The umbral moonshine module for the unique unimodular Niemeier root system

Duncan, John; Harvey, Jeffrey

doi:10.2140/ant.2017.11.505

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The umbral moonshine module for the unique unimodular Niemeier root system

John F. R. Duncan and Jeffrey A. Harvey

Vol. 11 (2017), No. 3, 505–535

DOI: 10.2140/ant.2017.11.505

Abstract

We use canonically twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan’s fifth-order mock theta functions appear as components.

Keywords

umbral moonshine, mock theta function, vertex operator algebra

Mathematical Subject Classification 2010

Primary: 17B69

Secondary: 11F22, 11F37

Milestones

Received: 27 January 2015

Revised: 5 October 2016

Accepted: 18 December 2016

Published: 6 May 2017

Authors

John F. R. Duncan
Department of Mathematics and Computer Science Emory University Atlanta, GA 30322 United States

Jeffrey A. Harvey
Enrico Fermi Institute and Department of Physics University of Chicago Chicago, IL 60637 United States

Vol. 11, No. 3, 2017