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

$$ Z_{X_2^{(1)}}=\left(\varphi_{0,1}(\tau,z)^2+2E_{4}\varphi_{-2,1}(\tau,z)^2\right)=144C_2(z;\tau)-\sum_{a=2}^{2}\Sigma_{X_2^{(1)}}^{(a)}B_2^{(a)}(z;\tau), $$ $$Z_{X_2^{(2)}}=\varphi^{(3)}=\left(\varphi_{0,1}(\tau,z)^2-E_{4}\varphi_{-2,1}(\tau,z)^2\right)/24=6C_2(z;\tau)-\sum_{a=2}^{2}\Sigma_{X_2^{(2)}}^{(a)}B_2^{(a)}(z;\tau)$$ 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

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}

related items

• Quantum black holes, wall crossing and mock modular forms
• Mathieu moonshine
• monstrous moonshine
• Characters of superconformal algebra and mock theta functions

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxNW5LZDU3eGU0aVk/edit

expositions

• Cheng Umbral moonshine
• Harvey Moonshine and mock modular forms

articles

• Cheng, Miranda C. N., John F. R. Duncan, and Jeffrey A. Harvey. 2012. “Umbral Moonshine”. ArXiv e-print 1204.2779. http://arxiv.org/abs/1204.2779.