Recursive representations of arbitrary Virasoro conformal blocks

  • Minjae ChoEmail author
  • Scott Collier
  • Xi Yin
Open Access
Regular Article - Theoretical Physics


We derive recursive representations in the internal weights of N -point Virasoro conformal blocks in the sphere linear channel and the torus necklace channel, and recursive representations in the central charge of arbitrary Virasoro conformal blocks on the sphere, the torus, and higher genus Riemann surfaces in the plumbing frame.


AdS-CFT Correspondence Conformal and W Symmetry Conformal Field Theory Field Theories in Lower Dimensions 


Open Access

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  1. [1]
    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].
  2. [2]
    G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.CrossRefGoogle Scholar
  5. [5]
    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, Little String Amplitudes (and the Unreasonable Effectiveness of 6D SYM), JHEP 12 (2014) 176 [arXiv:1407.7511] [INSPIRE].
  6. [6]
    Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, Interpolating the Coulomb Phase of Little String Theory, JHEP 12 (2015) 022 [arXiv:1502.01751] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, \( \mathcal{N}=4 \) superconformal bootstrap of the K3 CFT, JHEP 05 (2017) 126 [arXiv:1511.04065] [INSPIRE].
  8. [8]
    Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, (2, 2) superconformal bootstrap in two dimensions, JHEP 05 (2017) 112 [arXiv:1610.05371] [INSPIRE].
  9. [9]
    S. Collier, P. Kravchuk, Y.-H. Lin and X. Yin, Bootstrapping the Spectral Function: On the Uniqueness of Liouville and the Universality of BTZ, JHEP 09 (2018) 150 [arXiv:1702.00423] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Recursion representation of the Neveu-Schwarz superconformal block, JHEP 03 (2007) 032 [hep-th/0611266] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Elliptic recurrence representation of the N = 1 superconformal blocks in the Ramond sector, JHEP 11 (2008) 060 [arXiv:0810.1203] [INSPIRE].
  12. [12]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Recurrence relations for toric N = 1 superconformal blocks, JHEP 09 (2012) 122 [arXiv:1207.5740] [INSPIRE].
  13. [13]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    K.B. Alkalaev, R.V. Geiko and V.A. Rappoport, Various semiclassical limits of torus conformal blocks, JHEP 04 (2017) 070 [arXiv:1612.05891] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Graduate texts in contemporary physics, Springer, New York U.S.A. (1997).Google Scholar
  16. [16]
    K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].
  17. [17]
    X. Yin, Partition Functions of Three-Dimensional Pure Gravity, Commun. Num. Theor. Phys. 2 (2008) 285 [arXiv:0710.2129] [INSPIRE].
  18. [18]
    S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    P. Suchanek, Recursive methods of determination of 4-point blocks in N = 1 superconformal field theories, Ph.D. Thesis, Jagiellonian University, Kraków Poland (2009).Google Scholar
  21. [21]
    A. Zamolodchikov, Higher equations of motion in Liouville field theory, Int. J. Mod. Phys. A 19S2 (2004) 510 [hep-th/0312279] [INSPIRE].
  22. [22]
    J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    E. Perlmutter, Virasoro conformal blocks in closed form, JHEP 08 (2015) 088 [arXiv:1502.07742] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    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
  26. [26]
    V.A. Fateev and A.V. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Proving the AGT relation for N f = 0, 1, 2 antifundamentals, JHEP 06 (2010) 046 [arXiv:1004.1841] [INSPIRE].
  28. [28]
    L. Hollands, C.A. Keller and J. Song, Towards a 4d/2d correspondence for Sicilian quivers, JHEP 10 (2011) 100 [arXiv:1107.0973] [INSPIRE].
  29. [29]
    K.B. Alkalaev and V.A. Belavin, From global to heavy-light: 5-point conformal blocks, JHEP 03 (2016) 184 [arXiv:1512.07627] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    P. Kraus and A. Maloney, A cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    G. Mason and M.P. Tuite, On genus two Riemann surfaces formed from sewn tori, Commun. Math. Phys. 270 (2007) 587 [math/0603088] [INSPIRE].
  32. [32]
    I.R. Klebanov, String theory in two-dimensions, in Spring School on String Theory and Quantum Gravity (to be followed by Workshop), Trieste Italy (1991), pg. 30 [hep-th/9108019] [INSPIRE].
  33. [33]
    P.H. Ginsparg and G.W. Moore, Lectures on 2 − D gravity and 2 − D string theory, in Proceedings, Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles, Boulder U.S.A. (1992), pg. 277 [hep-th/9304011] [INSPIRE].
  34. [34]
    A. Giveon and D. Kutasov, Little string theory in a double scaling limit, JHEP 10 (1999) 034 [hep-th/9909110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    O. Aharony, B. Fiol, D. Kutasov and D.A. Sahakyan, Little string theory and heterotic/type-II duality, Nucl. Phys. B 679 (2004) 3 [hep-th/0310197] [INSPIRE].
  36. [36]
    O. Aharony, A. Giveon and D. Kutasov, LSZ in LST, Nucl. Phys. B 691 (2004) 3 [hep-th/0404016] [INSPIRE].
  37. [37]
    J.M. Maldacena and H. Ooguri, Strings in AdS 3 and the SL(2, ℝ) WZW model. Part 3. Correlation functions, Phys. Rev. D 65 (2002) 106006 [hep-th/0111180] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Jefferson Physical LaboratoryHarvard UniversityCambridgeU.S.A.

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