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

TL;DR

The paper constructs infinite families of 4d N=2 SCFTs hat-Gamma(G) labeled by ADE groups, obtained by gauging stacks of non-Lagrangian D_p(G) theories and (G,G) conformal matter. A large subclass satisfies $a=c$, with Schur indices matching those of N=4 SYM after fugacity rescaling, and, in specific odd-N D4 cases, connections to MacMahon’s generalized divisor-sum quasi-modular form emerge. When gcd$(\alpha_\Gamma, h_G^\vee)=1$, the central charges scale as $a=c \sim \frac{\Delta_\Gamma-1}{\Delta_\Gamma} \dim(G)$, revealing a Deligne–Cvitanović structure that mirrors the exceptional series. Beyond $a=c$, the authors explore Lagrangian affine quivers, generalized ADE links, and alternative bifundamental matter, discuss six-dimensional origins and VOAs, and outline potential holographic realizations and extensions to broader symmetry classes, suggesting rich future avenues in this line of non-Lagrangian 4d physics.

Abstract

We study a set of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) $\widehatΓ(G)$ labeled by a pair of simply-laced Lie groups $Γ$ 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 $Γ= D_4, E_6, E_7, E_8$ 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 quasi-modular. For generic choices of $Γ$ and $G$, it can be regarded as a generalization of the affine quiver gauge theory obtained from $D3$-branes probing an ALE singularity of type $Γ$. We also comment on a tantalizing connection regarding the theories labeled by $Γ$ in the Deligne--Cvitanović exceptional series.

Figures (4)

• Figure 2.1: When the gauge group $G$ appearing in the quiver in equation \ref{['eqn:trivone']} is an $SU(N)$ group such that each $p_i$ divides $N$, then one can use the description in equation \ref{['eqn:lagexp']} to rewrite \ref{['eqn:trivone']} as a Lagrangian quiver. We depict such Lagrangian quivers and observe that these are the standard affine quiver gauge theories that arise on the worldvolume of D3-branes probing $\mathbb{C}^2/\Gamma$ orbifolds Douglas:1996sw. Here, we introduce the shorthand notation of writing $N$ inside of a gauge node to represent an $SU(N)$ gauge group.
• Figure 2.2: 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 a 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 4.1: The SCFTs $\widehat{\Gamma}(SO(2\alpha_\Gamma\ell + 2))$ have Lagrangian quiver descriptions in terms of alternating $SO$ and $USp$ groups. We depict these quivers here.
• Figure 5.1: 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 $N \geq 1$ and $N \geq 0$ respectively, and we remind the reader that an integer $K$ inside a gauge node implies the gauge group $SU(K)$.