Communications in Number Theory and Physics

Volume 14 (2020)

Number 3

Properties of extremal CFTs with small central charge

Pages: 609 – 697



Francesca Ferrari (Institute of Physics, University of Amsterdam, The Netherlands; International School for Advanced Studies (SISSA), Trieste, Italy; and INFN Sezione Trieste, Italy)

Sarah M. Harrison (Department of Mathematics & Statistics, and Department of Physics, McGill University, Montreal, Quebec, Canada)


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 descendants. The most 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.

2010 Mathematics Subject Classification

Primary 11F37, 20C34, 81T40. Secondary 11F22, 20C35.

The first-named author is supported in part by the MIUR-SIR grant RBSI1471GJ “Quantum Field Theories at Strong Coupling: Exact Computations and Applications.”

The second-named author is supported by the National Science and Engineering Council of Canada.

Received 12 January 2018

Accepted 20 February 2020

Published 13 July 2020