Infinitely many 4D $\mathcal{N}=1$ SCFTs with $a=c$

We study a rich set of four-dimensional $\mathcal{N}=1$ superconformal field theories (SCFTs) with both central charges identical: $a=c$. We construct them via the diagonal $\mathcal{N}=1$ gauging of the flavor symmetry $G$ of a collection of $\mathcal{N}=2$ Argyres–Douglas theories of type ${\mathcal{D}}_p(G)$, with or without additional adjoint chiral multiplets. In this way, we construct infinitely many theories that flow to interacting SCFTs with $a=c$ in the infrared. Finally, we briefly highlight the features of the SCFTs without $a=c$ that arise from generalizing this construction.

Figure 1

RG flows triggered by the mass deformation of the $\mathcal{N}=2$ theory versus gauging an $\mathcal{N}=1$ vector multiplet. The red cross is to emphasize that there is no direct renormalization group flow between the $\mathcal{N}=2$ gauging and the $\mathcal{N}=1$ gauging.

Figure 2

Contour plots of $-{\epsilon }_1$ and $-{\epsilon }_3·p_3$. We see that ${\epsilon }_1$ lies in the range $(-\frac{1}{3},0)$, and thus, ${\epsilon }_2$ does also. In (b), we show that $-{\epsilon }_3·p_3$ is always positive, and since we are in a large $p_3$ limit, then we can see that ${\epsilon }_3$ must also lie in the range $(-\frac{1}{3},0)$.

Figure 3

Contours plot of ${\epsilon }_1$ and ${\epsilon }_3$ in the $(p_1,p_2)$ plane for $p_3=2$, 3, 4. They all satisfy the unitarity condition in Eq. (2.22).

Figure 4

Contour plot of $-{\epsilon }_1$ and $-{\epsilon }_3$ for $(p_1,p_2,p_3,p_4)=(2,2,p_3,p_4)$. Every $R$-symmetry mixing coefficient ${\epsilon }_i$ satisfies the unitarity condition in Eq. (2.22).

Figure 5

Plot of ${\epsilon }_{1,2,3,4}$ in gluing four ${\mathcal{D}}_{p_i}(G)$ theories with all $(p_1,p_2,p_3,p_4)$ listed in Table 1, except first case, $(2,2,p_3,p_4)$, which is numerically studied in Fig. 4. All the coefficients are in the range prescribed in Eq. (2.22).

Figure 6

Plots of ${\epsilon }_1$, ${\epsilon }_3$ and $R_{\varphi }$ of $(2,2,p_3)$ theories glued by $\mathcal{N}=1$ gauging with one adjoint chiral multiplet. We can see that ${\epsilon }_1={\epsilon }_2$ and ${\epsilon }_3$ lie in the range $(-\frac{1}{3},0)$. Furthermore, $R_{\varphi }$ is always larger than $1/3$. Thus, we conclude that for all values of $p_3$, the gauged theory flows to an SCFT in the infrared with $a=c$.

Figure 7

Contour plots of $-{\epsilon }_1$ and $R_{\varphi }$ of ${\mathcal{D}}_{p_1}(G)$ and ${\mathcal{D}}_{p_2}(G)$ theories glued by $\mathcal{N}=1$ gauging with one adjoint chiral multiplet. ${\epsilon }_1$ lies in the range $(-\frac{1}{3},0)$, as does ${\epsilon }_2$, and $R_{\varphi }$ is always larger than $1/3$; thus, any such gauging satisfies the unitarity bound.

Figure 8

All possible asymptotically free or conformal gaugings of an arbitrary collection of ${\mathcal{D}}_{p_i}(G)$ and $(G\times G)$ theories with two gauge nodes. A node with a solid border denotes an $\mathcal{N}=1$ gauge node, and a solid line between gauge nodes refers to the presence of $(G\times G)$ conformal matter. A line connecting a gauge node and ${\mathcal{C}}_n$ is a shorthand notation for all options of gaugings with the collections of ${\mathcal{D}}_{p_i}(G)$ theories as described in Table 5.

Figure 9

All possible gaugings of conformal matter and ${\mathcal{D}}_p(G)$ with three gauge nodes. See the caption of Fig. 8 for an explanation of the notation.

Figure 10

Deforming the theories with one ${\mathcal{D}}_p(G)$ and two adjoint chiral multiplets $({\varphi }_1,{\varphi }_2)$, which we refer to as $\widehat{O}$, via ADE-type superpotential. When the $\widehat{O}$ is deformed by $\mathrm{Tr}{\varphi }_2^2$, the $\mathrm{Tr}{\varphi }_1^2$ becomes free during the RG flow and must be flipped by the flip-operator $M$. The question marks represent non-ADE deformations.

Figure 11

The comet-shaped quiver that depicts the Lagrangian $\mathcal{N}=1$ quiver we consider. When we write a number $M$ inside a gauge node, we remind the reader that this corresponds to an $SU(M)$ gauge group. A dashed-bordered circular node denotes an $\mathcal{N}=2$ vector multiplet, a solid-bordered circular node denotes an $\mathcal{N}=1$ vector multiplet, and an undirected link between two nodes denotes an $\mathcal{N}=2$ bifundamental hypermultiplet. The dashed self-link corresponds to $n_a$ adjoint chiral multiplets.