Abstract
We give an algebraic proof of the formula on the norm of logarithmic primary of Virasoro algebra, which was proposed by Al. Zamolodchikov. This formula appears in the recursion formula for the norm of Gaiotto state, which guarantees the AGT relation for the four-dimensional SU(2) pure gauge theory.
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Astashkevich A., Fuchs D.: Asymptotics for singular vectors in Verma modules over the Virasoro algebra. Pac. J. Math. 177(2), 201–209 (1997)
Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010)
Awata H., Matsuo Y., Odake S., Shiraishi J.: Excited states of the Calogero-Sutherland model and singular vectors of the W N algebra. Nucl. Phys. B 449, 347–374 (1995)
Awata H., Yamada Y.: Five-dimensional AGT conjecture and the deformed Virasoro algebra. JHEP 1001, 125 (2010)
Bauer M., Di Francesco Ph., Itzykson C., Zuber J.B.: Singular vectors of the Virasoro algebra. Phys. Lett. B 260(3–4), 323–326 (1991)
Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241(2), 333–380 (1984)
Belavin A., Zamolodchikov Al.B.: Higher equations of motion in N = 1 SUSY Liouville field theory. JETP Lett. 84, 418–424 (2006) arXiv: hep-th/0610316
Benoit L., Saint-Aubin Y.: Degenerate conformal field theories and explicit expressions for some null vectors. Phys. Lett. B 215(3), 517–522 (1988)
Braverman, A., Feigin, B., Rybnikov, L., Finkelberg, M.: A finite analog of the AGT relation I: finite W-algebras and quasimaps’ spaces. arXiv:1008.3655
Etingof P., Styrkas K.: Algebraic integrability of Schrödinger operators and representations of Lie algebras. Compos. Math. 98(1), 91–112 (1995)
Fateev V.A., Litvinov A.V.: On AGT conjecture. JHEP 1002, 014 (2010)
Feigin B.L., Frenkel E.: Quantum \({\mathcal{W}}\) -algebras and elliptic algebras. Commun. Math. Phys. 178, 653–678 (1996)
Feigin, B.L., Fuchs, D.B.: Skew-symmetric invariant differential operators on the line and Verma modules over the Virasoro algebra. Funct. Anal. Appl. 16, 47–63, 96 (1982)
Feigin, B.L., Fuchs, D.B.: Representations of the Virasoro algebra. In: Representation of Lie groups and related topics. Adv. Stud. Contemp. Math., vol. 7, pp. 465–554. Gordon and Breach, New York (1990)
Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal field theory. In: Graduate Texts in Contemporary Physics. Springer, Berlin (1997)
Frenkel, E.: Langlands correspondence for loop groups. In: Cambridge Studies in Advanced Mathematics, vol. 103. Cambridge University Press, Cambridge (2007)
Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves, 2nd edn. In: Mathematical Surveys and Monographs, vol. 88. American Mathematical Society, Providence (2004)
Fuchs, D.B.L: Singular vectors over the Virasoro algebra and extended Verma modules. In: Unconventional Lie algebras. Adv. Soviet Math., vol. 17, pp. 65–74. American Mathematical Society, Providenec (1993)
Gaiotto, D.: Asymptotically free N = 2 theories and irregular conformal blocks. arXiv:0908.0307 [hep-th]
Hadasz L., Jaskólski Z., Suchanek P.: Recursion representation of the Neveu-Schwarz superconformal block. JHEP 0703, 032 (2007)
Hadasz L., Jaskólski Z., Suchanek P.: Elliptic recurrence representation of the N = 1 Neveu-Schwarz blocks. Nucl. Phys. B 798, 363–378 (2008)
Hadasz L., Jaskólski Z., Suchanek P.: Elliptic recurrence representation of the N = 1 superconformal blocks in the Ramond sector. JHEP 0811, 060 (2008)
Hadasz L., Jaskólski Z., Suchanek P.: Proving the AGT relation for N f = 0, 1, 2 antifundamentals. JHEP 1006, 1046 (2010)
Hanlon, P.J., Stanley, R.P., Stembridge, J.R.: Some combinatorial aspects of the spectra of normally distributed random matrices. In: Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991). Contemp. Math., vol. 138, pp. 151–174. American Mathematical Society. Providence (1992)
Imbimbo C., Mahapatra S., Mukhi S.: Construction of physical states of nontrivial ghost number in c < 1 string theory. Nucl. Phys. B 375(2), 399–420 (1992)
Itzykson, C., Drouffe, J.M.: Statistical field theory. In: Strong coupling, Monte Carlo methods, conformal field theory, and random systems. Cambridge Monographs on Mathematical Physics, vol. 2. Cambridge University Press, Cambridge (1989)
Jantzen, J.C.: Moduln mit einem höchsten Gewicht. In: Lect. Notes in Math., vol. 750. Springer, Berlin (1979)
Kac V.G.: Contravariant form for infinite-dimensional Lie algebras and superalgebras. Lect. Notes Phys. 94, 441–445 (1979)
Kent A.: Singular vectors of the Virasoro algebra. Phys. Lett. B 273(1–2), 56–62 (1991)
Kent A.: Projections of Virasoro singular vectors. Phys. Lett. B 278(4), 443–448 (1992)
Kato, M., Matsuda, S.: Null field construction in conformal and superconformal algebras. In: Conformal field theory and solvable lattice models (Kyoto, 1986). Adv. Stud. Pure Math., vol. 16, pp. 205–254. Academic Press, San Diego (1988)
Macdonald I.G.: Symmetric functions and Hall polynomials. Oxford Mathematical Monographs, 2nd edn. Oxford University Press, Oxford (1995)
Marshakov A., Mironov A., Morozov A.: On non-conformal limit of the AGT relations. Phys. Lett. B 682, 125–129 (2009)
Mimachi K., Yamada Y.: Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials. Commun. Math. Phys. 174, 447–455 (1995)
Nekrasov N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theoret. Math. Phys. 7(5), 831–864 (2003)
Poghossian R.: Recursion relations in CFT and N = 2 SYM theory. JHEP 0912, 038 (2009)
Sakamoto R., Shiraishi J., Arnaudon D., Frappat L., Ragoucy E.: Correspondence between conformal field theory and Calogero-Sutherland model. Nucl. Phys. B 704, 490–509 (2005)
Shiraishi, J.: Lectures on Quantum Integrable Systems. SGC Library, vol. 28 (in Japanese). Saiensu-sha, Japan (2003)
Shiraishi J., Kubo H., Awata H., Odake S.: A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38(1), 33–51 (1996)
Stanley R.P.: Some combinatorial properties of Jack symmetric functions. Adv. Math. 77(1), 76–115 (1989)
Taki., M.: On AGT Conjecture for Pure Super Yang-Mills and W-algebra. arXiv: 0912.4789 [hep-th]
Tsuchiya A., Kanie Y.: Fock space representations of the Virasoro algebra. Intertwining operators. Publ. Res. Inst. Math. Sci. 22(2), 259–327 (1986)
Wyllard N.: A N-1 conformal Toda field theory correlation functions from conformal \({\mathcal{N} = 2}\) SU(N) quiver gauge theories. JHEP 0911, 002 (2009)
Yanagida S.: Five-dimensional SU(2) AGT conjecture and recursive formula of deformed Gaiotto state. J. Math. Phys. 51, 123506 (2010)
Zamolodchikov Al.B.: Conformal symmetry in two dimensions: an explicit recurrence formula for the conformal partial wave amplitude. Commun. Math. Phys. 96(3), 419–422 (1984)
Zamolodchikov Al.B.: Conformal symmetry in two-dimensional space: on a recurrent representation of the conformal block. Teoret. Mat. Fiz. 73(1), 103–110 (1987)
Zamolodchikov, Al.B.: Higher equations of motion in Liouville field theory. Proceedings of 6th International Workshop on Conformal Field Theory and Integrable Models. Int. J. Modern Phys. A 19(May suppl.), 510–523 (2004). arXiv: hep-th/0312279
Zamolodchikov, A.B., Zamolodchikov, Al.B.: Conformal field theory and 2-D critical phenomena. 3. Conformal bootstrap and degenerate representations of conformal algebra. ITEP-90-31 (1990, preprint)
Zamolodchikov A.B., Zamolodchikov Al.B.: Conformal bootstrap in Liouville field theory. Nucl. Phys. B 477(2), 577–605 (1996)
Zamolodchikov, A.B., Zamolodchikov, Al.B.: Liouville field theory on a pseudosphere. arXiv: hep-th/0101152
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Yanagida, S. Norm of Logarithmic Primary of Virasoro Algebra. Lett Math Phys 98, 133–156 (2011). https://doi.org/10.1007/s11005-011-0502-0
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DOI: https://doi.org/10.1007/s11005-011-0502-0