has scaling dimension 2/3, $\delta u_4$ scaling dimension 4/3, and $\delta u_3$ is of scaling dimension 1. Therefore, we see that the mass parameter ${\mu }_1-{\mu }_2$ of the limiting Argyres-Douglas theory is now promoted to the vev $\delta u_3$ of a $U\left(1\right)$ multiplet in this realization. Equivalently, the $U\left(1\right)$ subgroup of the $SU\left(2\right)$ flavor symmetry of the limiting theory is weakly dynamically gauged, thus removing the Higgs branch.

Similarly, we saw here that the pure $SO\left(8\right)$ theory and the $SU\left(3\right)$ theory with $N_f=2$ flavors both give rise to the CFT $X_6$. In Sec. 10.5, we also learned that $SU\left(2\right)$ theory with $N_f=3$ flavors also has a point on the Coulomb branch where the low energy limit is described by the same theory $X_6$, see (10.5.1). The situation concerning their Higgs branches can also be studied similarly as above. The limiting theory itself has an operator $u$ of scaling dimension 3/2, a corresponding deformation parameter $m$ of dimension 1/2, and two mass parameters ${\mu }_1-{\mu }_2$ and ${\mu }_1-{\mu }_3$ for the $SU\left(3\right)$ flavor symmetry. This is most clearly seen in the description as a point on the Coulomb branch of the $SU\left(2\right)$ theory with $N_f=3$ flavors.

In terms of $SU\left(3\right)$ theory with $N_f=2$ flavors, we have two Coulomb branch operators $u_2$, $u_3$, the mass parameter $m$ for the $U\left(1\right)$ part of the flavor symmetry, and the mass parameter for the $SU\left(2\right)$ part ${\mu }_1-{\mu }_2$. We see that $u_2$ and $u_3$ has scaling dimensions $1$ and $3∕2$ respectively, $m$ has scaling dimension $1∕2$, and ${\mu }_1-{\mu }_2$ has dimension $1$. Then we see that $U\left(1\right)$ subgroup of the flavor symmetry $SU\left(3\right)$ is weakly gauged. The vev of this weakly-gauging $U\left(1\right)$ vector multiplet is $u_2$.

In terms of pure $SO\left(8\right)$ theory, we have four Coulomb branch operators $u_2$, $u_4$, $u_6$ and $u_8$, but as we discussed above, $u_8={ũ}_4^2$. Close to the Argyres-Douglas point, we see that $u_2$, $u_4$, $u_3$ and ${ũ}_4$ has scaling dimensions $1∕2$, $1$, $3∕2$ and $1$ respectively. We see that $U{\left(1\right)}^2$ subgroup of the flavor symmetry $SU\left(3\right)$ is weakly gauged by the two $U\left(1\right)$ vector multiplets with scalar components $u_4$ and ${ũ}_4$. The action of the outer automorphism $S_3$ of $SO\left(8\right)$ on the dimension-1 operators $u_4$ and ${ũ}_4$ are generated by the parity operation ${ũ}_4\to -{ũ}_4$ and a $120^{\circ }$ rotation acting on the $u_4$-${ũ}_4$ plane. This is exactly how the Weyl group of the flavor symmetry $SU\left(3\right)$ acts on the two mass parameters ${\mu }_1$, ${\mu }_2$, ${\mu }_3$ with ${\mu }_1+{\mu }_2+{\mu }_3=0$. Therefore we see that the outer-automorphism symmetry of $SO\left(8\right)$ can be identified with the Weyl group of the $SU\left(3\right)$ flavor symmetry.