10.1 Nf = 1 theory and the simplest Argyres-Douglas CFT

Let us come back to the curve of SU(2) gauge theory with Nf = 1 flavor again:

Σ : 2Λ(x μ) z + Λ2z = x2 u. (10.1.1)

With a generic choice of Λ and μ, there are three singularities on the u-plane. As we saw at the end of Sec. 5.2, two singularities collide at u = 3Λ2 when we set μ = 3 2Λ, see Fig. 10.1.

Figure 10.1: Argyres-Douglas point of Nf = 1 theory

When u 3Λ2, three branch points of x(z) collide at z = 1. Then, both the A-cycle and the B-cycle defining a and aD can be taken to be small loops around z = 0. This guarantees that both a and aD are small. Therefore we simultaneously have very light electric and magnetic particles. Such a point on the Coulomb branch is called the Argyres-Douglas point. This was first identified in the case of pure SU(3) theory in [57], and extended to SU(2) theories with flavors in [58].

The monodromy MAD1 around u = 3Λ2 can be found in various ways. One is to multiply the monodromies of the two colliding singularities of the Nf = 1 theory. Another is to follow how the three branch points move. Setting u = 3Λ2 + δu, we find that the three branch points are at z + 1 δu13. This determines how the cycles are mapped, resulting in the monodromy. In either method, we find

MAD1 1 11 0 . (10.1.2)

The transformation on the low energy coupling by MAD1 is

ττ = 1 1 τ. (10.1.3)

Note that τ = eπi3 is a fixed point of this transformation; by an explicit computation, we can check that τ eπi3 δu13. We find that the coupling is pinned at this strongly-coupled value.

The low energy limit is believed to be conformal. To isolate the physics in this limit, let us take

z = 1 + δz,x = Λ + δx,u = 3Λ2 + δu,μ = 3 2Λ + δμ (10.1.4)

and make all variables with δ to be very small. The curve in terms of the new variables is approximately given by

(δx)2 + δu = (δz)3 + δμδz. (10.1.5)

The differential is

λ = xdz z δxdδz. (10.1.6)

As the integral λ gives the mass of BPS particles, λ itself should have the scaling dimension 1. The relation (10.1.5) means that the scaling dimensions [δx] and [δz] should satisfy

[δx] : [δz] = 3 : 2. (10.1.7)

This fixes the scaling dimensions of all the variables involved:

[δx] = 3 5,[δx] = 2 5,[δu] = 6 5,[δμ] = 4 5. (10.1.8)

Note that the mass dimension, or equivalently the scaling dimension at the ultraviolet of the operator u = trΦ22 was 2. We find that the anomalous dimension is of order one, reducing [δu] significantly.

Figure 10.2: Argyres-Douglas theory ADNf=1(SU(2))

As we are taking the limit δu 0, we are zooming into the neighborhood of the u-plane around u = 3Λ2. In the limit, we can think of the low energy theory to be described by a theory with only a singularity at δu = 0, as shown on the left hand side of Fig. 10.2. We call the resulting theory the Argyres-Douglas CFT ADNf=1(SU(2)).14

Let us revisit this limiting procedure from the 6d point of view. We first write the original Nf = 1 curve in the form λ2 ϕ(z) = 0. Recall that ϕ(z) has one order-3 pole and one order-4 pole, as was studied in Sec. 9.6 and shown in Fig. 9.17. We also have three branch points of ϕ(z) on generic points. Suppose that we tune the parameters carefully so that two poles of ϕ(z) collide:

ϕ(z) ( P3(z) (z 𝜖)3 + P4(z) z4 )dz2 = P7(z) (z 𝜖)3z4dz2 Q7(z) z7 dz2 (10.1.9)

where Pd, Qd are generic polynomials of degree d at this stage. We end up having just one singularity with an order-7 pole, as shown on the right hand side of Fig. 10.2. To have no singularity at z = , we see that Q7(z) should be in fact of degree 3:

λ2 = c + cz + μz2 + uz3 z7 dz2. (10.1.10)

By the coordinate transformation z z(az b), we can set c = 1 and c = 0. We then have

λ2 = 1 + μz2 + uz3 z7 dz2. (10.1.11)

As the left hand side is of scaling dimension 2, we see that [z] = 25, and we conclude

[μ] = 4 5,[u] = 6 5 (10.1.12)

which agree with what we found above. Note that the variable z is auxiliary, and therefore there is no reason for its dimension to match.

In general for any conformal field theory, any dynamical scalar operator O should have scaling dimension larger than or equal to one:

[O] 1, (10.1.13)

and the equality is only attained when O describes a free decoupled scalar boson. Then the operator u with [u] = 65 is a genuine operator in the theory ADNf=1(SU(2)). The object μ is regarded as a parameter conjugate to u in the following sense. In an 𝒩=2 theory, we can consider a deformation of the prepotential

d4𝜃F d4𝜃(F + mO). (10.1.14)

Here, d4𝜃 is the chiral 𝒩=2 superspace integral we briefly mentioned at the end of Sec. 2.4, O is an operator and m is a parameter multiplying it. In an 𝒩=2 superconformal theory, the combination mO therefore needs to have a scaling dimension 2. Then we should have

[m] + [O] = 2. (10.1.15)

We see that the pair μ and u satisfies this condition, see (10.1.12). We therefore regard μ as the deformation parameter corresponding to the operator u.