Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 5

The Conway Moonshine Module is a reflected K3 theory

Pages: 1247 – 1323



Anne Taormina (Centre for Particle Theory, Department of Mathematical Sciences, University of Durham, United Kingdom)

Katrin Wendland (Mathematics Institute, University of Freiburg, Germany)


Recently, Duncan and Mack-Crane established an isomorphism, as Virasoro modules at central charges $c = 12$, between the space of states of the Conway Moonshine Module and the space of states of a special K3 theory that was extensively studied some time ago by Gaberdiel, Volpato and the two authors. In the present work, we lift this result to the level of modules of the extensions of these Virasoro algebras to $N = 4$ super Virasoro algebras. Moreover, we relate the super vertex operator algebra and module structure of the Conway Moonshine Module to the operator product expansion of this special K3 theory by a procedure we call reflection. This procedure can be applied to certain superconformal field theories, transforming all fields to holomorphic ones. It also allows to describe certain superconformal field theories within the language of super vertex operator algebras. We discuss reflection and its limitations in general, and we argue that through reflection, the Conway Moonshine Module inherits from the K3 theory a richer structure than anticipated so far. The comparison between the Conway Moonshine Module and the K3 theory is considerably facilitated by exploiting the free fermion description as well as the lattice vertex operator algebra description of both theories. We include an explicit construction of cocycles for the relevant charge lattices, which are half integral. The transition from the K3 theory to the Conway Moonshine Module via reflection promotes the latter to the role of a medium that collects the symmetries of K3 theories from distinct points of the moduli space, thus uncovering a version of symmetry surfing in this context.

Published 7 July 2021