Abstract
Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with 23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine have a uniform relation to symmetries of K3 string theories. Moreover, a supersymmetric vertex operator algebra with Conway sporadic symmetry also enjoys a close relation to the K3 elliptic genus. Inspired by the above two relations between moonshine and K3 string theory, we construct a chiral CFT by orbifolding the free theory of 24 chiral fermions and two pairs of fermionic and bosonic ghosts. In this paper we mainly focus on the case of umbral moonshine corresponding to the Niemeier lattice with root system given by 6 copies of D4 root system. This CFT then leads to the construction of an infinite-dimensional graded module for the umbral group {G^{D_4^{oplus 6}}} whose graded characters coincide with the umbral moonshine functions. We also comment on how one can recover all umbral moonshine functions corresponding to the Niemeier root systems {A_5^{oplus 4}D_4}, {A_7^{oplus 2}D_5^{oplus 2}}, {A_{11}D_7 E_6}, {A_{17}E_7}, and {D_{10}E_7^{oplus 2}}.
Highlights
The moonshine phenomenon, the study of which began with the discovery of monstrous moonshine [13], describes an interesting connection between modular objects and finite groups
Recent years have seen a surge of interest in moonshine, initiated by the observation [25] that the elliptic genus of K 3 surfaces has a close relation to the representation theory of the sporadic finite group M24
It was soon realised that this M24 connection is but one of the 23 instances of the umbral moonshine [4,5], which associates distinguished mock modular forms to elements of finite groups arising from the symmetries of specific lattices
Summary
The moonshine phenomenon, the study of which began with the discovery of monstrous moonshine [13], describes an interesting connection between modular objects and finite groups. It was conjectured that all the weak Jacobi forms arising from (the 4-plane preserving part of) Conway and umbral moonshine are realised as K 3 elliptic genera twined by a supersymmetry-preserving symmetry of the sigma model at certain points in the moduli space. As a result we are left with three Z-graded G D4⊕6 -modules, corresponding to the three components (H1X , H3X , H5X ) of the vector-valued mock modular forms in the X = D4⊕6 case of umbral moonshine. The first element of this construction is the following relation between the the weak Jacobi form φgX arising from umbral moonshine and Conway moonshine. 2 we review how umbral and Conway moonshine lead to the weak Jacobi forms φgX and φ±,g respectively, for every g ∈ G X in the former case and every 4-plane preserving element g of Co0 in the latter case. We collect details of the ghost theory and relevant data in the appendices
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