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.