Accessory parameters for Liouville theory on the torus



We give an implicit equation for the accessory parameter on the torus which is the necessary and sufficient condition to obtain the monodromy of the conformal factor. It is shown that the perturbative series for the accessory parameter in the coupling constant converges in a finite disk and give a rigorous lower bound for the radius of convergence. We work out explicitly the perturbative result to second order in the coupling for the accessory parameter and to third order for the one-point function. Modular invariance is discussed and exploited. At the non perturbative level it is shown that the accessory parameter is a continuous function of the coupling in the whole physical region and that it is analytic except at most a finite number of points. We also prove that the accessory parameter as a function of the modulus of the torus is continuous and real-analytic except at most for a zero measure set. Three soluble cases in which the solution can be expressed in terms of hypergeometric functions are explicitly treated.


Field Theories in Lower Dimensions Integrable Equations in Physics Conformal Field Models in String Theory 


  1. [1]
    T.L. Curtright and C.B. Thorn, Conformally Invariant Quantization of the Liouville Theory, Phys. Rev. Lett. 48 (1982) 1309 [Erratum ibid. 48 (1982) 1768] [INSPIRE].
  2. [2]
    H. Dorn and H. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  4. [4]
    J. Teschner, A Lecture on the Liouville vertex operators, Int. J. Mod. Phys. A 19S2 (2004) 436 [hep-th/0303150] [INSPIRE].MathSciNetADSGoogle Scholar
  5. [5]
    J. Teschner, On the Liouville three point function, Phys. Lett. B 363 (1995) 65 [hep-th/9507109] [INSPIRE].ADSGoogle Scholar
  6. [6]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
  8. [8]
    P. Olesen, Soliton condensation in some selfdual Chern-Simons theories, Phys. Lett. B 265 (1991) 361 [Erratum ibid. B 267 (1991) 541] [INSPIRE].
  9. [9]
    P. Olesen, Vacuum structure of the electroweak theory in high magnetic fields, Phys. Lett. B 268 (1991) 389 [INSPIRE].MathSciNetADSGoogle Scholar
  10. [10]
    R. Jackiw and S. Pi, Soliton solutions to the Gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett. 64 (1990) 2969 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  11. [11]
    N. Akerblom, G. Cornelissen, G. Stavenga and J.-W. van Holten, Nonrelativistic Chern-Simons Vortices on the Torus, J. Math. Phys. 52 (2011) 072901 [arXiv:0912.0718] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  13. [13]
    Y. Nakayama, Liouville field theory: A Decade after the revolution, Int. J. Mod. Phys. A 19 (2004) 2771 [hep-th/0402009] [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    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].MathSciNetADSMATHCrossRefGoogle Scholar
  15. [15]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    V. Alba and A. Morozov, Non-conformal limit of AGT relation from the 1-point torus conformal block, JETP Lett. 90 (2009) 708 [arXiv:0911.0363] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    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].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Modular bootstrap in Liouville field theory, Phys. Lett. B 685 (2010) 79 [arXiv:0911.4296] [INSPIRE].MathSciNetADSGoogle Scholar
  21. [21]
    P. Menotti and G. Vajente, Semiclassical and quantum Liouville theory on the sphere, Nucl. Phys. B 709 (2005) 465 [hep-th/0411003] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    V.A. Fateev, A. Litvinov, A. Neveu and E. Onofri, Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks, J. Phys. A 42 (2009) 304011 [arXiv:0902.1331] [INSPIRE].MathSciNetGoogle Scholar
  23. [23]
    F. Ferrari and M. Piatek, Liouville theory, N = 2 gauge theories and accessory parameters, JHEP 05 (2012) 025 [arXiv:1202.2149] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    L. Keen, H.E. Rauch and A.T. Vasquez, Moduli of punctured tori and the accessory parameter of Laméequation, Trans. Am. Math. Soc. 255 (1979) 201.MathSciNetMATHGoogle Scholar
  25. [25]
    P. Menotti, Riemann-Hilbert treatment of Liouville theory on the torus, J. Phys. A 44 (2011) 115403 [arXiv:1010.4946] [INSPIRE].MathSciNetADSGoogle Scholar
  26. [26]
    P. Menotti, Riemann-Hilbert treatment of Liouville theory on the torus: The general case, J. Phys. A 44 (2011) 335401 [arXiv:1104.3210] [INSPIRE].MathSciNetGoogle Scholar
  27. [27]
    R.S. Maier, On reducing the Heun equation to the hypergeometric equation: I. Polynomial transformations, J. Diff. Eq. 213 (2005) 171.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    E. Picard, Sur lequation Δu = ke u, Compt. Rend. 116 (1893) 1015.MATHGoogle Scholar
  29. [29]
    E. Picard, De lequation Δu = ke u sur une surface de Riemann fermée, J. Math. Pures Appl. 4 (1893) 273.Google Scholar
  30. [30]
    E. Picard, Sur lequation Δu = ke u, J. Math. Pures Appl. 4 (1898) 313.Google Scholar
  31. [31]
    E. Picard, De lequation Δu = ke u, Bull. Sci. Math. XXIV 1 (1900) 196.Google Scholar
  32. [32]
    H. Poincaré, Le fonctions fuchsiennes et lequation Δu = e u, J. Math. Pures Appl. 4 (1898) 137.Google Scholar
  33. [33]
    L. Lichtenstein, Integration der Differentialgleichung Δ2 u = ke u auf geschlossenen Flächen: Methode der unendlichen variabeln, Acta Math. 40 (1915) 1.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc. 324 (1991) 793.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    P.G. Zograf and L.A. Takhtajan, On Liouville equation, accessory parameters, and the geometry of Teichmüller space for Riemann surfaces of genus 0, Math. USSR Sbornik 60 (1988) 143.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    I. Kra, Accessory parameters for punctured spheres, Trans. Am. Math. Soc. 313 (1989) 589.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    L. Cantini, P. Menotti and D. Seminara, Proof of Polyakov conjecture for general elliptic singularities, Phys. Lett. B 517 (2001) 203 [hep-th/0105081] [INSPIRE].MathSciNetADSGoogle Scholar
  38. [38]
    L. Cantini, P. Menotti and D. Seminara, Liouville theory, accessory parameters and (2+1)-dimensional gravity, Nucl. Phys. B 638 (2002) 351 [hep-th/0203103] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    H. Whitney, Complex analytic varieties, Addison-Wesley, Reading Mass (1972).MATHGoogle Scholar
  40. [40]
    L.A. Takhtajan and P.G. Zograf Takhtajan, Hyperbolic 2-spheres with conical singularities, accessory parameters and Käler metric on \( {{\mathcal{M}}_{0,n }} \), Trans. Am. Math. Soc. 355 (2003) 1857.MATHCrossRefGoogle Scholar
  41. [41]
    A. Erdelyi (Ed.), Higher Transcendental Functions, vol.II McGraw-Hill, New York (1953).Google Scholar
  42. [42]
    NIST Digital Library of Mathematical Functions,
  43. [43]
    R.C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall Inc.Englewood Cliffs (1965).MATHGoogle Scholar
  44. [44]
    J.P. D’Angelo, Several complex variables and the geometry of real hypersurfaces, CRC Press, Ann Arbor, London, Tokio (1993).MATHGoogle Scholar
  45. [45]
    A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Springer, Berlin (1976).MATHCrossRefGoogle Scholar
  46. [46]
    W. Rudin, Principles of mathematical analysis, McGraw-Hill, New York (1976).MATHGoogle Scholar
  47. [47]
    B.L. van der Waerden, Algebra, Springer-Verlag, New York, Heidelberg, Berlin (1967).MATHGoogle Scholar
  48. [48]
    S. Lang, Algebra, Addison-Wesley, Reading Mass (1993)MATHGoogle Scholar
  49. [49]
    L.A. Takhtajan, Equivalence of geometric h < 1/2 and standard c > 25 approaches to two-dimensional quantum gravity, Mod. Phys. Lett. A 11 (1996) 93 [hep-th/9509026] [INSPIRE].MathSciNetADSGoogle Scholar
  50. [50]
    J-P Serre, A course in arithmetic, Springer-Verlag, New York, Heidelberg, Berlin (1996).Google Scholar
  51. [51]
    E.C. Titchmarsh, The theory of functions, Oxford University Press, London (1964).Google Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Dipartimento di Fisica, Università di Pisa and INFN, Sezione di PisaPisaItaly

Personalised recommendations