"Umbral moonshine"의 두 판 사이의 차이

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imported>Pythagoras0
(section 'expositions' updated)
 
(사용자 2명의 중간 판 5개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 
==introduction==
 
==introduction==
 
* generalization of [[Mathieu moonshine]]
 
* 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\}
 
\frac{24}{\ell-1}-1\in \{23,11,7,5,3,2\}
−
$$
+
</math>
 
* properties
 
* 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
 
* there exists a relation between all 23 cases of umbral moonshine and K3 sigma models
  
  
 
==examples==
 
==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
 
* 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),
 
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
 
where
−
$$
+
:<math>
 
\Sigma_{X_2^{(1)}}^{(1)}=q^{-1/12}(18-1872q-26070q^2-\cdots),
 
\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),
 
\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),
 
\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)
 
\Sigma_{X_2^{(2)}}^{(2)}=q^{-1/3}(-10q-44q^2-110q^3-\cdots)
−
$$
+
</math>
  
 
==Jacobi form==
 
==Jacobi form==
 
* [[Jacobi forms]]
 
* [[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_{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}
 
\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
 
* two types of representations : BPS and non-BPS
  
62번째 줄: 62번째 줄:
  
 
==umbral forms==
 
==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}
 
\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)}
 
H_{r,g}^{(\ell)}
−
$$
+
</math>
  
  
92번째 줄: 92번째 줄:
 
* [[Characters of superconformal algebra and mock theta functions]]
 
* [[Characters of superconformal algebra and mock theta functions]]
 
* [[K3 surfaces]]
 
* [[K3 surfaces]]
 +
* [[Mock Jacobi form]]
  
 
==computational resource==
 
==computational resource==
104번째 줄: 105번째 줄:
  
 
==articles==
 
==articles==
 +
* Miranda C. N. Cheng, John F. R. Duncan, Optimal Mock Jacobi Theta Functions, arXiv:1605.04480 [math.NT], May 14 2016, http://arxiv.org/abs/1605.04480
 
* Tohru Eguchi, Yuji Sugawara, Duality in N=4 Liouville Theory and Moonshine Phenomena, http://arxiv.org/abs/1603.02903v1
 
* 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.
 
* 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.
118번째 줄: 120번째 줄:
 
[[분류:Mock modular forms]]
 
[[분류:Mock modular forms]]
 
[[분류:Number theory and physics]]
 
[[분류:Number theory and physics]]
 +
[[분류:migrate]]
 +
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q7881385 Q7881385]
 +
===Spacy 패턴 목록===
 +
* [{'LOWER': 'umbral'}, {'LEMMA': 'moonshine'}]

2021년 2월 17일 (수) 02:01 기준 최신판

introduction

  • generalization of Mathieu moonshine
  • Let \(k\in \{1,2,3,4,6,8\}\) or \(\ell=k+1\in \{2,3,4,5,7,9\}\)

\[ \frac{24}{\ell-1}-1\in \{23,11,7,5,3,2\} \]

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


examples

\(k=1\)

  • 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

\[ Z_{X_2^{(1)}}=\left(\varphi_{0,1}(\tau,z)^2+2E_{4}\varphi_{-2,1}(\tau,z)^2\right)=144C_2(z;\tau)-\sum_{a=2}^{2}\Sigma_{X_2^{(1)}}^{(a)}B_2^{(a)}(z;\tau), \] \[Z_{X_2^{(2)}}=\varphi^{(3)}=\left(\varphi_{0,1}(\tau,z)^2-E_{4}\varphi_{-2,1}(\tau,z)^2\right)/24=6C_2(z;\tau)-\sum_{a=2}^{2}\Sigma_{X_2^{(2)}}^{(a)}B_2^{(a)}(z;\tau)\] where \[ \Sigma_{X_2^{(1)}}^{(1)}=q^{-1/12}(18-1872q-26070q^2-\cdots), \] \[ \Sigma_{X_2^{(1)}}^{(2)}=q^{-1/3}(3+510q+12804q^2+\cdots), \] \[ \Sigma_{X_2^{(2)}}^{(1)}=q^{-1/12}(1-16q-55q^2-144q^3-\cdots), \] \[ \Sigma_{X_2^{(2)}}^{(2)}=q^{-1/3}(-10q-44q^2-110q^3-\cdots) \]

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 with shadows

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

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

\[ H_{r,g}^{(\ell)} \]


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}

umbral moonshine conjecture

related items

computational resource


expositions

articles

  • Miranda C. N. Cheng, John F. R. Duncan, Optimal Mock Jacobi Theta Functions, arXiv:1605.04480 [math.NT], May 14 2016, http://arxiv.org/abs/1605.04480
  • 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.
  • Cheng, Miranda C. N., Sarah M. Harrison, Shamit Kachru, and Daniel Whalen. ‘Exceptional Algebra and Sporadic Groups at c=12’. arXiv:1503.07219 [hep-Th], 24 March 2015. http://arxiv.org/abs/1503.07219.
  • Duncan, John F. R., Michael J. Griffin, and Ken Ono. ‘Proof of the Umbral Moonshine Conjecture’. arXiv:1503.01472 [math], 4 March 2015. http://arxiv.org/abs/1503.01472.
  • Duncan, John F. R., and Jeffrey A. Harvey. “The Umbral Moonshine Module for the Unique Unimodular Niemeier Root System.” arXiv:1412.8191 [hep-Th], December 28, 2014. http://arxiv.org/abs/1412.8191.
  • Harvey, Jeffrey A., Sameer Murthy, and Caner Nazaroglu. “ADE Double Scaled Little String Theories, Mock Modular Forms and Umbral Moonshine.” arXiv:1410.6174 [hep-Th], October 22, 2014. http://arxiv.org/abs/1410.6174.
  • Cheng, Miranda C. N., and Sarah Harrison. “Umbral Moonshine and K3 Surfaces.” arXiv:1406.0619 [hep-Th], June 3, 2014. http://arxiv.org/abs/1406.0619.
  • Cheng, Miranda C. N., John F. R. Duncan, and Jeffrey A. Harvey. 2012. “Umbral Moonshine”. ArXiv e-print 1204.2779. http://arxiv.org/abs/1204.2779.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'umbral'}, {'LEMMA': 'moonshine'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Umbral_moonshine&oldid=51881"