Abstract
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.
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References
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].
A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett.B 103 (1981) 207 [INSPIRE].
E. Witten, The Feynman i𝜖 in String Theory, JHEP04 (2015) 055 [arXiv:1307.5124] [INSPIRE].
A. Sen, Equivalence of Two Contour Prescriptions in Superstring Perturbation Theory, JHEP04 (2017) 025 [arXiv:1610.00443] [INSPIRE].
S.R. Das, S. Naik and S.R. Wadia, Quantization of the Liouville Mode and String Theory, Mod. Phys. Lett.A 04 (1989) 1033.
A.B. Zamolodchikov, Three-point function in the minimal Liouville gravity, Theor. Math. Phys.142 (2005) 183 [hep-th/0505063] [INSPIRE].
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].
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].
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].
S. Ribault and R. Santachiara, Liouville theory with a central charge less than one, JHEP08 (2015) 109 [arXiv:1503.02067] [INSPIRE].
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].
D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
G. Giribet, On the timelike Liouville three-point function, Phys. Rev.D 85 (2012) 086009 [arXiv:1110.6118] [INSPIRE].
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].
T. Bautista, H. Erbin and M. Kudrna, BRST Cohomology of Timelike Liouville Theory, in progress.
R. Pius and A. Sen, Cutkosky rules for superstring field theory, JHEP10 (2016) 024 [Erratum ibid.1809 (2018) 122] [arXiv:1604.01783] [INSPIRE].
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].
A. Sen, Unitarity of Superstring Field Theory, JHEP12 (2016) 115 [arXiv:1607.08244] [INSPIRE].
A. Sen, One Loop Mass Renormalization of Unstable Particles in Superstring Theory, JHEP11 (2016) 050 [arXiv:1607.06500] [INSPIRE].
R. Pius and A. Sen, Unitarity of the Box Diagram, JHEP11 (2018) 094 [arXiv:1805.00984] [INSPIRE].
C. De Lacroix, H. Erbin and A. Sen, Analyticity and Crossing Symmetry of Superstring Loop Amplitudes, JHEP05 (2019) 139 [arXiv:1810.07197] [INSPIRE].
J. Polchinski, A Two-Dimensional Model for Quantum Gravity, Nucl. Phys.B 324 (1989) 123 [INSPIRE].
A. Dabholkar, Quantum Weyl Invariance and Cosmology, Phys. Lett.B 760 (2016) 31 [arXiv:1511.05342] [INSPIRE].
T. Bautista and A. Dabholkar, Quantum Cosmology Near Two Dimensions, Phys. Rev.D 94 (2016) 044017 [arXiv:1511.07450] [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Quantum black holes, localization and the topological string, JHEP06 (2011) 019 [arXiv:1012.0265] [INSPIRE].
A. Dabholkar, J. Gomes and S. Murthy, Localization & Exact Holography, JHEP04 (2013) 062 [arXiv:1111.1161] [INSPIRE].
A. Dabholkar, N. Drukker and J. Gomes, Localization in supergravity and quantum AdS4/CF T3holography, JHEP10 (2014) 090 [arXiv:1406.0505] [INSPIRE].
A. Strominger, Open string creation by S branes, Conf. Proc.C 0208124 (2002) 20 [hep-th/0209090] [INSPIRE].
M. Gutperle and A. Strominger, Time-like boundary Liouville theory, Phys. Rev.D 67 (2003) 126002 [hep-th/0301038] [INSPIRE].
A. Strominger and T. Takayanagi, Correlators in time-like bulk Liouville theory, Adv. Theor. Math. Phys.7 (2003) 369 [hep-th/0303221] [INSPIRE].
V. Schomerus, Rolling tachyons from Liouville theory, JHEP11 (2003) 043 [hep-th/0306026] [INSPIRE].
S. Fredenhagen and V. Schomerus, On minisuperspace models of S-branes, JHEP12 (2003) 003 [hep-th/0308205] [INSPIRE].
W. McElgin, Notes on Liouville Theory at c ≤ 1, Phys. Rev.D 77 (2008) 066009 [arXiv:0706.0365] [INSPIRE].
F. David, Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge, Mod. Phys. Lett.A 03 (1988) 1651.
J. Distler and H. Kawai, Conformal Field Theory and 2D Quantum Gravity, Nucl. Phys.B 321 (1989) 509 [INSPIRE].
N. Mavromatos and J. Miramontes, Regularizing the Functional Integral in 2D-Quantum Gravity, Mod. Phys. Lett.A 04 (1989) 1847.
E. D’Hoker and P. Kurzepa, 2D Quantum Gravity and Liouville Theory, Mod. Phys. Lett.A 05 (1990) 1411.
E. D’Hoker, Equivalence of Liouville Theory and 2-D Quantum Gravity, Mod. Phys. Lett.A 06 (1991) 745.
J. Polchinski, String Theory: Volume 1, An Introduction to the Bosonic String, Cambridge University Press, Cambridge U.K. (2005).
E. Silverstein, (A)dS backgrounds from asymmetric orientifolds, Clay Mat. Proc.1 (2002) 179 [hep-th/0106209] [INSPIRE].
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].
J. Teschner, Liouville theory revisited, Class. Quant. Grav.18 (2001) R153 [hep-th/0104158] [INSPIRE].
Y. Nakayama, Liouville field theory: A Decade after the revolution, Int. J. Mod. Phys.A 19 (2004) 2771 [hep-th/0402009] [INSPIRE].
A. Pakman, Liouville theory without an action, Phys. Lett.B 642 (2006) 263 [hep-th/0601197] [INSPIRE].
Al. Zamolodchikov and A. Zamolodchikov, Lectures on Liouville Theory and Matrix Models, (2007).
S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
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].
A. Kupiainen, Constructive Liouville Conformal Field Theory, 2016, arXiv:1611.05243 [INSPIRE].
R. Rhodes and V. vargas, Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity, arXiv:1602.07323 [INSPIRE].
H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys.B 429 (1994) 375 [hep-th/9403141] [INSPIRE].
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].
J. Teschner, On the Liouville three point function, Phys. Lett.B 363 (1995) 65 [hep-th/9507109] [INSPIRE].
J. Teschner, A Lecture on the Liouville vertex operators, Int. J. Mod. Phys.A 19S2 (2004) 436 [hep-th/0303150] [INSPIRE].
P. Gavrylenko and R. Santachiara, Crossing invariant correlation functions at c = 1 from isomonodromic τ functions, arXiv:1812.10362 [INSPIRE].
H. Sonoda, Sewing conformal field theories. 2., Nucl. Phys.B 311 (1988) 417 [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Modular bootstrap in Liouville field theory, Phys. Lett.B 685 (2010) 79 [arXiv:0911.4296] [INSPIRE].
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].
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].
A. Bilal, Remarks on the BRST cohomology for cM> 1 matter coupled to ’Liouville gravity’, Phys. Lett.B 282 (1992) 309 [hep-th/9202035] [INSPIRE].
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].
Al.B. Zamolodchikov, Conformal Symmetry in Two-Dimensional Space: Recursion Representation of Conformal Block, Theor. Math. Phys.73 (1987) 1088.
M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, JHEP04 (2019) 018 [arXiv:1703.09805] [INSPIRE].
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].
V.S. Dotsenko, Analytic continuations of 3-point functions of the conformal field theory, Nucl. Phys.B 907 (2016) 208 [arXiv:1601.07840] [INSPIRE].
T. Fulop, Reduced SL(2, ℝ) WZNW quantum mechanics, J. Math. Phys.37 (1996) 1617 [hep-th/9502145] [INSPIRE].
H. Kobayashi and I. Tsutsui, Quantum mechanical Liouville model with attractive potential, Nucl. Phys.B 472 (1996) 409 [hep-th/9601111] [INSPIRE].
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].
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].
T. Takayanagi, Matrix model and time-like linear dilaton matter, JHEP12 (2004) 071 [hep-th/0411019] [INSPIRE].
J. Maltz, Gauge Invariant Computable Quantities In Timelike Liouville Theory, JHEP01 (2013) 151 [arXiv:1210.2398] [INSPIRE].
E.J. Martinec and W.E. Moore, Modeling Quantum Gravity Effects in Inflation, JHEP07 (2014) 053 [arXiv:1401.7681] [INSPIRE].
A.R. Cooper, L. Susskind and L. Thorlacius, Two-dimensional quantum cosmology, Nucl. Phys.B 363 (1991) 132 [INSPIRE].
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].
I. Runkel and G.M.T. Watts, A Non-Rational CFT with Central Charge 1, Fortsch. Phys.50 (2002) 959 [hep-th/0201231].
B. Zwiebach, Closed string field theory: Quantum action and the B-V master equation, Nucl. Phys.B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
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Bautista, T., Dabholkar, A. & Erbin, H. Quantum gravity from timelike Liouville theory. J. High Energ. Phys. 2019, 284 (2019). https://doi.org/10.1007/JHEP10(2019)284
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DOI: https://doi.org/10.1007/JHEP10(2019)284