Virasoro singular vectors

introduction

• \(V(c,h)=\mathcal{U}(N_-) |\omega \rangle\)
• \(M(c,h)\) is a quotient of \(V(c,h)\) by modules generated by singular vectors

thm

Singular vector of grave $N=rs$ exists if and only if central charge $c$ and $h$ are $$ c=1-6(\beta-1)(1-\frac{1}{\beta}),\, h=\frac{(r\beta-s)^2-(\beta-1)^2}{4\beta} $$ with $\beta\in \mathbb{C}\backslash\{0\}$ and $r,s\in \mathbb{Z}_{>0}$, and if $\beta$ takes generic value, there is only one.

examples

• $|\chi_{11}\rangle=a_{-1}|\alpha_{11}\rangle$
• $|\chi_{12}\rangle=(a_{-2}+\sqrt{2\beta}a_{-1}^2)|\alpha_{12}\rangle$
• $|\chi_{22}\rangle=(a_{-4}+\frac{4\sqrt{2\beta}}{1-\beta}a_{-3}a_{-1}-2\frac{1+\beta+\beta^2}{\sqrt{2\beta}(1-\beta)}a_{-2}^2-4a_{-2}a_{-1}^2-\frac{2\sqrt{2\beta}}{1-\beta}a_{-1}^4)|\alpha_{22}\rangle$

thm [Mimachi-Yamada]

The Virasoro singular vector $|\chi_{rs}\rangle$ has one-to-one correspondence with the Jack symmetric polynomial with the rectangular diagram $\{s^r\}$

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxLUFFQWlRdHVjZDA/edit

articles

• Ridout, David, and Simon Wood. “From Jack Polynomials to Minimal Model Spectra.” arXiv:1409.4847 [hep-Th], September 16, 2014. http://arxiv.org/abs/1409.4847.
• Millionschikov, Dmitry. “Singular Virasoro Vectors and Lie Algebra Cohomology.” arXiv:1405.6734 [math], May 26, 2014. http://arxiv.org/abs/1405.6734.
• Awata, H., Y. Matsuo, S. Odake, and J. Shiraishi. “A Note on Calogero-Sutherland Model, W_n Singular Vectors and Generalized Matrix Models.” arXiv:hep-th/9503028, March 6, 1995. http://arxiv.org/abs/hep-th/9503028.
• Mimachi, Katsuhisa, and Yasuhiko Yamada. “Singular Vectors of the Virasoro Algebra in Terms of Jack Symmetric Polynomials.” Communications in Mathematical Physics 174, no. 2 (1995): 447–55.