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)

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

with the action on the coupling given by

The fixed point is at $\tau =e^{\pi i∕3}$.

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

The differential $\lambda$ has scaling dimension 1. Then $\left[z\right]=-1∕2$, and we find

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 $A{D}_{N_f=3}\left(SU\left(2\right)\right)$.