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| 10번째 줄: | 10번째 줄: | ||
==Fock space representations of Heisenberg algebra and Virasoro algebra== | ==Fock space representations of Heisenberg algebra and Virasoro algebra== | ||
| − | * set | + | * set <math>\alpha_0=\frac{\beta}{2}-\frac{1}{\beta}</math>, <math>\alpha_{r,s}=(1+r)\frac{\beta}{2}-(1+s)\frac{1}{\beta}</math> and <math>t=\frac{\beta^2}{2}</math> for <math>\beta\in \mathbb{C}</math> and <math>r,s\in \mathbb{Z}</math> |
* [[Heisenberg group and Heisenberg algebra|infinite dimensional Heisenberg algebra]] generated by | * [[Heisenberg group and Heisenberg algebra|infinite dimensional Heisenberg algebra]] generated by | ||
| − | + | :<math> | |
[a_m,a_n]=n\delta_{m,-n} | [a_m,a_n]=n\delta_{m,-n} | ||
| − | + | </math> | |
| − | * we define | + | * we define <math>F_{\alpha}, \alpha\in \mathbb{C}</math> a representation of the Heisenberg algebra defined by the generator <math>|\alpha \rangle</math> and the relations |
| − | + | :<math> | |
a_n|\alpha \rangle=0, \, n>0 \\ | a_n|\alpha \rangle=0, \, n>0 \\ | ||
a_0|\alpha \rangle=\alpha |\alpha \rangle | a_0|\alpha \rangle=\alpha |\alpha \rangle | ||
| − | + | </math> | |
===Virasoro algebra=== | ===Virasoro algebra=== | ||
* if we put | * if we put | ||
| − | + | :<math> | |
L_n=\frac{1}{2}\sum_{m\in \mathbb{Z}}:a_{n-m}a_m:-\alpha_0(n+1)a_n | L_n=\frac{1}{2}\sum_{m\in \mathbb{Z}}:a_{n-m}a_m:-\alpha_0(n+1)a_n | ||
| − | + | </math> | |
| − | for | + | for <math>n\in \mathbb{Z}</math> |
* we obtain the relations | * we obtain the relations | ||
| − | + | :<math> | |
[L_n,a_m]=-ma_{n+m}-\alpha_0n(n+1)\delta_{n+m,0} | [L_n,a_m]=-ma_{n+m}-\alpha_0n(n+1)\delta_{n+m,0} | ||
| − | + | </math> | |
and | and | ||
| − | + | :<math> | |
[L_n,L_m]=(n-m)L_{m+n}+\frac{c}{12}(n^3-n)\delta_{n+m,0} | [L_n,L_m]=(n-m)L_{m+n}+\frac{c}{12}(n^3-n)\delta_{n+m,0} | ||
| − | + | </math> | |
| − | * thus the Virasoro algebra acts on | + | * thus the Virasoro algebra acts on <math>F_{\alpha}</math> with the central charge <math>c=1-12\alpha_0^2=13-6(t+1/t)</math> |
* we also have | * we also have | ||
| − | + | :<math> | |
L_n|\alpha \rangle=0\, (n\in \mathbb{Z}_{> 0}) | L_n|\alpha \rangle=0\, (n\in \mathbb{Z}_{> 0}) | ||
| − | + | </math> | |
and | and | ||
| − | + | :<math> | |
L_0|\alpha \rangle=h_{\alpha}|\alpha \rangle | L_0|\alpha \rangle=h_{\alpha}|\alpha \rangle | ||
| − | + | </math> | |
| − | with | + | with <math>h_{\alpha}=\frac{\alpha^2}{2}-\alpha_0\alpha</math> |
==singular vectors in Fock space representations== | ==singular vectors in Fock space representations== | ||
;def | ;def | ||
| − | An element | + | An element <math>|\chi \rangle\in F_{\alpha}</math> is called the singular vectors of degree <math>N\in \mathbb{Z}</math> if |
| − | + | :<math> | |
L_n|\chi \rangle=0\, (n\in \mathbb{Z}_{> 0}) | L_n|\chi \rangle=0\, (n\in \mathbb{Z}_{> 0}) | ||
| − | + | </math> | |
and | and | ||
| − | + | :<math> | |
L_0|\chi \rangle=(h_{\alpha}+N)|\chi \rangle | L_0|\chi \rangle=(h_{\alpha}+N)|\chi \rangle | ||
| − | + | </math> | |
;thm | ;thm | ||
| − | If | + | If <math>\alpha\notin \{\alpha_{r,s}:r,s\in \mathbb{Z}_{>0} \text{or } r,s\in \mathbb{Z}_{<0} \}</math>, <math>F_{\alpha}</math> is irreducible. |
| − | If | + | If <math>\alpha=\alpha_{r,s}</math> for some <math>r,s\in \mathbb{Z}_{>0}</math>, there exists a unique singular vector <math>|\chi_{r,s}\rangle \in F_{\alpha}</math> of degree <math>N=rs</math> up to a constant factor. |
===examples=== | ===examples=== | ||
| − | * | + | * <math>|\chi_{1,1}\rangle=a_{-1}|\alpha_{1,1}\rangle</math> |
| − | * | + | * <math>|\chi_{1,2}\rangle=(a_{-2}+\sqrt{2t}a_{-1}^2)|\alpha_{1,2}\rangle</math> |
| − | * | + | * <math>|\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</math> |
| − | * if we use the substitutions | + | * if we use the substitutions <math>a_{-n}\mapsto \sqrt{\frac{t}{2}}p_n,\, |\alpha_{r,s}\rangle\mapsto 1</math>, we get the Jack symmetric polynomials |
;thm [Mimachi-Yamada] | ;thm [Mimachi-Yamada] | ||
| − | The Virasoro singular vector | + | The Virasoro singular vector <math>|\chi_{r,s}\rangle</math> has one-to-one correspondence with the Jack symmetric polynomial <math>J_{\{s^r\}}(x;t)</math> with the rectangular diagram <math>\{s^r\}</math> |
==computational resource== | ==computational resource== | ||
| 76번째 줄: | 76번째 줄: | ||
==articles== | ==articles== | ||
| + | * Schechtman, Vadim, and Alexander Varchenko. “Rational Differential Forms on Line and Singular Vectors in Verma Modules over <math>\widehat {sl}_2</math>.” arXiv:1511.09014 [math-Ph], November 29, 2015. http://arxiv.org/abs/1511.09014. | ||
| + | * Yanagida, Shintarou. “Singular Vectors of <math>N=1</math> 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. | * 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. | * 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. | ||
| 90번째 줄: | 92번째 줄: | ||
[[분류:conformal field theory]] | [[분류:conformal field theory]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 13일 (금) 08:50 기준 최신판
introduction
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
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.