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

Abstract

We study a set of four-dimensional N = 2 superconformal field theories (SCFTs) Γ(G) labeled by a pair of simply-laced Lie groups Γ and G. They are constructed out of gauging a number of 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 N = 4 super Yang-Mills theory upon rescaling fugacities. Especially, we find that the Schur index of 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.

Snowmass White Paper on SCFTs

“…This conjecture, which has been shown [278] to follow from a previous conjecture by Arakawa [330], carries deep implication; the VOAs which arise from the cohomological reduction of N = 2 SCFTs would then be of a special type known as "quasi-lisse" [330], a property which ensures that their vacuum characters satisfy a linear modular differential equation [331][332][333]. The representation theory of χ[T ], which is seldomly rational, can be complicated and it remains an open problem to understand which characters participate in the modular property of the vacuum character; these and related issues have been recently investigated in [334][335][336][337]. In recent years a varieties of techniques have been employed to compute χ[T ] in a large set of examples [279,331,332,[338][339][340][341][342][343][344][345][346][347][348][349] and it is worthwhile noticing that surprisingly often, in the Argyres-Douglas case, χ[T ] is an affine Kac-Moody at boundary admissible level [342,343,350,351].…”

The exact Schur index in closed form

“…Unflavored indices of N = 4 super Yang-Mills theories with gauge group SU (N ) were obtained in terms of elliptic integrals in [27]. In [39], it was further pointed out that when the gauge groups are SU (2N + 1), the unflavored indices are given by the generating function M N of MacMahon's generalized "sum-of-divisor" function,…”

“…Another series of non-Lagrangian theories whose Schur indices are related to those of Lagrangian theories were introduced in [39]. The theories in question are defined by conformally gauging different sets of D p (G) superconformal field theories [45,46].…”

“…The theories in question are defined by conformally gauging different sets of D p (G) superconformal field theories [45,46]. 14 Due to the restrictions considered in [39,46], at most four D p i (G) theories can be gauged along their common G-symmetry forming a quiver structure Γ[G] with one gauge node and four (or less) D p i (G)-legs, where Γ = D 4 , E 6,7,8 . It was pointed out in [39] that for a set of Γ and G such that Γ[G] is flavorless, the Schur index I Γ[G] is actually related to the one of N = 4 super Yang-Mills with gauge group G as,…”

Modular Anomaly Equation for Schur Index of $\mathcal{N}=4$ Super-Yang-Mills

“…Some remarkable (quasi)-modular properties of the index are studied recently in [14,15] in the context of a larger class of theories, based on some earlier works in e.g. [16,17].…”