Umbral moonshine
imported>Pythagoras0님의 2013년 8월 5일 (월) 05:06 판 (→expositions)
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 $\varphi_{0,1}(\tau,z)$
$k=2$
- $k=2$ moonshine with $2.M_{12}$
- decomposition of $\varphi^{(3)}=\left(\varphi_{0,1}(\tau,z)^2-E_{4}\varphi_{-2,1}(\tau,z)^2\right)/24$
Jacobi form
$$ \varphi_{0,1}(\tau,z)=4\left[\left(\frac{\theta_{10}(z;\tau)}{\theta_{10}(0;\tau)}\right)+\left(\frac{\theta_{00}(z;\tau)}{\theta_{00}(0;\tau)}\right)+\left(\frac{\theta_{01}(z;\tau)}{\theta_{01}(0;\tau)}\right)\right],\\ \varphi_{-2,1}(\tau,z)=\frac{-\theta_{11}(z;\tau)^2}{\eta(\tau)^6} $$
$\mathcal{N}=4$ 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
umbral forms
umbral groups
umbral moonshine conjecture
- Quantum black holes, wall crossing and mock modular forms
- Mathieu moonshine
- monstrous moonshine
- Characters of superconformal algebra and mock theta functions
computational resource