Umbral moonshine - 편집 역사

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메타데이터: 새 문단

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imported>Pythagoras0: /* related items */

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section 'articles' updated

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articles

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articles

@@ 122번째 줄: / 122번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q7881385 Q7881385]

← 이전 판		2020년 12월 28일 (월) 08:05 판
121번째 줄:		121번째 줄:
	[[분류:Number theory and physics]]		[[분류:Number theory and physics]]
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q7881385 Q7881385]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
 * generalization of [[Mathieu moonshine]]
-* Let $k\in \{1,2,3,4,6,8\}$ or $\ell=k+1\in \{2,3,4,5,7,9\}$
+* 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 $|M_{24}|=244823040$
+** primes dividing <math>|M_{24}|=244823040</math>
-** $(p+1)|24$
+** <math>(p+1)|24</math>
-** $\rm{PSL}(2,\mathbb{F}_p)\subset M_{24}$
+** <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==
-===$k=1$===
+===<math>k=1</math>===
-* [[Mathieu moonshine]] corresponds to $k=1$ case
+* [[Mathieu moonshine]] corresponds to <math>k=1</math> case
-* decomposition of $Z_{K3}=2\varphi_{0,1}(\tau,z)$
+* decomposition of <math>Z_{K3}=2\varphi_{0,1}(\tau,z)</math>
-===$k=2$===
+===<math>k=2</math>===
-* $k=2$ moonshine with $2.M_{12}$
+* <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>
-$$Z_{X_2^{(2)}}=\varphi^{(3)}=\left(\varphi_{0,1}(\tau,z)^2-E_{4}\varphi_{-2,1}(\tau,z)^2\right)/24=6C_2(z;\tau)-\sum_{a=2}^{2}\Sigma_{X_2^{(2)}}^{(a)}B_2^{(a)}(z;\tau)$$
+:<math>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>
-==$\mathcal{N}=4$ super conformal algebra==
+==<math>\mathcal{N}=4</math> super conformal algebra==
-* $c=6k$, $k\in \mathbb{Z}_{\geq 1}$
+* <math>c=6k</math>, <math>k\in \mathbb{Z}_{\geq 1}</math>
 * two types of representations : BPS and non-BPS
@@ 62번째 줄: / 62번째 줄: @@
 ==umbral forms==
-* $H^{(\ell)}=(H_r^{\ell})_{1\leq r \leq \ell-1}$ is a vector valued mock modular form with shadows
+* <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 $\chi^{(\ell)}=24/(\ell-1)$
+where <math>\chi^{(\ell)}=24/(\ell-1)</math>
-* For example, $H^{(2)}$ is a mock modular form with shadow $24\eta(\tau)^3$
+* 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 $g\in G^{\ell}$
+* More generally, we have Mckay-Thompson series for each conjugacy class <math>g\in G^{\ell}</math>
-$$
+:<math>
 H_{r,g}^{(\ell)}
-$$
+</math>

← 이전 판		2020년 11월 13일 (금) 10:24 판
120번째 줄:		120번째 줄:
	[[분류:Mock modular forms]]		[[분류:Mock modular forms]]
	[[분류:Number theory and physics]]		[[분류:Number theory and physics]]
		+	[[분류:migrate]]

@@ 92번째 줄: / 92번째 줄: @@
 * [[Characters of superconformal algebra and mock theta functions]]
 * [[K3 surfaces]]
 ==computational resource==

@@ 104번째 줄: / 104번째 줄: @@
 ==articles==
 * Tohru Eguchi, Yuji Sugawara, Duality in N=4 Liouville Theory and Moonshine Phenomena, http://arxiv.org/abs/1603.02903v1
 * Cheng, Miranda C. N., Francesca Ferrari, Sarah M. Harrison, and Natalie M. Paquette. “Landau-Ginzburg Orbifolds and Symmetries of K3 CFTs.” arXiv:1512.04942 [hep-Th], December 15, 2015. http://arxiv.org/abs/1512.04942.

@@ 99번째 줄: / 99번째 줄: @@
 ==expositions==
 * [https://docs.google.com/file/d/0B8XXo8Tve1cxMTBuZWJ0RUVzSmc/edit Cheng Umbral moonshine]
 * [https://docs.google.com/file/d/0B8XXo8Tve1cxVWxBV3VGbWNZclk/edit Harvey Moonshine and mock modular forms]
 ==articles==