"Characters of superconformal algebra and mock theta functions"의 두 판 사이의 차이

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2번째 줄: 2번째 줄:
  
  
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==$\mathcal{N}=4$ superconformal algebra==
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==<math>\mathcal{N}=4</math> superconformal algebra==
 
===generators and relations===
 
===generators and relations===
 
* [[Virasoro algebra]]
 
* [[Virasoro algebra]]
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$$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}$$
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:<math>[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}</math>
 
* [[Affine sl(2)]]
 
* [[Affine sl(2)]]
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$$[J_m^i,J_n^j]=\epsilon_{ijk}J_{m+n}^k+\delta_{m+n}\delta^{i,j}\frac{c}{3},\quad i,j,k\in \{1,2,3\},\quad m,n\in \mathbb{Z}$$
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:<math>[J_m^i,J_n^j]=\epsilon_{ijk}J_{m+n}^k+\delta_{m+n}\delta^{i,j}\frac{c}{3},\quad i,j,k\in \{1,2,3\},\quad m,n\in \mathbb{Z}</math>
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$$[L_m,J_n^i]=-nJ_{m+n}^i,\quad m,n\in \mathbb{Z}$$
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:<math>[L_m,J_n^i]=-nJ_{m+n}^i,\quad m,n\in \mathbb{Z}</math>
 
* fermionic operators
 
* fermionic operators
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$$
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:<math>
 
G_r^a,\overline{G}_s^b,\quad a,b\in \{1,2\}
 
G_r^a,\overline{G}_s^b,\quad a,b\in \{1,2\}
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$$
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</math>
  
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===$c=6k$ with $k=1$ case===
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===<math>c=6k</math> with <math>k=1</math> case===
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* non-BPS characters : $h>k/4,\ell=1/2$
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* non-BPS characters : <math>h>k/4,\ell=1/2</math>
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$$
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:<math>
 
\operatorname{ch}^{\tilde R}_{h=1/4+n,\ell=0}=q^{h-3/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}=q^{n-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}
 
\operatorname{ch}^{\tilde R}_{h=1/4+n,\ell=0}=q^{h-3/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}=q^{n-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}
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$$
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</math>
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* BPS characters : $h=1/4,\ell=0,1/2$
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* BPS characters : <math>h=1/4,\ell=0,1/2</math>
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$$
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:<math>
 
\operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}\mu(z;\tau)\\
 
\operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}\mu(z;\tau)\\
 
\operatorname{ch}^{\tilde R}_{h=1/4,\ell=1/2}+2\operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=q^{-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}
 
\operatorname{ch}^{\tilde R}_{h=1/4,\ell=1/2}+2\operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=q^{-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}
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$$
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</math>
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where $\mu(z;\tau)$ is the [[Appell-Lerch sums]] which is a holomorphic part of a mock modular form
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where <math>\mu(z;\tau)</math> is the [[Appell-Lerch sums]] which is a holomorphic part of a mock modular form
 
* this is related to [[Mathieu moonshine]] and the [[elliptic genus]] of K3 surface
 
* this is related to [[Mathieu moonshine]] and the [[elliptic genus]] of K3 surface
  
  
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===$k\geq 2$ case===
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===<math>k\geq 2</math> case===
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* this is related to [[Umbral moonshine]] and elliptic genus of hyperKahler manifolds of complex dimension $2k$
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* this is related to [[Umbral moonshine]] and elliptic genus of hyperKahler manifolds of complex dimension <math>2k</math>
  
  
37번째 줄: 37번째 줄:
  
 
==history==
 
==history==
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* 1986 Eguchi-Taoimina $\mathcal{N}=4$ superconformal algebra
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* 1986 Eguchi-Taoimina <math>\mathcal{N}=4</math> superconformal algebra
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* 1990 Odake, $\mathcal{N}=2$ superconformal algebra
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* 1990 Odake, <math>\mathcal{N}=2</math> superconformal algebra
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
  

2020년 11월 13일 (금) 07:35 판

introduction

\(\mathcal{N}=4\) superconformal algebra

generators and relations

\[[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}\]

\[[J_m^i,J_n^j]=\epsilon_{ijk}J_{m+n}^k+\delta_{m+n}\delta^{i,j}\frac{c}{3},\quad i,j,k\in \{1,2,3\},\quad m,n\in \mathbb{Z}\] \[[L_m,J_n^i]=-nJ_{m+n}^i,\quad m,n\in \mathbb{Z}\]

  • fermionic operators

\[ G_r^a,\overline{G}_s^b,\quad a,b\in \{1,2\} \]

\(c=6k\) with \(k=1\) case

  • non-BPS characters \[h>k/4,\ell=1/2\]

\[ \operatorname{ch}^{\tilde R}_{h=1/4+n,\ell=0}=q^{h-3/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}=q^{n-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3} \]

  • BPS characters \[h=1/4,\ell=0,1/2\]

\[ \operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}\mu(z;\tau)\\ \operatorname{ch}^{\tilde R}_{h=1/4,\ell=1/2}+2\operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=q^{-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3} \] where \(\mu(z;\tau)\) is the Appell-Lerch sums which is a holomorphic part of a mock modular form


\(k\geq 2\) case

  • this is related to Umbral moonshine and elliptic genus of hyperKahler manifolds of complex dimension \(2k\)




history



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