Mock Modular Mathieu Moonshine Modules

TL;DR

This paper constructs super vertex operator algebras that establish new moonshine relations linking Mathieu groups with mock modular forms, providing explicit modules for these sporadic groups.

Contribution

It introduces the first explicit modules connecting Mathieu groups to mock modular forms via super vertex operator algebras and superconformal algebra decompositions.

Findings

Constructed super vertex operator algebras for Mathieu moonshine

Derived mock modular forms from twined partition functions

Identified modules with specific moonshine properties

Abstract

We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of Co_0 that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain N=4 superconformal algebra. Similarly, any subgroup of Co_0 that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain N=2 superconformal algebra. Through the decomposition of the corresponding twined partition functions into characters of the N=4 (resp. N=2) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensional Z-graded module for the corresponding group. The Mathieu groups are singled out amongst various other possibilities by the moonshine property: requiring the corresponding weak Jacobi forms to have certain asymptotic behaviour near cusps. Our constructions constitute the first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman--Sims, are also discussed.