We analyze aspects of extant examples of 2d extremal chiral (super)conformal field theories with $c\leq 24$. These are theories whose only operators with dimension smaller or equal to $c/24$ are the vacuum and its (super)Virasoro descendents. The prototypical example is the monster CFT, whose famous genus zero property is intimately tied to the Rademacher summability of its twined partition functions, a property which also distinguishes the functions of Mathieu and umbral moonshine. However, there are now several additional known examples of extremal CFTs, all of which have at least $\mathcal N=1$ supersymmetry and global symmetry groups connected to sporadic simple groups. We investigate the extent to which such a property, which distinguishes the monster moonshine module from other $c=24$ chiral CFTs, holds for the other known extremal theories. We find that in most cases, the special Rademacher summability property present for monstrous and umbral moonshine does not hold for the other extremal CFTs, with the exception of the Conway module and two $c=12, ~\mathcal N=4$ superconformal theories with $M_{11}$ and $M_{22}$ symmetry. This suggests that the connection between extremal CFT, sporadic groups, and mock modular forms transcends strict Rademacher summability criteria.

中文翻译：

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具有小中心电荷的极值 CFT 的特性

我们用 $c\leq 24$ 分析了二维极值手征（超）共形场论的现存例子的各个方面。这些理论的维数小于或等于 $c/24$ 的唯一算子是真空及其（超级）Virasoro 后代。典型的例子是怪物 CFT，其著名的零属属性与其缠绕分区函数的 Rademacher 可和性密切相关，该属性也区分了 Mathieu 和本影月光的功能。但是，现在还有几个已知的极值 CFT 示例，所有这些示例都至少具有 $\mathcal N=1$ 超对称性和全局对称性群连接到零星单群。我们调查了这种将怪物月光模块与其他 $c=24$ 手性 CFT 区分开来的特性在多大程度上适用于其他已知的极值理论。我们发现，在大多数情况下，除了 Conway 模块和两个 $c=12, ~\mathcal N=4$ 超共形理论之外，对于其他极值 CFT 而言，存在于 monstrous 和 umbral Moonshine 的特殊 Rademacher 可和性属性不成立$M_{11}$ 和 $M_{22}$ 对称。这表明极值 CFT、零星组和模拟模块化形式之间的联系超越了严格的 Rademacher 可总结性标准。