#### 10.3 Argyres-Douglas CFT from the ${N}_{f}=3$
theory

The special limit of ${N}_{f}=3$
theory can be found in exactly the same way. We start from the curve (8.5.1)

$$\frac{{\left(x-\mu -\Lambda \right)}^{2}}{z}+2\Lambda \left(x-\mu -\Lambda \right)z={x}^{2}-u$$ | (10.3.1) |

with the same mass for three ﬂavors. On the
$u$-plane,
we have one singularity with the Higgs branch, and two singularities without. We tune
$\mu $ so that
singularity with the Higgs branch collides with another without, in a way that their monodromies
do not commute. See Fig. 10.5.

The monodromy around the resulting singularities is

$${M}_{AD3}=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill -1\hfill \end{array}\right)$$ | (10.3.2) |

with the action on the coupling given by

$$\tau \mapsto {\tau}^{\prime}=\frac{-\tau +1}{-\tau}.$$ | (10.3.3) |

The ﬁxed point is at $\tau ={e}^{\pi i\u22153}$.

In the 6d description, we had two poles of order two and one pole of order four. We collide an
order-2 pole and an order-4 pole, ending up with a pole of order six. The curve is then

$${\lambda}^{2}=\frac{1+\delta \mu z+{\mu}^{\prime}z+\delta u{z}^{2}+{\left(\frac{{\mu}_{1}-{\mu}_{2}}{2}\right)}^{2}{z}^{3}}{{z}^{6}}d{z}^{2}$$ | (10.3.4) |

The diﬀerential $\lambda $ has
scaling dimension 1. Then $\left[z\right]=-1\u22152$,
and we ﬁnd

$$\left[\delta \mu \right]=\frac{1}{2},\phantom{\rule{1em}{0ex}}\left[\delta u\right]=\frac{3}{2},\phantom{\rule{1em}{0ex}}\left[{\mu}^{\prime}\right]=1,\left[\Delta \mu \right]=1,$$ | (10.3.5) |

where we deﬁned $\Delta \mu ={\mu}_{1}-{\mu}_{2}$.
Two parameters ${\mu}^{\prime}$
and $\Delta \mu $
are of scaling dimension 1, and we identify them with the mass parameters associated to the
$SU\left(3\right)$ ﬂavor symmetry. We
also see $\left[\delta \mu \right]+\left[\delta u\right]=2$ again. We call
this resulting theory $A{D}_{{N}_{f}=3}\left(SU\left(2\right)\right)$.