Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models

  • Nina JaverzatEmail author
  • Raoul Santachiara
  • Omar Foda
Open Access
Regular Article - Theoretical Physics


We study Virasoro minimal-model 4-point conformal blocks on the sphere and 0-point conformal blocks on the torus (the Virasoro characters), as solutions of Zamolodchikov-type recursion relations. In particular, we study the singularities due to resonances of the dimensions of conformal fields in minimal-model representations, that appear in the intermediate steps of solving the recursion relations, but cancel in the final results.


Conformal Field Theory Lattice Integrable Models 


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]
    V. Belavin, X. Cao, B. Estienne and R. Santachiara, Second level semi-degenerate fields in \( {\mathcal{W}}_3 \) Toda theory: matrix element and differential equation, JHEP 03 (2017) 008 [arXiv:1610.07993] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  2. [2]
    V. Belavin, B. Estienne, O. Foda and R. Santachiara, Correlation functions with fusion-channel multiplicity in \( {\mathcal{W}}_3 \) Toda field theory, JHEP 06 (2016) 137 [arXiv:1602.03870] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    V.A. Fateev and A.V. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    B.L. Feigin and D.B. Fuks, Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra, Funct. Anal. Appl. 16 (1982) 114.CrossRefzbMATHGoogle Scholar
  5. [5]
    B.L. Feigin and D.B. Fuks, Verma modules over the Virasoro algebra, Funct. Anal. Appl. 17 (1982) 241.CrossRefzbMATHGoogle Scholar
  6. [6]
    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
  7. [7]
    E. Perlmutter, Virasoro conformal blocks in closed form, JHEP 08 (2015) 088 [arXiv:1502.07742] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Picco, S. Ribault and R. Santachiara, A conformal bootstrap approach to critical percolation, SciPost 1 (2016) 009.CrossRefGoogle Scholar
  9. [9]
    M. Picco, S. Ribault and R. Santachiara, A conformal bootstrap solution for critical Potts clusters, in progress.Google Scholar
  10. [10]
    R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    R. Poghossian, Recurrence relations for the \( {\mathcal{W}}_3 \) conformal blocks and \( \mathcal{N}=2 \) SYM partition functions, JHEP 11 (2017) 053 [Erratum ibid. 01 (2018) 088] [arXiv:1705.00629] [INSPIRE].
  12. [12]
    S. Ribault and R. Santachiara, Liouville theory with a central charge less than one, JHEP 08 (2015) 109.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
  14. [14]
    Al.B. Zamolodchikov, Conformal symmetry in two dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419.Google Scholar
  15. [15]
    Al.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.Google Scholar

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© The Author(s) 2018

Authors and Affiliations

  1. 1.LPTMS, CNRS, Université Paris-Sud, Université Paris-SaclayOrsayFrance
  2. 2.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia

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