Properties of extremal CFTs with small central charge

TL;DR

This paper investigates the unique properties of extremal 2d chiral conformal field theories with small central charge, focusing on their connection to sporadic groups, moonshine phenomena, and Rademacher summability, revealing that most do not exhibit this summability.

Contribution

It analyzes known extremal CFTs with small central charge, comparing their moonshine properties and Rademacher summability, and identifies exceptions like the Conway module and specific superconformal theories.

Findings

Most extremal CFTs lack Rademacher summability property.

The Conway module and certain superconformal theories are exceptions.

Connections to sporadic groups and moonshine extend beyond Rademacher summability.

Abstract

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.