"Umbral moonshine"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 10번째 줄: | 10번째 줄: | ||
===$k=1$=== | ===$k=1$=== | ||
* [[Mathieu moonshine]] corresponds to $k=1$ case | * [[Mathieu moonshine]] corresponds to $k=1$ case | ||
| − | * decomposition of $\varphi_{0,1}(\tau,z)$ | + | * decomposition of $Z_{K3}=2\varphi_{0,1}(\tau,z)$ |
| 16번째 줄: | 16번째 줄: | ||
===$k=2$=== | ===$k=2$=== | ||
* $k=2$ moonshine with $2.M_{12}$ | * $k=2$ moonshine with $2.M_{12}$ | ||
| − | * decomposition of $\varphi^{(3)}=\left(\varphi_{0,1}(\tau,z)^2-E_{4}\varphi_{-2,1}(\tau,z)^2\right)/24$ | + | * decomposition of weight 0 and index 2 Jacobi forms |
| + | $$ | ||
| + | Z_{X_2^{(1)}}=\left(\varphi_{0,1}(\tau,z)^2+E_{4}\varphi_{-2,1}(\tau,z)^2\right), | ||
| + | $$ | ||
| + | $$Z_{X_2^{(2)}}=\varphi^{(3)}=\left(\varphi_{0,1}(\tau,z)^2-E_{4}\varphi_{-2,1}(\tau,z)^2\right)/24$$ | ||
| 38번째 줄: | 42번째 줄: | ||
==mock modular form== | ==mock modular form== | ||
| + | * [[Mock modular forms]] | ||
==umbral forms== | ==umbral forms== | ||
* $H^{(\ell)}=(H_r^{\ell})_{1\leq r \leq \ell-1}$ is a vector valued mock modular form | * $H^{(\ell)}=(H_r^{\ell})_{1\leq r \leq \ell-1}$ is a vector valued mock modular form | ||
| + | * $H^{(2)}$ is a mock modular form with shadow $24\eta(\tau)^3$ | ||
| + | |||
==umbral groups== | ==umbral groups== | ||
| 48번째 줄: | 55번째 줄: | ||
\hline | \hline | ||
G & M_{24} & M_{12} & & & &\\ | G & M_{24} & M_{12} & & & &\\ | ||
| − | \overline{G} & M_{24} & 2.M_{12} & & \\ | + | \overline{G} & M_{24} & 2.M_{12} & 2.AGL_{3}(2) & GL_2(5)/2 & SL_2(3) & Z/4 \\ |
\end{array} | \end{array} | ||
| + | |||
==umbral moonshine conjecture== | ==umbral moonshine conjecture== | ||
2013년 8월 5일 (월) 04:28 판
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
$$ Z_{X_2^{(1)}}=\left(\varphi_{0,1}(\tau,z)^2+E_{4}\varphi_{-2,1}(\tau,z)^2\right), $$ $$Z_{X_2^{(2)}}=\varphi^{(3)}=\left(\varphi_{0,1}(\tau,z)^2-E_{4}\varphi_{-2,1}(\tau,z)^2\right)/24$$
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
- $H^{(\ell)}=(H_r^{\ell})_{1\leq r \leq \ell-1}$ is a vector valued mock modular form
- $H^{(2)}$ is a mock modular form with shadow $24\eta(\tau)^3$
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) & Z/4 \\ \end{array}
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