10.3 Argyres-Douglas CFT from the Nf = 3 theory

The special limit of Nf = 3 theory can be found in exactly the same way. We start from the curve (8.5.1)

(x μ Λ)2 z + 2Λ(x μ Λ)z = x2 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 μ so that singularity with the Higgs branch collides with another without, in a way that their monodromies do not commute. See Fig. 10.5.

Figure 10.5: Argyres-Douglas point of Nf = 3 theory

The monodromy around the resulting singularities is

MAD3 = 0 1 1 1 (10.3.2)

with the action on the coupling given by

ττ = τ + 1 τ . (10.3.3)

The fixed point is at τ = eπi3.

Figure 10.6: Argyres-Douglas theory ADNf=3(SU(2))

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

λ2 = 1 + δμz + μz + δuz2 + (μ1μ2 2 )2z3 z6 dz2 (10.3.4)

The differential λ has scaling dimension 1. Then [z] = 12, and we find

[δμ] = 1 2,[δu] = 3 2,[μ] = 1, [Δμ] = 1, (10.3.5)

where we defined Δμ = μ1 μ2. Two parameters μ and Δμ are of scaling dimension 1, and we identify them with the mass parameters associated to the SU(3) flavor symmetry. We also see [δμ] + [δu] = 2 again. We call this resulting theory ADNf=3(SU(2)).