The Moonshine Anomaly

TL;DR

This paper investigates the order and nature of a specific cohomology class associated with the Monster group in moonshine theory, revealing it has order 24 and is not a Chern class, using a novel finite-group T-duality approach.

Contribution

It introduces a finite-group version of T-duality to analyze the Moonshine anomaly and establishes its order and non-Chern class nature.

Findings

The anomaly class has order exactly 24.

The anomaly class is not a Chern class.

Finite-group T-duality relates the anomaly to the Leech lattice CFT.

Abstract

The anomaly for the Monster group $\mathbb{M}$ acting on its natural (aka moonshine) representation $V^\natural$ is a particular cohomology class $\omega^\natural \in \mathrm{H}^3(\mathbb{M},\mathrm{U}(1))$ that arises as a conformal field theoretic generalization of the second Chern class of a representation. This paper shows that $\omega^\natural$ has order exactly $24$ and is not a Chern class. In order to perform this computation, this paper introduces a finite-group version of T-duality, which is used to relate $\omega^\natural$ to the anomaly for the Leech lattice CFT.