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

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$$
 
$$
 
\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}
+
\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
 
where $\mu(z;\tau)$ 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
  
  

2013년 8월 9일 (금) 15:56 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 $2k$




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