More general Argyres-Douglas CFTs: XN and YN

10.5 More general Argyres-Douglas CFTs: $X_{N}$ and $Y_{N}$

Figure 10.7: A higher Argyres-Douglas theory

Let us switch gears and consider other Argyres-Douglas CFTs obtained from more complicated gauge theories with gauge group of the form $S U {(2)}^{n}$ . As an example, consider a rather complicated theory with gauge group $S U {(2)}^{4}$ studied at the end of Sec. 9.6. By performing the same limiting procedure we did in the $S U (2)$ theory with $N_{f} = 1, 2, 3$ , we can consider the theory described by $λ^{2} - ϕ (z) = 0$ where $ϕ (z)$ can have poles of very high order. The examples shown in Fig. 10.7 have either just one pole of order 13 or one order-9 pole and an order-11 pole. They describe complicated 4d $𝒩 = 2$ supersymmetric conformal ﬁeld theories.


X˙N	Y˙N

Figure 10.8: The theory

X_{N}

and the theory

Y_{N}

Let us introduce names to these theories. The $X_{N}$ theory is the superconformal ﬁeld theory corresponding to a sphere with one regular puncture and a puncture with an order- $N$ pole, and the $Y_{N}$ theory is the superconformal ﬁeld theory corresponding to a sphere with just a puncture with an order- $N$ pole, see Fig. 10.8. As can be seen from Fig. 10.2, Fig. 10.4 and Fig. 10.6, we know

Y_{7} = A D_{N_{f} = 1} (S U (2)), Y_{8} = A D_{N_{f} = 2} (S U (2)) = X_{5}, A D_{N_{f} = 3} (S U (2)) = X_{6} .

(10.5.1)

Also, recall the construction of the $S U (2)$ theory with one ﬂavor given in Fig. 9.17. There, a sphere with a regular puncture and a puncture of pole order 3 served as an empty boundary condition, and a sphere with a regular puncture and a puncture of pole order 4 behaves as a free hypermultiplet in the doublet of $S U (2)$ . Equivalently, we see

X_{3} = an empty theory, X_{4} = free hypermultiplet in the doublet of S U (2) .

(10.5.2)

We depicted them in the ﬁrst row of Fig. 10.9.

Figure 10.9: Ordinary and wild gauge theories

More generally, we can have a two-punctured sphere with poles of arbitrary order $N$ and $N^{'}$ . One example with $N = 6$ and $N^{'} = 5$ is shown in the second row of Fig. 10.9. It can be understood as an $S U (2)$ gauge theory with somewhat unusual matter contents, described by two strongly-interacting CFTs $X_{N}$ and $X_{N^{'}}$ . Note that an order-2 pole always carries an $S U (2)$ ﬂavor symmetry, and therefore the $X_{N}$ theory always has an $S U (2)$ ﬂavor symmetry. The $S U (2)$ gauge symmetry coming from the tube couples these two theories. This type of gauge theory with $X_{N}$ as part of its matter contents is often called a wild gauge theory.

It is straightforward to ﬁnd the running of the coupling of this theory. Assume $a$ is very big, as always. The branch points of $λ^{2} = ϕ (z)$ is around where

\frac{Λ^{2}}{z^{N}} d z^{2} \sim \frac{u d z^{2}}{z^{2}} or Λ^{2} z^{N^{'}} d z^{2} \sim \frac{u d z^{2}}{z^{2}} .

(10.5.3)

Then they are around

z_{-} \sim {(\frac{Λ}{a})}^{2 ∕ (N - 2)}, z_{+} \sim {(\frac{a}{Λ})}^{2 ∕ (N^{'} - 2)} .

(10.5.4)

We ﬁnd

a_{D} \sim \frac{2}{2 π i} \int_{z_{+}}^{z_{-}} x \frac{d z}{z} \sim - \frac{2}{2 π i} (\frac{2}{N - 2} + \frac{2}{N^{'} - 2}) a log \frac{a}{Λ} .

(10.5.5)

This means that the one-loop running is given by

Λ \frac{d}{d Λ} τ = \frac{2}{2 π i} (b_{N} + b_{N^{'}} - 4)

(10.5.6)

where

b_{N} = 2 - \frac{2}{N - 2} .

(10.5.7)

Figure 10.10: The running of the coupling measures the two-point correlator of the currents.

The contribution to the one-loop running from one doublet hypermultiplet is given by $b = 1$ . Then this $b_{N}$ can be roughly thought of as an eﬀective number of doublet hypermultiplets, carried by the theory $X_{N}$ . More precisely, $b_{N}$ measures the coeﬃcient of the correlator of the symmetry current $j_{μ}$ of the $S U (2)$ ﬂavor symmetry, see Fig 10.10. As shown there, for $S U (2)$ with ﬂavors, the running of the gauge coupling is caused by the loop of gauge multiplets (shown as wavy lines) or of hypermultiplets (shown as straight lines) coupled to the gauge ﬁelds via the $S U (2)$ current operator $j_{μ}$ . Then the contribution to the one-loop running measures $⟨ j_{μ} j_{ν} ⟩$ . The fact that the $X_{N}$ theory contributes $b_{N}$ times a free ﬂavor does means that

{⟨ j_{μ} j_{ν} ⟩}_{X_{N}} = b_{N} {⟨ j_{μ} j_{ν} ⟩}_{free hyper in a doublet of S U (2)} .

(10.5.8)

Recall that $X_{3}$ is just empty and $X_{4}$ is one free hypermultiplet in the doublet of ﬂavor $S U (2)$ . Our general formula correctly reproduces $b_{3} = 0$ and $b_{4} = 1$ .

In the next section we will see that a singular limit of the pure $S U (N)$ gauge theories becomes the theory $Y_{N + 4}$ , whereas a singular limit of the pure $S O (2 N)$ gauge theories becomes the theory $X_{N + 2}$ . We will also see that $S U (N)$ gauge theories with two ﬂavors have a singular limit given by $X_{N + 3}$ .

Let us denote the Argyres-Douglas CFTs obtained from the pure $G$ gauge theory as $A D_{N_{f} = 0} (G)$ , and the Argyres-Douglas CFTs obtained from the $S U (N)$ with two ﬂavors as $A D_{N_{f} = 2} (S U (N))$ . Then we can succinctly express these equivalences as

\begin{matrix} A D_{N_{f} = 0} & (S U (N)) & = Y_{N + 4}, \\ A D_{N_{f} = 0} & (S O (2 N)) & = X_{N + 2}, \\ A D_{N_{f} = 2} & (S U (N)) & = X_{N + 3} . \end{matrix}

(10.5.9)

We have already seen in (10.5.1) that $A D_{N_{f} = 2} (S U (2))$ , the Argyres-Douglas theories arising from $S U (2)$ with $N_{f} = 2$ ﬂavors, is equivalent to both $X_{5}$ and $Y_{8}$ . This coincidence is a manifestation of the equivalence $S U (4) ≃ S O (6)$ from the point of view of (10.5.9). Also, consider the pure $S O (4)$ theory, which is two copies of the pure $S U (2)$ theory. Its most singular point is where both copies are at the monopole point, thus realizing two free hypermultiplets. Indeed, this has an $S U (2)$ ﬂavor symmetry, and is a doublet under it, realizing the fact

A D_{N_{f} = 0} (S O (4)) = X_{4} = a free hypermultiplet in the doublet of S U (2) .

(10.5.10)

These equations are rather interesting to the author, in the sense that they are equalities among the quantum ﬁeld theories, not among the physical quantities in a single quantum ﬁeld theory.

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10.5 More general Argyres-Douglas CFTs: XN and Y N

10.5 More general Argyres-Douglas CFTs: $X_{N}$ and $Y_{N}$