Supersymmetric minimal models

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 $$ M_j = m_j +m_{j+1}+\ldots+m_{2k-2}, \tilde{M}=m_1+m_3+\ldots+m_{2k-3} $$

• the matrix $A_k$ for $k$ has size $2(k-1)$

related items

• Minimal models
• Ramanujan-Göllnitz-Gordon continued fraction

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxZl9xd1p3aUFVSUU/edit

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.
  • 파일:Supersymmetric Analogs of the Gordon-Andrews Identities, and Related TBA Systems.tex
• 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