The DOZZ formula from the path integral

  • Antti Kupiainen
  • Rémi Rhodes
  • Vincent Vargas
Open Access
Regular Article - Theoretical Physics


We present a rigorous proof of the Dorn, Otto, Zamolodchikov, Zamolodchikov formula (the DOZZ formula) for the 3 point structure constants of Liouville Conformal Field Theory (LCFT) starting from a rigorous probabilistic construction of the functional integral defining LCFT given earlier by the authors and David. A crucial ingredient in our argument is a probabilistic derivation of the reflection relation in LCFT based on a refined tail analysis of Gaussian multiplicative chaos measures.


Conformal Field Theory Conformal Field Models in String Theory 


Open Access

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  1. [1]
    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].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    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].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Bytsko and J. Teschner, The Integrable structure of nonrational conformal field theory, Adv. Theor. Math. Phys. 17 (2013) 701 [arXiv:0902.4825] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  4. [4]
    X. Cao, P. Le Doussal, A. Rosso and R. Santachiara, Liouville field theory and log-correlated Random Energy Models, Phys. Rev. Lett. 118 (2017) 090601 [arXiv:1611.02193] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    F. David, A. Kupiainen, R. Rhodes and V. Vargas, Liouville Quantum Gravity on the Riemann sphere, Commun. Math. Phys. 342 (2016) 869 [arXiv:1410.7318] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    F. David, A. Kupiainen, R. Rhodes and V. Vargas, Renormalizability of Liouville Quantum Gravity at the Seiberg bound, arXiv:1506.01968 [INSPIRE].
  7. [7]
    H. Dorn and H.J. Otto, On correlation functions for noncritical strings with c ≤ 1 d ≥ 1, Phys. Lett. B 291 (1992) 39 [hep-th/9206053] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    M. Goulian and M. Li, Correlation functions in Liouville theory, Phys. Rev. Lett. 66 (1991) 2051 [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    V.A. Fateev and A.V. Litvinov, Multipoint correlation functions in Liouville field theory and minimal Liouville gravity, Theor. Math. Phys. 154 (2008) 454 [arXiv:0707.1664] [INSPIRE].CrossRefGoogle Scholar
  11. [11]
    V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory II, JHEP 01 (2009) 033 [arXiv:0810.3020] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    J.-P. Kahane, Sur le chaos multiplicatif, Ann. Sci. Math. Québec 9 (1985) 105.MathSciNetzbMATHGoogle Scholar
  14. [14]
    V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal Structure of 2D Quantum Gravity, Mod. Phys. Lett. A 3 (1988) 819 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    A. Kupiainen, R. Rhodes and V. Vargas, Conformal Ward and BPZ Identities for Liouville quantum field theory, arXiv:1512.01802 [INSPIRE].
  16. [16]
    A. Kupiainen, R. Rhodes and V. Vargas, Integrability of Liouville theory: proof of the DOZZ Formula, arXiv:1707.08785 [INSPIRE].
  17. [17]
    A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
  19. [19]
    N. Seiberg, Notes on Quantum Liouville Theory and Quantum Gravity, Prog. Theor. Phys. Suppl. 102 (1990) 319.ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    J. Teschner, On the Liouville three point function, Phys. Lett. B 363 (1995) 65 [hep-th/9507109] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
  22. [22]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    J. Teschner, A lecture on the Liouville vertex operators, Int. J. Mod. Phys. A19S2 (2004) 436.Google Scholar
  24. [24]
    Troyanov M.: Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc. 324 (1991) 793.MathSciNetCrossRefGoogle Scholar
  25. [25]
    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].ADSMathSciNetCrossRefGoogle Scholar

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© The Author(s) 2018

Authors and Affiliations

  1. 1.University of Helsinki, Department of Mathematics and Statistics, P.O. FinlandHelsinkiFinland
  2. 2.Université Paris-Est Marne la Vallée, LAMAChamps sur MarneFrance
  3. 3.ENS Ulm, DMAParisFrance

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