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}