Open Access

Mock modular Mathieu moonshine modules

  • Miranda C N Cheng1Email author,
  • Xi Dong2,
  • John F R Duncan3,
  • Sarah Harrison2,
  • Shamit Kachru2 and
  • Timm Wrase2
Research in the Mathematical Sciences20152:13

DOI: 10.1186/s40687-015-0034-9

Received: 1 July 2014

Accepted: 15 June 2015

Published: 28 July 2015

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 \(\textit{Co}_0\) that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain \({\mathcal {N}}=4\) superconformal algebra. Similarly, any subgroup of \(\textit{Co}_0\) that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain \({\mathcal {N}}=2\) superconformal algebra. Through the decomposition of the corresponding twined partition functions into characters of the \({\mathcal {N}}=4\) (resp. \({\mathcal {N}}=2\)) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensional \({\mathbb {Z}}\)-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.