Virasoro singular vectors

Verma modules

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

Fock space representations of Heisenberg algebra and Virasoro algebra

• set \(\alpha_0=\frac{\beta}{2}-\frac{1}{\beta}\), \(\alpha_{r,s}=(1+r)\frac{\beta}{2}-(1+s)\frac{1}{\beta}\) and \(t=\frac{\beta^2}{2}\) for \(\beta\in \mathbb{C}\) and \(r,s\in \mathbb{Z}\)
• infinite dimensional Heisenberg algebra generated by

\[ [a_m,a_n]=n\delta_{m,-n} \]

• we define \(F_{\alpha}, \alpha\in \mathbb{C}\) a representation of the Heisenberg algebra defined by the generator \(|\alpha \rangle\) and the relations

\[ a_n|\alpha \rangle=0, \, n>0 \\ a_0|\alpha \rangle=\alpha |\alpha \rangle \]

Virasoro algebra

• if we put

\[ L_n=\frac{1}{2}\sum_{m\in \mathbb{Z}}:a_{n-m}a_m:-\alpha_0(n+1)a_n \] for \(n\in \mathbb{Z}\)

• we obtain the relations

\[ [L_n,a_m]=-ma_{n+m}-\alpha_0n(n+1)\delta_{n+m,0} \] and \[ [L_n,L_m]=(n-m)L_{m+n}+\frac{c}{12}(n^3-n)\delta_{n+m,0} \]

• thus the Virasoro algebra acts on \(F_{\alpha}\) with the central charge \(c=1-12\alpha_0^2=13-6(t+1/t)\)
• we also have

\[ L_n|\alpha \rangle=0\, (n\in \mathbb{Z}_{> 0}) \] and \[ L_0|\alpha \rangle=h_{\alpha}|\alpha \rangle \] with \(h_{\alpha}=\frac{\alpha^2}{2}-\alpha_0\alpha\)

singular vectors in Fock space representations

def

An element \(|\chi \rangle\in F_{\alpha}\) is called the singular vectors of degree \(N\in \mathbb{Z}\) if \[ L_n|\chi \rangle=0\, (n\in \mathbb{Z}_{> 0}) \] and \[ L_0|\chi \rangle=(h_{\alpha}+N)|\chi \rangle \]

thm

If \(\alpha\notin \{\alpha_{r,s}:r,s\in \mathbb{Z}_{>0} \text{or } r,s\in \mathbb{Z}_{<0} \}\), \(F_{\alpha}\) is irreducible.

If \(\alpha=\alpha_{r,s}\) for some \(r,s\in \mathbb{Z}_{>0}\), there exists a unique singular vector \(|\chi_{r,s}\rangle \in F_{\alpha}\) of degree \(N=rs\) up to a constant factor.

examples

• \(|\chi_{1,1}\rangle=a_{-1}|\alpha_{1,1}\rangle\)
• \(|\chi_{1,2}\rangle=(a_{-2}+\sqrt{2t}a_{-1}^2)|\alpha_{1,2}\rangle\)
• \(|\chi_{2,2}\rangle=(a_{-4}+\frac{4\sqrt{2t}}{1-t}a_{-3}a_{-1}-2\frac{1+t+t^2}{\sqrt{2t}(1-t)}a_{-2}^2-4a_{-2}a_{-1}^2-\frac{2\sqrt{2t}}{1-t}a_{-1}^4)|\alpha_{2,2}\rangle\)
• if we use the substitutions \(a_{-n}\mapsto \sqrt{\frac{t}{2}}p_n,\, |\alpha_{r,s}\rangle\mapsto 1\), we get the Jack symmetric polynomials

thm [Mimachi-Yamada]

The Virasoro singular vector \(|\chi_{r,s}\rangle\) has one-to-one correspondence with the Jack symmetric polynomial \(J_{\{s^r\}}(x;t)\) with the rectangular diagram \(\{s^r\}\)

computational resource

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

articles

• Schechtman, Vadim, and Alexander Varchenko. “Rational Differential Forms on Line and Singular Vectors in Verma Modules over \(\widehat {sl}_2\).” arXiv:1511.09014 [math-Ph], November 29, 2015. http://arxiv.org/abs/1511.09014.
• Yanagida, Shintarou. “Singular Vectors of \(N=1\) Super Virasoro Algebra via Uglov Symmetric Functions.” arXiv:1508.06036 [math-Ph], August 25, 2015. http://arxiv.org/abs/1508.06036.
• Kirillov, Anatol N. ‘Notes on Schubert, Grothendieck and Key Polynomials’. arXiv:1501.07337 [math], 28 January 2015. http://arxiv.org/abs/1501.07337.
• 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.
• Tsuchiya, Akihiro, and Simon Wood. “On the Extended W-Algebra of Type sl_2 at Positive Rational Level.” International Mathematics Research Notices, June 19, 2014. doi:10.1093/imrn/rnu090.
• Fuchs, Dmitry. “Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras.” Symmetry, Integrability and Geometry: Methods and Applications, August 27, 2008. doi:10.3842/SIGMA.2008.059.
• 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.
• Kent, A. “Singular Vectors of the Virasoro Algebra.” Physics Letters B 273, no. 1–2 (December 1991): 56–62. doi:10.1016/0370-2693(91)90553-3.
• Feigin, B. L., and D. B. Fuchs. "8. Representations of the Virasoro algebra." Representation of Lie groups and related topics (1990): 465.
• Malikov, F. G., B. L. Feigin, and D. B. Fuks. “Singular Vectors in Verma Modules over Kac—Moody Algebras.” Functional Analysis and Its Applications 20, no. 2 (April 1, 1986): 103–13. doi:10.1007/BF01077264.