10.2 Argyres-Douglas CFT from the $N_f=2$ theory

Consider the curve of the $N_f=2$ theory,

where we set the masses of the two flavors the same. We can also use the curve of the alternative form

Its moduli space for generic $\mu$ was shown in Fig. 8.9.

When $\mu =0$, two singularities without the Higgs branch attached collide. Instead, let us tune the parameter $\mu$ so that the singularity with the Higgs branch collides with a singularity without, see Fig. 10.3. For definiteness let us use the first form of the curve. Then this collision happens when $\mu =2\Lambda$, at $u=4{\Lambda }^2$. The four branch points then collide at $z=1$.

We find that the monodromy around the resulting singularity is

acting on the coupling as

The strong coupling value $\tau =i$ is the fixed point of this transformation, and the low-energy coupling approaches this value when we let $u\to 4{\Lambda }^2$.

Expanding the variables as before,

we find that the curve in the limit is

with the differential $\lambda \sim \delta xd\delta z$. Here we reinstated a small difference $\Delta \mu ={\mu }_1-{\mu }_2$ between the bare masses ${\mu }_1$, ${\mu }_2$ of two doublet hypermultiplets. Demanding $\lambda$ to have scaling dimension 1, we see that

Then we find

We see again that $\left[\delta u\right]+\left[\delta \mu \right]=2$, and therefore $\delta \mu$ is a deformation parameter corresponding to the operator $\delta u$. $\Delta \mu$ is a mass parameter for the non-Abelian flavor symmetry $SU{\left(2\right)}_F$. In general, in a conformal theory, a non-Abelian flavor symmetry current $J^a$ should have scaling dimension 3. The $\mathcal{𝒩}=2$ supersymmetry relates it to the mass term, which is given for a Lagrangian theory by the familiar term $Q\tilde{Q}$ and has scaling dimension 2. Therefore, the non-Abelian mass parameter of $\mathcal{𝒩}=2$ superconformal theory should always have scaling dimension 1. Our computation of $\left[\Delta \mu \right]$ is consistent with this general argument. We call this resulting theory $A{D}_{N_f=2}\left(SU\left(2\right)\right)$.

Let us study the limiting procedure of the $N_f=2$ theory from the 6d point of view. Before taking the limit, the curve in the first form (10.2.1) was ${\lambda }^2=\varphi \left(z\right)$ with two order-4 poles of $\varphi \left(z\right)$. We collide them, and we end up with a singularity of order 8. Just as in the analysis before, we conclude that the curve in the limit is given by

We easily see that $\left[z\right]=-1∕3$. Then we find the same scaling dimensions as in (10.2.8).

The curve in the second form (10.2.2), when written as ${\lambda }^2=\varphi \left(z\right)$, had two poles of order 2, and another of order 3. At the two order-two poles, the residues of $xdz∕z$ are $\pm \left({\mu }_1+{\mu }_2\right)∕2$ and $\pm \left({\mu }_1-{\mu }_2\right)∕2$, respectively. Let us collide an order-2 pole with the residue $\pm \left({\mu }_1+{\mu }_2\right)∕2$ and an order-3 pole to form a pole of order 5. We end up having a $\varphi \left(z\right)$ with one pole of order 5, say at $z=0$, and another pole of order 2, with the residue $\pm \left({\mu }_1-{\mu }_2\right)∕2$, see Fig. 10.4. The curve in the limit can also be easily found:

The last coefficient was fixed by the condition at $z=\infty$. Demanding $\lambda$ to have scaling dimension 1, we see that $\left[z\right]=-2∕3$, and

It is reassuring to find the same answer.