"Supersymmetric minimal models"의 두 판 사이의 차이

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imported>Pythagoras0
1번째 줄: 1번째 줄:
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==introduction==
 +
* The (normalized) characters of a generic  $N$=1 superconformal
 +
minimal model $\cal{SM}(p,p')$ are given by
 +
$$
 +
\hat{\chi}_{r,s}^{(p,p')}(q) = \hat{\chi}_{p-r,p'-s}^{(p,p')}(q) =
 +
{(-q^{\varepsilon_{r-s}})_\infty \over (q)_\infty}
 +
~\sum_{\ell\in \ZZ} \left( q^{\ell(\ell pp'+rp'-sp)/2}
 +
          -q^{(\ell p+r)(\ell p'+s)/2}  \right)~,
 +
$$
 +
 +
where
 +
$$
 +
\varepsilon_a=
 +
\begin{cases} 1/2, & \text{if $a$ is even$\leftrightarrow$ NS sector}\\ 1, & \text{if $a$ is odd$\leftrightarrow$ ~R~ sector} \\ \end{cases}
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$$
 +
 +
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==the first type==
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* for $s=1,3\cdots, 2k-1$
 +
$$
 +
\begin{aligned}
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\hat{\chi}_{1,s}^{(2,4k)}&(q) ~=
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  \sum_{m_1,\ldots,m_{k-1}=0}^\infty
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  {(-q^{1/2})_{N_1} ~q^{{1\over 2}N_1^2+N_2^2+\ldots+N_{k-1}^2
 +
        +N_{(s+1)/2}+N_{(s+3)/2}+\ldots+N_{k-1}} \over
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  (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}}  \\
 +
  &=    \sum_{m_1,\ldots,m_{k}=0}^\infty
 +
  {q^{N_1^2+N_2^2+\ldots+N_{k-1}^2
 +
        +N_{(s+1)/2}+N_{(s+3)/2}+\ldots+N_{k-1}-N_1 m_k+{1\over 2}m_k^2} \over
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  (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} {N_1 \choose m_k}_q
 +
\end{aligned}
 +
$$
 +
* for $s=2$ and $s=2k$
 +
$$
 +
\eqalign{\hat{\chi}_{1,2}^{(2,4k)}&(q) ~=
 +
  \sum_{m_1,\ldots,m_{k-1}=0}^\infty
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  {(-q)_{N_1}~q^{{1\over 2}N_1(N_1+1)+N_2(N_2+1)+\ldots+N_{k-1}(N_{k-1}+1)} \over
 +
  (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}}    \cr
 +
  &=  \sum_{m_1,\ldots,m_{k}=0}^\infty
 +
  {q^{N_1(N_1+1)+N_2(N_2+1)+\ldots+N_{k-1}(N_{k-1}+1)-N_1 m_k
 +
      +{1\over 2}m_k(m_k-1)} \over
 +
  (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} {N_1 \choose m_k}_q ~~,\cr
 +
\hat{\chi}_{1,2k}^{(2,4k)}&(q) ~=
 +
  \sum_{m_1,\ldots,m_{k-1}=0}^\infty
 +
  {(-1)_{N_1} ~q^{{1\over 2}N_1(N_1+1)+N_2^2+\ldots+N_{k-1}^2} \over
 +
    (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} \cr
 +
  &=    \sum_{m_1,\ldots,m_{k}=0}^\infty
 +
  {q^{N_1^2+N_2^2+\ldots+N_{k-1}^2
 +
        -N_1 m_k+{1\over 2}m_k(m_k+1)} \over
 +
  (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} {N_1 \choose m_k}_q ~~.\cr}
 +
$$
 +
 +
 +
==second type==
 +
* The second type of fermionic forms for the characters of the same family of models $\cal{SM}(2,4k)$ is presented in the
 +
following conjecture:
 +
For $k=2,3,4,\ldots$ and $s=1,2,\ldots,2k$
 +
* $s$ is odd
 +
$$
 +
\hat{\chi}_{1,s}^{(2,4k)}(q) =
 +
  \sum_{m_1,\ldots,m_{2k-2}=0}^\infty
 +
  {q^{{1\over 2}(M_1^2+M_2^2+\ldots+M_{2k-2}^2)
 +
        +M_s+M_{s+2}+\ldots+M_{2k-3}} \over
 +
  (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{2k-2}}}
 +
$$
 +
* $s$ is even
 +
$$
 +
\hat{\chi}_{1,s}^{(2,4k)}(q)=
 +
\sum_{m_1,\ldots,m_{2k-2}=0}^\infty
 +
  {q^{{1\over 2}(M_1^2+M_2^2+\ldots+M_{2k-2}^2)
 +
        +M_s+M_{s+2}+\ldots+M_{2k-2} +{1\over 2}\tilde{M}} \over
 +
  (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{2k-2}}}
 +
$$
 +
where
 +
$$
 +
M_j = m_j +m_{j+1}+\ldots+m_{2k-2},
 +
  \tilde{M}=m_1+m_3+\ldots+m_{2k-3}
 +
$$
 +
 +
 +
 
==related items==
 
==related items==
 
* [[Minimal models]]
 
* [[Minimal models]]

2013년 7월 14일 (일) 14:06 판

introduction

  • The (normalized) characters of a generic $N$=1 superconformal

minimal model $\cal{SM}(p,p')$ are given by $$ \hat{\chi}_{r,s}^{(p,p')}(q) = \hat{\chi}_{p-r,p'-s}^{(p,p')}(q) = {(-q^{\varepsilon_{r-s}})_\infty \over (q)_\infty} ~\sum_{\ell\in \ZZ} \left( q^{\ell(\ell pp'+rp'-sp)/2} -q^{(\ell p+r)(\ell p'+s)/2} \right)~, $$

where $$ \varepsilon_a= \begin{cases} 1/2, & \text{if $a$ is even$\leftrightarrow$ NS sector}\\ 1, & \text{if $a$ is odd$\leftrightarrow$ ~R~ sector} \\ \end{cases} $$


the first type

  • for $s=1,3\cdots, 2k-1$

$$ \begin{aligned} \hat{\chi}_{1,s}^{(2,4k)}&(q) ~= \sum_{m_1,\ldots,m_{k-1}=0}^\infty {(-q^{1/2})_{N_1} ~q^{{1\over 2}N_1^2+N_2^2+\ldots+N_{k-1}^2 +N_{(s+1)/2}+N_{(s+3)/2}+\ldots+N_{k-1}} \over (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} \\ &= \sum_{m_1,\ldots,m_{k}=0}^\infty {q^{N_1^2+N_2^2+\ldots+N_{k-1}^2 +N_{(s+1)/2}+N_{(s+3)/2}+\ldots+N_{k-1}-N_1 m_k+{1\over 2}m_k^2} \over (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} {N_1 \choose m_k}_q \end{aligned} $$

  • for $s=2$ and $s=2k$

$$ \eqalign{\hat{\chi}_{1,2}^{(2,4k)}&(q) ~= \sum_{m_1,\ldots,m_{k-1}=0}^\infty {(-q)_{N_1}~q^{{1\over 2}N_1(N_1+1)+N_2(N_2+1)+\ldots+N_{k-1}(N_{k-1}+1)} \over (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} \cr &= \sum_{m_1,\ldots,m_{k}=0}^\infty {q^{N_1(N_1+1)+N_2(N_2+1)+\ldots+N_{k-1}(N_{k-1}+1)-N_1 m_k +{1\over 2}m_k(m_k-1)} \over (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} {N_1 \choose m_k}_q ~~,\cr \hat{\chi}_{1,2k}^{(2,4k)}&(q) ~= \sum_{m_1,\ldots,m_{k-1}=0}^\infty {(-1)_{N_1} ~q^{{1\over 2}N_1(N_1+1)+N_2^2+\ldots+N_{k-1}^2} \over (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} \cr &= \sum_{m_1,\ldots,m_{k}=0}^\infty {q^{N_1^2+N_2^2+\ldots+N_{k-1}^2 -N_1 m_k+{1\over 2}m_k(m_k+1)} \over (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{k-1}}} {N_1 \choose m_k}_q ~~.\cr} $$


second type

  • The second type of fermionic forms for the characters of the same family of models $\cal{SM}(2,4k)$ is presented in the

following conjecture: For $k=2,3,4,\ldots$ and $s=1,2,\ldots,2k$

  • $s$ is odd

$$ \hat{\chi}_{1,s}^{(2,4k)}(q) = \sum_{m_1,\ldots,m_{2k-2}=0}^\infty {q^{{1\over 2}(M_1^2+M_2^2+\ldots+M_{2k-2}^2) +M_s+M_{s+2}+\ldots+M_{2k-3}} \over (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{2k-2}}} $$

  • $s$ is even

$$ \hat{\chi}_{1,s}^{(2,4k)}(q)= \sum_{m_1,\ldots,m_{2k-2}=0}^\infty {q^{{1\over 2}(M_1^2+M_2^2+\ldots+M_{2k-2}^2) +M_s+M_{s+2}+\ldots+M_{2k-2} +{1\over 2}\tilde{M}} \over (q)_{m_1}(q)_{m_2} \ldots (q)_{m_{2k-2}}} $$ where $$ M_j = m_j +m_{j+1}+\ldots+m_{2k-2}, \tilde{M}=m_1+m_3+\ldots+m_{2k-3} $$


related items


computational resource


articles

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