Super vertex algebras, meromorphic Jacobi forms and umbral moonshine

Abstract

The vector-valued mock modular forms of umbral moonshine may be repackaged into meromorphic Jacobi forms of weight one. In this work we constructively solve two cases of the meromorphic module problem for umbral moonshine. Specifically, for the type A Niemeier root systems with Coxeter numbers seven and thirteen, we construct corresponding bigraded super vertex operator algebras, equip them with actions of the corresponding umbral groups, and verify that the resulting trace functions on canonically twisted modules recover the meromorphic Jacobi forms that are specified by umbral moonshine. We also obtain partial solutions to the meromorphic module problem for the type A Niemeier root systems with Coxeter numbers four and five, by constructing super vertex operator algebras that recover the meromorphic Jacobi forms attached to maximal subgroups of the corresponding umbral groups.

A Borcherds–Kac–Moody Superalgebra with Conway Symmetry

“…Of course, for Monstrous and Conway moonshines, we have explicit moonshine modules in hand. Notably, a uniform construction of the umbral moonshine modules is still lacking (see, however, [1,17,29,32]).…”

K3 Elliptic Genus and an Umbral Moonshine Module

“…In this paper we construct a module for the D ⊕6 4 case of umbral moonshine. This is the first time that the module is constructed for a case of umbral moonshine with a sizeable umbral finite group (with |G D ⊕6 4 | ∼ 10 3 , this group is larger than the cases of umbral moonshine where modules have been constructed previously in [7,12,13], where the groups have order dividing 24). This is also the first construction of the umbral module which utilises the connection to symmetries of K3 string theory.…”

“…As a result, one can construct modules forG < 2.M 12 compatible with the corresponding case of umbral moonshine, for three of the maximal subgroups of 2.M 12 . For completeness we list the explicit generators of these three maximal subgroups in terms of permutation groups on 24 objects: 18,5,9,24,16) (2,6,8,11,17,20) (3,12,23,13,22,14) (4,10,19,15,21,7), (1,9)…”

“…is not yet clear. That said, encouraging results have been obtained for certain cases of umbral moonshine with fairly small groups 1 : the cases where the Niemeier root systems X are given by E ⊕3 8 [12], A ⊕4 6 and A ⊕2 12 [13], as well as D ⊕4 6 , D ⊕3 8 , D ⊕2 12 and D 24 [7]. The construction in [12] is rather different from those in [7,13]: the former relies on special identities satisfied by the mock modular forms H E ⊕3 8 g,r and the latter constructs H X g,r through the mock Jacobi forms Ψ X g and their associated meromorphic Jacobi forms.…”

Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine Modules

“…In the more recent work [14], and in the present paper, the focus is shifted from the vectorvalued mock modular forms H (ℓ) g to certain meromorphic Jacobi forms. The former may be obtained from the latter once a canonically defined polar part that captures the poles of the latter is removed.…”