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 $|M_{24}|$
  • $(p+1)|24$
  • $\rm{PSL}(2,\mathbb{F}_p)\subset M_{24}$
• Mathieu moonshine corresponds to $k=1$ case
• $k=2$ moonshine with $2.M_{12}$

Jacobi form

• Jacobi forms

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

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

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

• https://docs.google.com/file/d/0B8XXo8Tve1cxMTBuZWJ0RUVzSmc/edit