Umbral moonshine

introduction

generalization of Mathieu moonshine
Let <math>k\in \{1,2,3,4,6,8\}</math> or <math>\ell=k+1\in \{2,3,4,5,7,9\}</math>

<math>

\frac{24}{\ell-1}-1\in \{23,11,7,5,3,2\} </math>

properties
- primes dividing <math>|M_{24}|=244823040</math>
- <math>(p+1)|24</math>
- <math>\rm{PSL}(2,\mathbb{F}_p)\subset M_{24}</math>
there exists a relation between all 23 cases of umbral moonshine and K3 sigma models

examples

<math>k=1</math>

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

<math>k=2</math>

<math>k=2</math> moonshine with <math>2.M_{12}</math>
decomposition of weight 0 and index 2 Jacobi forms

<math>

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), </math>

<math>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)</math>

where

<math>

\Sigma_{X_2^{(1)}}^{(1)}=q^{-1/12}(18-1872q-26070q^2-\cdots), </math>

<math>

\Sigma_{X_2^{(1)}}^{(2)}=q^{-1/3}(3+510q+12804q^2+\cdots), </math>

<math>

\Sigma_{X_2^{(2)}}^{(1)}=q^{-1/12}(1-16q-55q^2-144q^3-\cdots), </math>

<math>

\Sigma_{X_2^{(2)}}^{(2)}=q^{-1/3}(-10q-44q^2-110q^3-\cdots) </math>

Jacobi form

Jacobi forms

<math>

\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} </math>

<math>\mathcal{N}=4</math> super conformal algebra

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

extremal Jacobi forms

mock modular form

Mock modular forms

umbral forms

<math>H^{(\ell)}=(H_r^{\ell})_{1\leq r \leq \ell-1}</math> is a vector valued mock modular form with shadows

<math>

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

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

<math>

H_{r,g}^{(\ell)} </math>

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}