Mock modular Mathieu moonshine modules

• Published in: Research in the mathematical sciences Vol. 2; no. 1
• Main Authors: Cheng, Miranda C N, Dong, Xi, Duncan, John F R, Harrison, Sarah, Kachru, Shamit, Wrase, Timm
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-12-2015
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Mathematics; Mathematics and Statistics; Elliptic Genus; Sporadic Group; Superconformal Algebra; Jacobi Form; Global Symmetry Group
• ISSN: 2197-9847; 2197-9847
• DOI: 10.1186/s40687-015-0034-9

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.