Journal of High Energy Physics

, 2019:284 | Cite as

Quantum gravity from timelike Liouville theory

  • Teresa BautistaEmail author
  • Atish Dabholkar
  • Harold Erbin
Open Access
Regular Article - Theoretical Physics


A proper definition of the path integral of quantum gravity has been a long- standing puzzle because the Weyl factor of the Euclidean metric has a wrong-sign kinetic term. We propose a definition of two-dimensional Liouville quantum gravity with cos- mological constant using conformal bootstrap for the timelike Liouville theory coupled to supercritical matter. We prove a no-ghost theorem for the states in the BRST cohomology. We show that the four-point function constructed by gluing the timelike Liouville three- point functions is well defined and crossing symmetric (numerically) for external Liouville energies corresponding to all physical states in the BRST cohomology with the choice of the Ribault-Santachiara contour for the internal energy.


Conformal Field Theory Models of Quantum Gravity 2D Gravity BRST Quantization 


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]
    G.W. Gibbons, S.W. Hawking and M. Perry, Path Integrals and the Indefiniteness of the Gravitational Action, Nucl. Phys.B 138 (1978) 141 [INSPIRE].ADSMathSciNetGoogle Scholar
  2. [2]
    A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett.B 103 (1981) 207 [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    E. Witten, The Feynman i𝜖 in String Theory, JHEP04 (2015) 055 [arXiv:1307.5124] [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    A. Sen, Equivalence of Two Contour Prescriptions in Superstring Perturbation Theory, JHEP04 (2017) 025 [arXiv:1610.00443] [INSPIRE].ADSzbMATHGoogle Scholar
  5. [5]
    S.R. Das, S. Naik and S.R. Wadia, Quantization of the Liouville Mode and String Theory, Mod. Phys. Lett.A 04 (1989) 1033.ADSMathSciNetGoogle Scholar
  6. [6]
    A.B. Zamolodchikov, Three-point function in the minimal Liouville gravity, Theor. Math. Phys.142 (2005) 183 [hep-th/0505063] [INSPIRE].zbMATHGoogle Scholar
  7. [7]
    I.K. Kostov and V.B. Petkova, Bulk correlation functions in 2 − D quantum gravity, Theor. Math. Phys.146 (2006) 108 [hep-th/0505078] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  8. [8]
    I.K. Kostov and V.B. Petkova, Non-rational 2 − D quantum gravity. I. World sheet CFT, Nucl. Phys.B 770 (2007) 273 [hep-th/0512346] [INSPIRE].
  9. [9]
    I.K. Kostov and V.B. Petkova, Non-Rational 2D Quantum Gravity II. Target Space CFT, Nucl. Phys.B 769 (2007) 175 [hep-th/0609020] [INSPIRE].
  10. [10]
    S. Ribault and R. Santachiara, Liouville theory with a central charge less than one, JHEP08 (2015) 109 [arXiv:1503.02067] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  11. [11]
    Y. Ikhlef, J.L. Jacobsen and H. Saleur, Three-Point Functions in c ≤ 1 Liouville Theory and Conformal Loop Ensembles, Phys. Rev. Lett.116 (2016) 130601 [arXiv:1509.03538] [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP12 (2011) 071 [arXiv:1108.4417] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  13. [13]
    G. Giribet, On the timelike Liouville three-point function, Phys. Rev.D 85 (2012) 086009 [arXiv:1110.6118] [INSPIRE].
  14. [14]
    P. Bouwknegt, J.G. McCarthy and K. Pilch, BRST analysis of physical states for 2 − D gravity coupled to c ≤ 1 matter, Commun. Math. Phys.145 (1992) 541 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  15. [15]
    T. Bautista, H. Erbin and M. Kudrna, BRST Cohomology of Timelike Liouville Theory, in progress.Google Scholar
  16. [16]
    R. Pius and A. Sen, Cutkosky rules for superstring field theory, JHEP10 (2016) 024 [Erratum ibid.1809 (2018) 122] [arXiv:1604.01783] [INSPIRE].
  17. [17]
    C. de Lacroix, H. Erbin, S.P. Kashyap, A. Sen and M. Verma, Closed Superstring Field Theory and its Applications, Int. J. Mod. Phys.A 32 (2017) 1730021 [arXiv:1703.06410] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  18. [18]
    A. Sen, Unitarity of Superstring Field Theory, JHEP12 (2016) 115 [arXiv:1607.08244] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  19. [19]
    A. Sen, One Loop Mass Renormalization of Unstable Particles in Superstring Theory, JHEP11 (2016) 050 [arXiv:1607.06500] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  20. [20]
    R. Pius and A. Sen, Unitarity of the Box Diagram, JHEP11 (2018) 094 [arXiv:1805.00984] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    C. De Lacroix, H. Erbin and A. Sen, Analyticity and Crossing Symmetry of Superstring Loop Amplitudes, JHEP05 (2019) 139 [arXiv:1810.07197] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  22. [22]
    J. Polchinski, A Two-Dimensional Model for Quantum Gravity, Nucl. Phys.B 324 (1989) 123 [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    A. Dabholkar, Quantum Weyl Invariance and Cosmology, Phys. Lett.B 760 (2016) 31 [arXiv:1511.05342] [INSPIRE].ADSzbMATHGoogle Scholar
  24. [24]
    T. Bautista and A. Dabholkar, Quantum Cosmology Near Two Dimensions, Phys. Rev.D 94 (2016) 044017 [arXiv:1511.07450] [INSPIRE].
  25. [25]
    A. Dabholkar, J. Gomes and S. Murthy, Quantum black holes, localization and the topological string, JHEP06 (2011) 019 [arXiv:1012.0265] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  26. [26]
    A. Dabholkar, J. Gomes and S. Murthy, Localization & Exact Holography, JHEP04 (2013) 062 [arXiv:1111.1161] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  27. [27]
    A. Dabholkar, N. Drukker and J. Gomes, Localization in supergravity and quantum AdS 4/CF T 3holography, JHEP10 (2014) 090 [arXiv:1406.0505] [INSPIRE].ADSzbMATHGoogle Scholar
  28. [28]
    A. Strominger, Open string creation by S branes, Conf. Proc.C 0208124 (2002) 20 [hep-th/0209090] [INSPIRE].Google Scholar
  29. [29]
    M. Gutperle and A. Strominger, Time-like boundary Liouville theory, Phys. Rev.D 67 (2003) 126002 [hep-th/0301038] [INSPIRE].ADSGoogle Scholar
  30. [30]
    A. Strominger and T. Takayanagi, Correlators in time-like bulk Liouville theory, Adv. Theor. Math. Phys.7 (2003) 369 [hep-th/0303221] [INSPIRE].MathSciNetGoogle Scholar
  31. [31]
    V. Schomerus, Rolling tachyons from Liouville theory, JHEP11 (2003) 043 [hep-th/0306026] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    S. Fredenhagen and V. Schomerus, On minisuperspace models of S-branes, JHEP12 (2003) 003 [hep-th/0308205] [INSPIRE].ADSMathSciNetGoogle Scholar
  33. [33]
    W. McElgin, Notes on Liouville Theory at c ≤ 1, Phys. Rev.D 77 (2008) 066009 [arXiv:0706.0365] [INSPIRE].
  34. [34]
    F. David, Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge, Mod. Phys. Lett.A 03 (1988) 1651.ADSMathSciNetGoogle Scholar
  35. [35]
    J. Distler and H. Kawai, Conformal Field Theory and 2D Quantum Gravity, Nucl. Phys.B 321 (1989) 509 [INSPIRE].ADSGoogle Scholar
  36. [36]
    N. Mavromatos and J. Miramontes, Regularizing the Functional Integral in 2D-Quantum Gravity, Mod. Phys. Lett.A 04 (1989) 1847.ADSGoogle Scholar
  37. [37]
    E. D’Hoker and P. Kurzepa, 2D Quantum Gravity and Liouville Theory, Mod. Phys. Lett.A 05 (1990) 1411.ADSzbMATHGoogle Scholar
  38. [38]
    E. D’Hoker, Equivalence of Liouville Theory and 2-D Quantum Gravity, Mod. Phys. Lett.A 06 (1991) 745.ADSMathSciNetzbMATHGoogle Scholar
  39. [39]
    J. Polchinski, String Theory: Volume 1, An Introduction to the Bosonic String, Cambridge University Press, Cambridge U.K. (2005).zbMATHGoogle Scholar
  40. [40]
    E. Silverstein, (A)dS backgrounds from asymmetric orientifolds, Clay Mat. Proc.1 (2002) 179 [hep-th/0106209] [INSPIRE].
  41. [41]
    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].ADSMathSciNetzbMATHGoogle Scholar
  42. [42]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav.18 (2001) R153 [hep-th/0104158] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  43. [43]
    Y. Nakayama, Liouville field theory: A Decade after the revolution, Int. J. Mod. Phys.A 19 (2004) 2771 [hep-th/0402009] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  44. [44]
    A. Pakman, Liouville theory without an action, Phys. Lett.B 642 (2006) 263 [hep-th/0601197] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  45. [45]
    Al. Zamolodchikov and A. Zamolodchikov, Lectures on Liouville Theory and Matrix Models, (2007).Google Scholar
  46. [46]
    S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
  47. [47]
    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].ADSMathSciNetzbMATHGoogle Scholar
  48. [48]
    A. Kupiainen, Constructive Liouville Conformal Field Theory, 2016, arXiv:1611.05243 [INSPIRE].
  49. [49]
    R. Rhodes and V. vargas, Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity, arXiv:1602.07323 [INSPIRE].
  50. [50]
    H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys.B 429 (1994) 375 [hep-th/9403141] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  51. [51]
    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].ADSMathSciNetzbMATHGoogle Scholar
  52. [52]
    J. Teschner, On the Liouville three point function, Phys. Lett.B 363 (1995) 65 [hep-th/9507109] [INSPIRE].ADSGoogle Scholar
  53. [53]
    J. Teschner, A Lecture on the Liouville vertex operators, Int. J. Mod. Phys.A 19S2 (2004) 436 [hep-th/0303150] [INSPIRE].
  54. [54]
    P. Gavrylenko and R. Santachiara, Crossing invariant correlation functions at c = 1 from isomonodromic τ functions, arXiv:1812.10362 [INSPIRE].
  55. [55]
    H. Sonoda, Sewing conformal field theories. 2., Nucl. Phys.B 311 (1988) 417 [INSPIRE].
  56. [56]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Modular bootstrap in Liouville field theory, Phys. Lett.B 685 (2010) 79 [arXiv:0911.4296] [INSPIRE].ADSMathSciNetGoogle Scholar
  57. [57]
    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].
  58. [58]
    X. Cao, P. Le Doussal, A. Rosso and R. Santachiara, Operator Product Expansion in Liouville Field Theory and Seiberg type transitions in log-correlated Random Energy Models, Phys. Rev.E 97 (2018) 042111 [arXiv:1801.09991] [INSPIRE].
  59. [59]
    A. Bilal, Remarks on the BRST cohomology for c M> 1 matter coupled to ’Liouville gravity’, Phys. Lett.B 282 (1992) 309 [hep-th/9202035] [INSPIRE].ADSGoogle Scholar
  60. [60]
    M. Asano and M. Natsuume, The No ghost theorem for string theory in curved backgrounds with a flat timelike direction, Nucl. Phys.B 588 (2000) 453 [hep-th/0005002] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  61. [61]
    Al.B. Zamolodchikov, Conformal Symmetry in Two-Dimensional Space: Recursion Representation of Conformal Block, Theor. Math. Phys.73 (1987) 1088.Google Scholar
  62. [62]
    M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, JHEP04 (2019) 018 [arXiv:1703.09805] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  63. [63]
    D. Kutasov and N. Seiberg, Number of degrees of freedom, density of states and tachyons in string theory and CFT, Nucl. Phys.B 358 (1991) 600 [INSPIRE].ADSMathSciNetGoogle Scholar
  64. [64]
    V.S. Dotsenko, Analytic continuations of 3-point functions of the conformal field theory, Nucl. Phys.B 907 (2016) 208 [arXiv:1601.07840] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  65. [65]
    T. Fulop, Reduced SL(2, ℝ) WZNW quantum mechanics, J. Math. Phys.37 (1996) 1617 [hep-th/9502145] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  66. [66]
    H. Kobayashi and I. Tsutsui, Quantum mechanical Liouville model with attractive potential, Nucl. Phys.B 472 (1996) 409 [hep-th/9601111] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  67. [67]
    I.K. Kostov, B. Ponsot and D. Serban, Boundary Liouville theory and 2 − D quantum gravity, Nucl. Phys.B 683 (2004) 309 [hep-th/0307189] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  68. [68]
    B. Carneiro da Cunha and E.J. Martinec, Closed string tachyon condensation and world sheet inflation, Phys. Rev.D 68 (2003) 063502 [hep-th/0303087] [INSPIRE].
  69. [69]
    T. Takayanagi, Matrix model and time-like linear dilaton matter, JHEP12 (2004) 071 [hep-th/0411019] [INSPIRE].ADSMathSciNetGoogle Scholar
  70. [70]
    J. Maltz, Gauge Invariant Computable Quantities In Timelike Liouville Theory, JHEP01 (2013) 151 [arXiv:1210.2398] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  71. [71]
    E.J. Martinec and W.E. Moore, Modeling Quantum Gravity Effects in Inflation, JHEP07 (2014) 053 [arXiv:1401.7681] [INSPIRE].ADSGoogle Scholar
  72. [72]
    A.R. Cooper, L. Susskind and L. Thorlacius, Two-dimensional quantum cosmology, Nucl. Phys.B 363 (1991) 132 [INSPIRE].ADSMathSciNetGoogle Scholar
  73. [73]
    I. Runkel and G.M.T. Watts, A Nonrational CFT with c = 1 as a limit of minimal models, JHEP09 (2001) 006 [hep-th/0107118] [INSPIRE].ADSGoogle Scholar
  74. [74]
    I. Runkel and G.M.T. Watts, A Non-Rational CFT with Central Charge 1, Fortsch. Phys.50 (2002) 959 [hep-th/0201231].ADSMathSciNetzbMATHGoogle Scholar
  75. [75]
    B. Zwiebach, Closed string field theory: Quantum action and the B-V master equation, Nucl. Phys.B 390 (1993) 33 [hep-th/9206084] [INSPIRE].ADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)Mühlenberg 1PotsdamGermany
  2. 2.Abdus Salam International Centre for Theoretical PhysicsTriesteItaly
  3. 3.Sorbonne UniversitéParisFrance
  4. 4.CNRSParisFrance
  5. 5.Ludwig-Maximilians-UniversitätMünchenGermany

Personalised recommendations