Umbral moonshine

introduction

generalization of Mathieu moonshine
Let $k\in \{1,2,3,4,6,8\}$ 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 $|M_{24}|=244823040$
- $(p+1)|24$
- $\rm{PSL}(2,\mathbb{F}_p)\subset M_{24}$
there exists a relation between all 23 cases of umbral moonshine and K3 sigma models

examples

$k=1$

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

$$ Z_{X_2^{(1)}}=\left(\varphi_{0,1}(\tau,z)^2+2E_{4}\varphi_{-2,1}(\tau,z)^2\right)=144C_2(z;\tau)-\sum_{a=2}^{2}\Sigma_{X_2^{(1)}}^{(a)}B_2^{(a)}(z;\tau), $$ $$Z_{X_2^{(2)}}=\varphi^{(3)}=\left(\varphi_{0,1}(\tau,z)^2-E_{4}\varphi_{-2,1}(\tau,z)^2\right)/24=6C_2(z;\tau)-\sum_{a=2}^{2}\Sigma_{X_2^{(2)}}^{(a)}B_2^{(a)}(z;\tau)$$ where $$ \Sigma_{X_2^{(1)}}^{(1)}=q^{-1/12}(18-1872q-26070q^2-\cdots), $$ $$ \Sigma_{X_2^{(1)}}^{(2)}=q^{-1/3}(3+510q+12804q^2+\cdots), $$ $$ \Sigma_{X_2^{(2)}}^{(1)}=q^{-1/12}(1-16q-55q^2-144q^3-\cdots), $$ $$ \Sigma_{X_2^{(2)}}^{(2)}=q^{-1/3}(-10q-44q^2-110q^3-\cdots) $$

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 with shadows

$$ \chi^{(\ell)}\cdot S_{r}^{(\ell)}=\sum_{n\in \mathbb{Z}}(2\ell n+r)q^{(2\ell n+r)^2/4\ell} $$ where $\chi^{(\ell)}=24/(\ell-1)$

For example, $H^{(2)}$ is a mock modular form with shadow $24\eta(\tau)^3$
More generally, we have Mckay-Thompson series for each conjugacy class $g\in G^{\ell}$

$$ H_{r,g}^{(\ell)} $$

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) & \mathbb{Z}/4 \\ \end{array}