Journal of High Energy Physics

, 2016:181 | Cite as

Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations

  • P. GavrylenkoEmail author
  • A. Marshakov
Open Access
Regular Article - Theoretical Physics


We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist fields are identified with particular primary fields of the W-algebra, and we propose a straightforward way to compute their W-charges. We demonstrate how these exact conformal blocks can be effectively computed using the technique arisen from the gauge theory/CFT correspondence. We discuss also their direct relation with the isomonodromic tau-function for the quasipermutation monodromy data, which can be an encouraging step on the way of definition of generic conformal blocks for W-algebra using the isomonodromy/CFT correspondence.


Conformal and W Symmetry Integrable Hierarchies Integrable Equations in Physics 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    A.B. Zamolodchikov, Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys. 65 (1985) 1205 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  2. [2]
    V.A. Fateev and A.B. Zamolodchikov, Conformal quantum field theory models in two-dimensions having Z(3) symmetry, Nucl. Phys. B 280 (1987) 644 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    V.A. Fateev and S.L. Lukyanov, The models of two-dimensional conformal quantum field theory with Z(n) symmetry, Int. J. Mod. Phys. A 3 (1988) 507 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    P. Bowcock and G.M.T. Watts, Null vectors, three point and four point functions in conformal field theory, Theor. Math. Phys. 98 (1994) 350 [hep-th/9309146] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, hep-th/0302191 [INSPIRE].
  6. [6]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
  10. [10]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
  11. [11]
    I.M. Krichever, The tau function of the universal Whitham hierarchy, matrix models and topological field theories, Commun. Pure Appl. Math. 47 (1994) 437 [hep-th/9205110] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Marshakov, Tau-functions for quiver gauge theories, JHEP 07 (2013) 068 [arXiv:1303.0753] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    P. Gavrylenko and A. Marshakov, Residue formulas for prepotentials, instanton expansions and conformal blocks, JHEP 05 (2014) 097 [arXiv:1312.6382] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Al. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, J. Exp. Theor. Phys. 90 (1986) 1808.Google Scholar
  16. [16]
    A.B. Zamolodchikov, Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions, Nucl. Phys. B 285 (1987) 481 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    S. Apikyan and Al. Zamolodchikov, Conformal blocks related to conformally invariant Ramond states of a free scalar field, J. Exp. Theor. Phys. 92 (1987) 34 [Zh. Eksp. Teor. Fiz. 92 (1987) 34] [INSPIRE].
  18. [18]
    O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP 10 (2012) 038 [Erratum ibid. 1210 (2012) 183] [arXiv:1207.0787] [INSPIRE].
  19. [19]
    V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory. I., JHEP 11 (2007) 002 [arXiv:0709.3806] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Sato T. Miwa and M. Jimbo, Holonomic quantum fields I, Publ. RIMS Kyoto Univ. 14 (1978) 223.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. Sato T. Miwa and M. Jimbo, Holonomic quantum fields II, Publ. RIMS Kyoto Univ. 15 (1979) 201.CrossRefzbMATHGoogle Scholar
  22. [22]
    M. Sato T. Miwa and M. Jimbo, Holonomic quantum fields III, Publ. RIMS Kyoto Univ. 15 (1979) 577.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    M. Sato T. Miwa and M. Jimbo, Holonomic quantum fields IV, Publ. RIMS Kyoto Univ. 15 (1979) 871.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    M. Sato T. Miwa and M. Jimbo, Holonomic quantum fields V, Publ. RIMS Kyoto Univ. 16 (1980) 531.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    P. Gavrylenko, Isomonodromic τ-functions and W N conformal blocks, JHEP 09 (2015) 167 [arXiv:1505.00259] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    V. Knizhnik, Analytic fields on Riemann surfaces. II, Comm. Math. Phys. 112 (1987) 567.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    V. Knizhnik, Multiloop amplitudes in the theory of quantum strings and complex geometry, Russ. Phys. Usp. 159 (1989) 401.MathSciNetGoogle Scholar
  28. [28]
    D. Novikov, The 2 × 2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system, Theor. Math. Phys. 161 (2009) 1485.CrossRefzbMATHGoogle Scholar
  29. [29]
    M. Bershadsky and A. Radul, Conformal field theories with additional Z(n) symmetry, Int. J. Mod. Phys. A 02 (1987) 165.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Bershadsky and A. Radul, Fermionic fields on Z(n)-curves, Comm. Math. Phys. 116 (1988) 689.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The conformal field theory of orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    B. Dubrovin, Theta functions and non-linear equations, Russ. Math. Surv. 36 (1981) 11.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    D. Mumford, Tata lectures on theta (1988).Google Scholar
  34. [34]
    J. Fay, Theta-functions on Riemann surfaces, Lecture Notes in Mathematics volume 352, Springer (1973).Google Scholar
  35. [35]
    A. Bilal, A remark on the Ninfinity limit of W(n) algebras, Phys. Lett. B 227 (1989) 406 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    A. Marshakov and A. Morozov, A note on on W 3-algebra, Nucl. Phys. B 339 (1990) 79 [Sov. Phys. JETP 70 (1990) 403] [INSPIRE].
  37. [37]
    V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP 01 (2012) 051 [arXiv:1109.4042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    A. Kokotov and D. Korotkin, τ-function on Hurwitz spaces, Math. Phys. Anal. Geom. 7 (2004) 1 [math-ph/0202034].MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    P. Gavrylenko, N. Iorgov and O. Lisovyy, Higher rank isomonodromic deformations and W-algebras, to appear.Google Scholar
  40. [40]
    D. Korotkin, Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices, Math. Ann. 329 (2004) 335 [math-ph/0306061].MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    A. Kokotov and D. Korotkin, Isomonodromic τ-function of Hurwitz Frobenius manifolds and its applications, Int. Math. Res. Not. (2006) 1 [math-ph/0310008].
  42. [42]
    J.D. Fay, Kernel functions analytic torsion and moduli spaces, Memoirs of American Mathematical Society volume 96, American Mathematical Society, U.S.A. (1992).Google Scholar
  43. [43]
    G.L. Cardoso, B. de Wit and S. Mahapatra, Deformations of special geometry: in search of the topological string, JHEP 09 (2014) 096 [arXiv:1406.5478] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    N. Iorgov, O. Lisovyy and J. Teschner, Isomonodromic τ -functions from Liouville conformal blocks, Commun. Math. Phys. 336 (2015) 671 [arXiv:1401.6104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics and Laboratory of Mathematical Physics, NRU HSEMoscowRussia
  2. 2.Bogolyubov Institute for Theoretical PhysicsKyivUkraine
  3. 3.Theory Department, Lebedev Physics Institute and Institute for Theoretical and Experimental PhysicsMoscowRussia

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