Module constructions for certain subgroups of the largest Mathieu group

Abstract

For certain subgroups of M 24 , we give vertex operator algebraic module constructions whose associated trace functions are meromorphic Jacobi forms. These meromorphic Jacobi forms are canonically associated to the mock modular forms of Mathieu moonshine. The construction is related to the Conway moonshine module and employs a technique introduced by Anagiannis-Cheng-Harrison. With this construction we are able to give concrete vertex algebraic realizations of certain cuspidal Hecke eigenforms of weight two. In particular, we give explicit realizations of trace functions whose integralities are equivalent to divisibility conditions on the number of Fp points on the Jacobians of modular curves.

Cone Vertex Algebras, Mock Theta Functions, and Umbral Moonshine Modules

“…while (1.6) appears in the construction of the optimal meromorphic Jacobi forms associated to the umbral McKay-Thompson series [11], [14]. The latter fact allows us to draw a connection between the construction of modules for the McKay-Thompson series (as considered in this paper) and the meromorphic module problem considered in [22], [24]. Furthermore, the specialized Appell-Lerch sum (1.6) is also interesting because it captures the non-modular part of the elliptic genus of non-compact supersymmetric coset models, as featured in [27], [28], [29].…”

“…For some instances of umbral moonshine it has already been shown that suitable (super) vertex operator algebras can be used to explicitly construct the modules ǨX [20], [21] or to solve the so called "meromorphic module problem", i.e. building modules such that specific trace functions give the meromorphic Jacobi forms associated to the H g of Umbral Moonshine [22], [23], [24]. In particular, in [20] the authors built the module ǨE 3 8 through the means of particular vertex operator algebras obtained from lattice cones.…”