Infinitely many 4D $\mathcal{N}=2$ SCFTs with $a=c$ and beyond

We study a set of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) $\widehat{\mathrm{\Gamma }}(G)$ labeled by a pair of simply laced Lie groups $\mathrm{\Gamma }$ and $G$. They are constructed out of gauging a number of ${\mathcal{D}}_p(G)$ and $(G,G)$ conformal matter SCFTs; therefore, they do not have Lagrangian descriptions in general. For $\mathrm{\Gamma }=D_4,E_6,E_7,E_8$, and some special choices of $G$, the resulting theories have identical central charges ($a=c$) without taking any large $N$ limit. Moreover, we find that the Schur indices for such theories can be written in terms of that of $\mathcal{N}=4$ super-Yang-Mills theory upon rescaling fugacities. Especially, we find that the Schur index of ${\widehat{D}}_4(SU(N))$ theory for $N$ odd is written in terms of MacMahon’s generalized sum-of-divisor function, which is quasimodular. For generic choices of $\mathrm{\Gamma }$ and $G$, it can be regarded as a generalization of the affine quiver gauge theory obtained from $\text{D}3$-branes probing singularity of type $\mathrm{\Gamma }$. We also comment on a tantalizing connection regarding the theories labeled by $\mathrm{\Gamma }$ in the Deligne-Cvitanović exceptional series.

Figure 1

When the gauge group $G$ appearing in the quiver in Eq. (2.6) is an $SU(N)$ group such that each $p_i$ divides $N$, then one can use the description in Eq. (2.8) to rewrite (2.6) as a Lagrangian quiver. We depict such Lagrangian quivers and observe that these are the standard affine quiver gauge theories that arise on the world volume of D3-branes probing ${\mathbb{C}}^2/\mathrm{\Gamma }$ orbifolds [7]. Here, we introduce the shorthand notation of writing $N$ inside of a gauge node to represent an $SU(N)$ gauge group.

Figure 2

The trivalent gauging of the common $G$ flavor symmetry of ${\mathcal{D}}_{p_1}(G)$, ${\mathcal{D}}_{p_2}(G)$, and ${\mathcal{D}}_{p_3}(G)$ is depicted in (a). Each of the three theories has an interpretation as a sphere with an irregular puncture and a regular puncture, where the latter contributes the flavor symmetry, $G$. The regular punctures are glued together to gauge $G$, and the three irregular punctures, denoted by crosses, remain. In known examples, this is equivalent to a one-punctured sphere, depicted in (b), where the puncture (denoted as a boxed cross) is formed by coalescing the three irregular punctures.

Figure 3

The SCFTs $\widehat{\mathrm{\Gamma }}(SO(2{\alpha }_{\mathrm{\Gamma }}\ell +2))$ have Lagrangian quiver descriptions in terms of alternating $SO$ and $USp$ groups. We depict these quivers here.

Figure 4

We show the affine quivers associated to the affine Dynkin diagrams ${\widehat{A}}_{N-1}$ and ${\widehat{D}}_{N+4}$, together with the generalizations that we consider. We can take $N\ge 1$ and $N\ge 0$, respectively, and we remind the reader that an integer $K$ inside a gauge node implies the gauge group $SU(K)$.