Vol. 11, No. 3, 2017

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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

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.

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umbral moonshine, mock theta function, vertex operator algebra
Mathematical Subject Classification 2010
Primary: 17B69
Secondary: 11F22, 11F37
Received: 27 January 2015
Revised: 5 October 2016
Accepted: 18 December 2016
Published: 6 May 2017
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