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

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

$$\frac{{\Lambda}^{N}}{z}+{\Lambda}^{N}z={x}^{N}+\cdots +{u}_{N}$$ | (11.5.1) |

with the diﬀerential $\lambda =xdz\u2215z$. 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

$$c\phantom{\rule{0.3em}{0ex}}\delta {z}^{2}={x}^{N}+{u}_{2}{x}^{N-2}+\cdots +\delta {u}_{N}$$ | (11.5.2) |

where $c$ is an unimportant constant. The diﬀerential is now given by $\lambda =xd\delta z\sim \delta zdx$. Introducing $\stackrel{\u0303}{z}=1\u2215x$, 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

$${\lambda}^{2}=\varphi \left(\stackrel{\u0303}{z}\right)$$ | (11.5.4) |

where $\varphi \left(\stackrel{\u0303}{z}\right)$ 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

$$A{D}_{{N}_{f}=0}\left(SU\left(N\right)\right)={Y}_{N+4}.$$ | (11.5.5) |

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

$$\left[{u}_{k}\right]=\frac{2k}{N+2}.$$ | (11.5.6) |

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

$$\int {d}^{4}\mathit{\theta}{u}_{k}{u}_{N+2-k}$$ | (11.5.7) |

where ${d}^{4}\mathit{\theta}$ is the chiral $\mathcal{\mathcal{N}}=2$ 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 $\left[{u}_{3}\right]=6\u22155$ and a corresponding parameter with $\left[{u}_{2}\right]=4\u22155$. 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:

$$A{D}_{{N}_{f}=0}\left(SU\left(3\right)\right)={Y}_{7}=A{D}_{{N}_{f}=1}\left(SU\left(2\right)\right).$$ | (11.5.8) |

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

$$\left(x-{\mu}_{1}\right)\frac{{\Lambda}^{N-1}}{z}+\left(x-{\mu}_{2}\right){\Lambda}^{N-1}z={x}^{N}+\cdots +{u}_{N}.$$ | (11.5.9) |

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

$${u}_{N-1}=2{\Lambda}^{N-1}+\delta {u}_{N-1},\phantom{\rule{1em}{0ex}}z=1+\delta z$$ | (11.5.10) |

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 $x={x}^{\prime}-\left({\mu}_{1}+{\mu}_{2}\right)\u22152$, shift $\delta z$ by $\delta z\to \delta z-\left({c}^{\prime}\u2215c\right)\left({\mu}_{1}-{\mu}_{2}\right)\u2215\left(2{x}^{\prime}\right)$, and introduce $\stackrel{\u0303}{z}=1\u2215{x}^{\prime}$. 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

$$A{D}_{{N}_{f}=2}\left(SU\left(N\right)\right)={X}_{N+3}.$$ | (11.5.13) |

The most singular point of $SU\left(N\right)$ theory with odd number of ﬂavors gives an $\mathcal{\mathcal{N}}=2$ 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 $\mathcal{\mathcal{N}}=2$ CFT, as we will see in Sec. 12.4.4.

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

$${x}^{2}\left(\frac{{\Lambda}^{2N-2}}{z}+{\Lambda}^{2N-2}z\right)={x}^{2N}+{u}_{2}{x}^{2N-2}+\cdots +{u}_{2N}.$$ | (11.5.14) |

Take

$${u}_{2N-2}=2{\Lambda}^{2N-2}+\delta {u}_{2N-2},\phantom{\rule{2em}{0ex}}z=1+\delta z$$ | (11.5.15) |

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

$$c\phantom{\rule{0.3em}{0ex}}\delta {z}^{2}={x}^{2N-2}+\cdots +\delta {u}_{2N-2}+\frac{{u}_{2N}}{{x}^{2}}$$ | (11.5.16) |

where $c$ is an unimportant constant. The diﬀerential is given by $\lambda =xd\delta z\sim \delta zdz$. In terms of $\stackrel{\u0303}{z}=1\u2215{x}^{2}$, 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

$$A{D}_{{N}_{f}=0}\left(SO\left(2N\right)\right)={X}_{N+2}.$$ | (11.5.18) |

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

$${Y}_{8}=A{D}_{{N}_{f}=0}\left(SU\left(4\right)\right)=A{D}_{{N}_{f}=0}\left(SO\left(6\right)\right)={X}_{5}.$$ | (11.5.19) |

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).

The $SU\left(2\right)$ theory with ${N}_{f}=2$ ﬂavors has a Higgs branch of the form ${\u2102}^{2}\u2215{\mathbb{Z}}_{2}$, but the pure $SU\left(4\right)$ theory does not have it in the ultraviolet. We just claimed

$$A{D}_{{N}_{f}=0}\left(SU\left(4\right)\right)=A{D}_{{N}_{f}=2}\left(SU\left(2\right)\right).$$ | (11.5.20) |

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 $2\u22153$ 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 $3\u22152$ respectively, $m$ has scaling dimension $1\u22152$, 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, ${u}_{8}={\u0169}_{4}^{2}$. Close to the Argyres-Douglas point, we see that ${u}_{2}$, ${u}_{4}$, ${u}_{3}$ and ${\u0169}_{4}$ has scaling dimensions $1\u22152$, $1$, $3\u22152$ 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 ${\u0169}_{4}$. The action of the outer automorphism ${S}_{3}$ of $SO\left(8\right)$ on the dimension-1 operators ${u}_{4}$ and ${\u0169}_{4}$ are generated by the parity operation ${\u0169}_{4}\to -{\u0169}_{4}$ and a $12{0}^{\circ}$ rotation acting on the ${u}_{4}$-${\u0169}_{4}$ 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.