Umbral moonshine

introduction

$k\in \{1,2,3,4,6,8\}$ or $\ell=k+1\in \{2,3,4,5,7,9\}$

$$ \frac{24}{\ell-1}-1\in \{23,11,7,5,3,2\} $$

properties
- primes dividing $|M_{24}|$
- $(p+1)|24$
- $\rm{PSL}(2,\mathbb{F}_p)\subset M_{24}$

$k=1$

Mathieu moonshine corresponds to $k=1$ case
decomposition of $Z_{K3}=2\varphi_{0,1}(\tau,z)$

$k=2$

$k=2$ moonshine with $2.M_{12}$
decomposition of weight 0 and index 2 Jacobi forms

$$ Z_{X_2^{(1)}}=\left(\varphi_{0,1}(\tau,z)^2+E_{4}\varphi_{-2,1}(\tau,z)^2\right), $$ $$Z_{X_2^{(2)}}=\varphi^{(3)}=\left(\varphi_{0,1}(\tau,z)^2-E_{4}\varphi_{-2,1}(\tau,z)^2\right)/24$$

Jacobi form

Jacobi forms

$$ \varphi_{0,1}(\tau,z)=4\left[\left(\frac{\theta_{10}(z;\tau)}{\theta_{10}(0;\tau)}\right)+\left(\frac{\theta_{00}(z;\tau)}{\theta_{00}(0;\tau)}\right)+\left(\frac{\theta_{01}(z;\tau)}{\theta_{01}(0;\tau)}\right)\right],\\ \varphi_{-2,1}(\tau,z)=\frac{-\theta_{11}(z;\tau)^2}{\eta(\tau)^6} $$

$\mathcal{N}=4$ super conformal algebra

$c=6k$, $k\in \mathbb{Z}_{\geq 1}$
two types of representations : BPS and non-BPS

extremal Jacobi forms

mock modular form

Mock modular forms

umbral forms

$H^{(\ell)}=(H_r^{\ell})_{1\leq r \leq \ell-1}$ is a vector valued mock modular form
$H^{(2)}$ is a mock modular form with shadow $24\eta(\tau)^3$

umbral groups

\begin{array}{l|l|l|I|I} \ell & 2 & 3 & 4 & 5 & 7 & 9 \\ \hline G & M_{24} & M_{12} & & & &\\ \overline{G} & M_{24} & 2.M_{12} & 2.AGL_{3}(2) & GL_2(5)/2 & SL_2(3) & Z/4 \\ \end{array}