"Supersymmetric minimal models"의 두 판 사이의 차이
imported>Pythagoras0 |
imported>Pythagoras0 |
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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 | ||
+ | $$ | ||
+ | \varepsilon_a= | ||
+ | \begin{cases} 1/2, & \text{if $a$ is even$\leftrightarrow$ NS sector}\\ 1, & \text{if $a$ is odd$\leftrightarrow$ ~R~ sector} \\ \end{cases} | ||
+ | $$ | ||
+ | |||
+ | |||
+ | ==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== | ==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 $$ M_j = m_j +m_{j+1}+\ldots+m_{2k-2}, \tilde{M}=m_1+m_3+\ldots+m_{2k-3} $$
computational resource
articles
- Melzer, Ezer. 1994. “Supersymmetric Analogs of the Gordon-Andrews Identities, and Related TBA Systems”. ArXiv e-print hep-th/9412154. http://arxiv.org/abs/hep-th/9412154.
- A. Meurman and A. Rocha-Caridi, Highest weight representations of the Neveu-Schwarz and Ramond algebras Commun. Math. Phys. 107:263 (1986) http://link.springer.com/article/10.1007/BF01209395