Communications in Mathematical Physics

, Volume 365, Issue 1, pp 17–60 | Cite as

Rigid Fuchsian Systems in 2-Dimensional Conformal Field Theories

  • Vladimir Belavin
  • Yoshishige Haraoka
  • Raoul SantachiaraEmail author


We investigate Fuchsian equations arising in the context of 2-dimensional conformal field theory (CFT) and we apply the Katz theory of Fucshian rigid systems to solve some of these equations. We show that the Katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions. We focus on the case of \({\mathcal{W}_{3}}\) Toda CFT: we argue that the differential equations arising for four-point conformal blocks with one nth level semi-degenerate field and a fully-degenerate one in the fundamental sl3 representation are associated to Fuchsian rigid systems. We show how to apply Katz theory to determine the explicit form of the differential equations, the integral expression of solutions and the monodromy group representation. The theory of twisted homology is also used in the analysis of the integral expression. The computation of the connection coefficients is done for the first time in the case of a Katz system with multiplicities, thus extending the work done by Oshima in the multiplicity free case. This approach allows us to construct the corresponding fusion matrices and to perform the whole bootstrap program: new explicit factorization of \({\mathcal{W}_{3}}\) correlation functions as well as shift relations between structure constants for general Toda theories are also provided.


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We thank T. Dupic, P. Gavrylenko, N. Iorgov, O. Lisovyy, Y. Matsuo, and S. Ribault for discussions and X. Cao, B. Estienne, and O. Foda for preliminary contributions and previous results on which the study of this manuscript is based. The second author is supported by the JSPS grants-in-aid for scientific research B, No. 15H03628.


  1. 1.
    Zamolodchikov A.B.: Infinite additional symmetries in two-dimensional conformal quantum field theory. Theor. Math. Phys. 65, 1205–1213 (1985)CrossRefGoogle Scholar
  2. 2.
    Frenkel E., Ben-Zvi D.: Vertex Algebras and Algebraic Curves, Second Edition Mathematical Surveys and Monographs.American Mathematical Society, Providence (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Zhu Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9, 237–302 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Verlinde E.P.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360–376 (1988)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Fuchs, J.: On non-semisimple fusion rules and tensor categories. arXiv:hep-th/0602051
  6. 6.
    Dorn H., Otto H.J.: Two and three point functions in Liouville theory. Nucl. Phys. B 429, 375–388 (1994) arXiv:hep-th/9403141 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zamolodchikov A.B., Zamolodchikov A.B.: Structure constants and conformal bootstrap in Liouville field theory. Nucl. Phys. B 477, 577–605 (1996) arXiv:hep-th/9506136 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Belavin V., Estienne B., Foda O., Santachiara R.: Correlation functions with fusion-channel multiplicity in \({\mathcal{W}_{3}}\) Toda field theory. J. High Energy Phys. 6, 137 (2016) arXiv:1602.03870 ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Katz N.: Rigid Local Systems, Volume 139 of Annals of Mathematics Studies. Princeton University Press, Princeton (1996)Google Scholar
  10. 10.
    Zamolodchikov A.B.: Conformal symmetry in two-dimensional space: recursion representation of conformal block. Theor. Math. Phys. 73, 1088–1093 (1987)CrossRefGoogle Scholar
  11. 11.
    Ponsot B., Teschner J.: Clebsch–Gordan and Racah–Wigner coefficients for a continuous series of representations of U q (sl(2, R)). Commun. Math. Phys. 224, 613–655 (2001) ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Teschner J.: On the Liouville three-point function. Phys. Lett.B 363, 65–70 (1995) arXiv:hep-th/9507109 ADSCrossRefGoogle Scholar
  14. 14.
    Kupiainen, A., Rhodes, R., Vargas, V.: Integrability of Liouville theory: proof of the DOZZ formula. arXiv:1707.08785 (2017)
  15. 15.
    Bauer M., Bernard D.: Conformal field theories of stochastic Loewner evolutions. Commun. Math. Phys. 239, 493–521 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Estienne B., Pasquier V., Santachiara R., Serban D.: Conformal blocks in Virasoro and \({\mathcal{W}}\) theories: duality and the Calogero–Sutherland model. Nucl. Phys. B 860, 377–420 (2012) arXiv:1110.1101 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fateev V.A., Lukyanov S.L.: The models of two-dimensional conformal quantum field theory with Z(n) symmetry. Int. J. Mod. Phys. A 3, 507 (1988)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Bouwknegt P., Schoutens K.: \({\mathcal{W}}\) symmetry. Adv. Ser. Math. Phys. 22, 1–875 (1995)Google Scholar
  19. 19.
    Read N., Rezayi E.: Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level. Phys. Rev. B. 59, 8084–8092 (1999) arXiv:cond-mat/9809384 ADSCrossRefGoogle Scholar
  20. 20.
    Estienne B., Santachiara R.: Relating Jack wavefunctions to \({\mathcal{W}{A}_{k-1}}\) theories. J. Phys. Math. Gener. 42, 5209 (2009) arXiv:0906.1969 ADSzbMATHGoogle Scholar
  21. 21.
    Dupic T., Estienne B., Ikhlef Y.: The fully packed loop model as a non-rational \({\mathcal{W}_{3}}\) conformal field theory. J. Phys. Math. Gener. 49, 505202 (2016) arXiv:1606.05376 CrossRefzbMATHGoogle Scholar
  22. 22.
    Fateev V.A., Litvinov A.V.: Coulomb integrals in Liouville theory and Liouville gravity. JETP Lett. 84, 531–536 (2007)ADSCrossRefGoogle Scholar
  23. 23.
    Wyllard N.: A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories. JHEP 11, 002 (2009) arXiv:0907.2189 ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Mironov A., Morozov A.: On AGT relation in the case of U(3). Nucl. Phys. B. 825, 1–37 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Alkalaev K.B., Belavin V.A.: Conformal blocks of \({\mathcal{W}_{N}}\) minimal models and AGT correspondence. JHEP 07, 024 (2014) arXiv:1404.7094 ADSCrossRefGoogle Scholar
  26. 26.
    Belavin V., Foda O., Santachiara R.: AGT, N-Burge partitions and \({\mathcal{W}_{N}}\) minimal models. JHEP 10, 073 (2015) arXiv:1507.03540 ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Poghossian, R.: Recurrence relations for the \({\mathcal{W}_{3}}\) conformal blocks and \({\mathcal{N} = 2 SYM}\) partition functions. arXiv:1705.00629
  28. 28.
    Fateev V.A., Litvinov A.V.: Correlation functions in conformal Toda field theory I. JHEP 11, 002 (2007) arXiv:0709.3806 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ribault S.: On sl 3 Knizhnik–Zamolodchikov equations and \({\mathcal{W}_{3}}\) null-vector equations. JHEP 10, 002 (2009) arXiv:0811.4587 ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Gavrylenko P.: Isomonodromic \({\tau}\)-functions and \({\mathcal{W}_{N}}\) conformal blocks. J. High Energy Phys. 9, 167 (2015) arXiv:1505.00259 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Bershtein, M., Gavrylenko, P., Marshakov, A.: Twist-field representations of \({\mathcal{W}}\)-algebras, exact conformal blocks and character identities. arXiv:1705.00957
  32. 32.
    Belavin V., Cao X., Estienne B., Santachiara R.: Second level semi-degenerate fields in \({\mathcal{W}_{3}}\) Toda theory: matrix element and differential equation. J. High Energy Phys. 3, 8 (2017) arXiv:1610.07993 ADSCrossRefzbMATHGoogle Scholar
  33. 33.
    Dettweiler M., Reiter S.: An algorithm of Katz and its application to the inverse Galois problem. J. Symb. Comput. 30, 761–798 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Dettweiler M., Reiter S.: Rigid local systems and motives of type G 2. With an appendix by Michale Dettweiler and Nicholas M. Katz. Compos. Math. 146, 929–963 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Dettweiler, M., Sabbah, C.: Hodge theory of the middle convolution. arXiv:1209.4185 (2012)
  36. 36.
    Oshima T.: Fractional calculus of Weyl algebra and Fuchsian differential equations. Math. Soc. Jpn. 28, 1–203 (2012)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Lukyanov, S.L., Fateev, V.A.: Physics reviews: additional symmetries and exactly soluble models in two-dimensional conformal field theory (1990)Google Scholar
  38. 38.
    Bouwknegt, P., Schoutens, K.: \({\mathcal{W}}\) symmetry in conformal field theory. (1993)Google Scholar
  39. 39.
    Kanno S., Matsuo Y., Shiba S.: Analysis of correlation functions in Toda theory and AGT-\({\mathcal{W}}\) relation for SU(3) quiver. Phys. Rev. D 82, 066009 (2010) arXiv:1007.0601 ADSCrossRefGoogle Scholar
  40. 40.
    Bowcock P., Watts G.M.T.: Null vectors of the \({\mathcal{W}_{3}}\) algebra. Phys. Lett. B 297, 282–288 (1992) arXiv:hep-th/9209105 ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Ribault, S.: Conformal field theory on the plane. arXiv:1406.4290
  42. 42.
    Di Francesco P., Mathieu P., Sénéchal D.: Conformal Field Theory Graduate Texts in Contemporary Physics. Springer, Berlin (1997)zbMATHGoogle Scholar
  43. 43.
    Dettweiler M., Reiter S.: Middle convolution of Fuchsian systems and the construction of rigid differential systems. J. Algebra 318, 1–24 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Haraoka Y., Hamaguchi S.: Topological theory for Selberg type integral associated with rigid Fuchsian systems. Math. Ann. 353, 1239–1271 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Haraoka Y.: Canonical forms of differential equations free from accessory parameters. SIAM J. Math. Anal. 25, 1203–1226 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Haraoka Y.: Monodromy representations of systems of differential equations free from accessory parameters. SIAM J. Math. Anal. 25, 1595–1621 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Kita M.: On hypergeometric functions in several variables 1. New integral representations of Euler type. Jpn. J. Math. New Ser. 18, 25–74 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Haraoka Y., Mimachi K.: A connection problem for Simpson’s even family of rank four. Funkc. Ekvacioj 54(3), 495–515 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Mimachi, K., Yoshida, M.: Regularizable cycles associated with a Selberg type integral under some resonance condition. arXiv:math/0408272
  50. 50.
    Mimachi K.: Intersection numbers for twisted cycles and the connection problem associated with the generalized hypergeometric function n+1F n. Int. Math. Res. Not. 8, 1757–1781 (2011)zbMATHGoogle Scholar
  51. 51.
    Fateev V.A., Litvinov A.V.: Correlation functions in conformal Toda field theory II. JHEP 01, 033 (2009) arXiv:0810.3020 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Kiritsis E.B.: Analytic aspects of rational conformal field theories. Nucl. Phys. B. 329, 591–627 (1990)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    Fuchs J.: Operator algebra from fusion rules (II) Implementing apparent singularities. Nucl. Phys. B. 386, 343–382 (1992)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Mukhi, S., Muralidhara, G.: Universal RCFT Correlators from the Holomorphic Bootstrap. arXiv:1708.06772 (2017)
  55. 55.
    Furlan P., Petkova V.B.: On some 3-point functions in the \({\mathcal{W}_{4}}\) CFT and related braiding matrix. JHEP 12, 079 (2015) arXiv:1504.07556 ADSGoogle Scholar
  56. 56.
    Furlan P., Petkova V.B.: \({\mathcal{W}_{4}}\) Toda example as hidden Liouville CFT. Phys. Part. Nucl. Lett. 14, 286–290 (2017) arXiv:1606.02535 CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Particle Physics and AstrophysicsWeizmann Institute of ScienceRehovotIsrael
  2. 2.Department of MathematicsKumamoto UniversityKumamotoJapan
  3. 3.LPTMS, CNRS (UMR 8626), Université Paris-SaclayOrsayFrance

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