The Moonshine Anomaly

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.