The umbral moonshine module conjecture

Let $\Lambda=\{2,3,4,5,7,13\}$. For each $\ell\in\Lambda$, let $G^{(\ell)}$ be the corresponding umbral group and let $H^{(\ell)}=(H^{(\ell)}_r)$ and $H^{(\ell)}_g=(H^{(\ell)}_{g,r})$ be the associated vector-valued mock modular forms. A naturally defined $\mathbb{Z}\times\mathbb{Q}$-graded module is written

$$K^{(\ell)}=\bigoplus_{0<r<\ell}K^{(\ell)}_r=\bigoplus_{\substack{r,k\in\mathbb{Z}\\0<r<\ell}}K^{(\ell)}_{r,k-r^2/4\ell}.$$

Umbral moonshine module conjecture. For every $\ell\in\Lambda$, such a $G^{(\ell)}$-module exists and satisfies

H^{(\ell)}_r(\tau)=-2\delta_{r,1}q^{-1/4\ell}+\sum_{\substack{k\in\mathbb{Z}\r^2-4k\ell<0}}\dim\left(K^{(\ell)}_{r,k-r^2/4\ell}\right)q^{k-r^2/4\ell},

where $q=\operatorname{e}(\tau)$, and

H^{(\ell)}_{g,r}(\tau)=-2\delta_{r,1}q^{-1/4\ell}+\sum_{\substack{k\in\mathbb{Z}\r^2-4k\ell<0}}\operatorname{tr}_{K^{(\ell)}_{r,k-r^2/4\ell}}(g)q^{k-r^2/4\ell}.

This conjecture proposes the infinite-dimensional graded modules underlying the McKay–Thompson series of umbral moonshine; the existence and graded-trace interpretation are not established in the source.