Letters in Mathematical Physics

, Volume 106, Issue 1, pp 29–56 | Cite as

Coupling of Two Conformal Field Theories and Nakajima–Yoshioka Blow-Up Equations

  • Mikhail Bershtein
  • Boris Feigin
  • Alexei Litvinov


We study the conformal vertex algebras which naturally arise in relation to the Nakajima–Yoshioka blow-up equations.


vertex algebras AGT relation Virasoro algebra quantum Hamiltonian reduction coset construction bilinear equations 

Mathematics Subject Classification

17B68 81R10 


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  1. 1.
    Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010) arXiv:0906.3219
  2. 2.
    Alba, V.A., Fateev, V.A., Litvinov, A.V., Tarnopolsky, G.M.: On combinatorial expansion of the conformal blocks arising from AGT conjecture. Lett. Math. Phys. 98, 33–64 (2011) arXiv:1012.1312
  3. 3.
    Belavin A., Polyakov A., Zamolodchikov A.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333 (1984)zbMATHMathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Belavin, A., Bershtein, M., Feigin, B., Litvinov, A., Tarnopolsky, G.: Instanton moduli spaces and bases in coset conformal field theory. Commun. Math. Phys. 319(1), 269–301 (2013) arXiv:1111.2803
  5. 5.
    Bernstein, J., Gel’fand, I., Gel’fand, S.: Category of \({\mathfrak{g}}\)-modules Funkts. Anal. Prilozh. 10(2), 18 (1976)Google Scholar
  6. 6.
    Bershtein, M., Shchechkin, A.: Bilinear equations on Painlevé \({\tau}\) functions from CFT. Commun. Math. Phys. 339(3), 1021–1061 (2015) arXiv:1406.3008
  7. 7.
    Bruzzo, U., Poghossian, R.,, Tanzini, A.: Poincaré polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces. Commun. Math. Phys. 304, 395–409 (2011) arXiv:0909.1458
  8. 8.
    Fateev, V.A., Litvinov, A.V.: On AGT conjecture JHEP 1002, 014 (2010) arXiv:0912.0504
  9. 9.
    Feigin, B., Foda, O., Welsh, T.: Andrews-Gordon type identities from combinations of Virasoro characters. Ramanujan J. 17 (1), 33–52 (2008) arXiv:math-ph/0504014.
  10. 10.
    Feigin B., Frenkel E.: Quantization of the Drinfeld–Sokolov reduction. Phys. Lett. B 246(1–2), 75 (1990)zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Feigin B., Frenkel E.: Affine Kac–Moody algebras, bosonization and resolutions. Lett. Math. Phys. 19, 307–317 (1990)zbMATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Feigin, B., Frenkel, E.: Coinvariants of nilpotent subalgebras of the Virasoro algebra and partition identities. Adv. Sov. Math. 16, 139–148 (1993) arXiv:hep-th/9301039
  13. 13.
    Feigin, B., Fuchs, D.: Representations of the Virasoro algebra. Representations of Lie Groups and Related Topics, vol. 465. Adv. Stud. Contemp. Math., 7, Gordon and Breach, New York (1990)Google Scholar
  14. 14.
    Feigin, B.L., Stoyanovsky, A.V.: Quasi-particles models for the representation of Lie algebras and geometry of flag manifold. Funct. Anal. Appl. 28, 68–90 (1994) arXiv:hep-th/9308079
  15. 15.
    Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves. Mathematical Surveys and Monographs vol. 88. American Mathematical Society (2004)Google Scholar
  16. 16.
    Frenkel, I.B., Kac, V.G.: Basic representations of affine lie algebras and dual resonance models. Invent. Math. 62(1) (1980/81)Google Scholar
  17. 17.
    Fujitsu, A.: ope.math: operator product expansions in free field realizations of conformal field theory. Comput. Phys. Commun. 79, 78–99 (1994)Google Scholar
  18. 18.
    Goddard P., Kent A., Olive D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103(1), 105 (1986)zbMATHMathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Iohara, K., Koga, Y.: Representation theory of the Virasoro algebra. Springer Monographs in Mathematics. Springer-Verlag, London Ltd, London (2011)Google Scholar
  20. 20.
    Kac V.G., Kazhdan D.A.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. Math. 34(1), 97–108 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Kac V.G., Wakimoto M.: Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. Proc. Natl. Acad. Sci. USA 85, 4956–4960 (1988)zbMATHMathSciNetCrossRefADSGoogle Scholar
  22. 22.
    Maulik, D., Okounkov, A.: Quantum Groups and Quantum Cohomology. arXiv:1211.1287
  23. 23.
    Nakajima, H., Yoshioka, K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. Inventiones mathematicae 162(2), 313–355 (2005) arXiv:math/0306198
  24. 24.
    Nakajima, H., Yoshioka, K.: Perverse coherent sheaves on blow-up. III. Blow-up formula from wall-crossing. Kyoto J. Math. 51(2), 263 (2011) arXiv:0911.1773
  25. 25.
    Schiffmann, O., Vasserot, E.: Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on \({\mathbb{A}^2}\). arXiv:1202.2756

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Mikhail Bershtein
    • 1
    • 2
    • 3
    • 4
  • Boris Feigin
    • 1
    • 3
    • 4
  • Alexei Litvinov
    • 1
    • 5
  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Institute for Information Transmission ProblemsMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.Independent University of MoscowMoscowRussia
  5. 5.NHETC, Department of Physics and AstronomyRutgers UniversityPiscatawayUSA

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