Abstract
Correlation functions in Liouville theory are meromorphic functions of the Liouville momenta, as is shown explicitly by the DOZZ formula for the three-point function on S2. In a certain physical region, where a real classical solution exists, the semiclassical limit of the DOZZ formula is known to agree with what one would expect from the action of the classical solution. In this paper, we ask what happens outside of this physical region. Perhaps surprisingly we find that, while in some range of the Liouville momenta the semiclassical limit is associated to complex saddle points, in general Liouville’s equations do not have enough complex-valued solutions to account for the semiclassical behavior. For a full picture, we either must include “solutions” of Liouville’s equations in which the Liouville field is multivalued (as well as being complex-valued), or else we can reformulate Liouville theory as a Chern-Simons theory in three dimensions, in which the requisite solutions exist in a more conventional sense. We also study the case of “timelike” Liouville theory, where we show that a proposal of Al. B. Zamolodchikov for the exact three-point function on S2 can be computed by the original Liouville path integral evaluated on a new integration cycle.
Similar content being viewed by others
References
A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].
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].
B. Freivogel, Y. Sekino, L. Susskind and C.-P. Yeh, A Holographic framework for eternal inflation, Phys. Rev. D 74 (2006) 086003 [hep-th/0606204] [INSPIRE].
Y. Sekino and L. Susskind, Census Taking in the Hat: FRW/CFT Duality, Phys. Rev. D 80 (2009) 083531 [arXiv:0908.3844] [INSPIRE].
D. Harlow and L. Susskind, Crunches, Hats and a Conjecture, arXiv:1012.5302 [INSPIRE].
H. Dorn and H. 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].
N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl. 102 (1990) 319 [INSPIRE].
L. Hadasz and Z. Jaskolski, Polyakov conjecture for hyperbolic singularities, Phys. Lett. B 574 (2003) 129 [hep-th/0308131] [INSPIRE].
L. Hadasz and Z. Jaskolski, Classical Liouville action on the sphere with three hyperbolic singularities, Nucl. Phys. B 694 (2004) 493 [hep-th/0309267] [INSPIRE].
E. Witten, Analytic Continuation Of Chern-Simons Theory, arXiv:1001.2933 [INSPIRE].
S. Pasquetti and R. Schiappa, Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models, Annales Henri Poincaré 11 (2010) 351 [arXiv:0907.4082] [INSPIRE].
M.V. Berry, Infinitely many stokes smoothings in the gamma function, Proc. Roy. Soc. Lond. A 434 (1991) 465.
W.G.C. Boyd, Gamma function asymptotics by an extension of the method of steepest descent, Proc. Roy. Soc. Lond. A 447 (1994) 609.
E. Braaten, T. Curtright, G. Ghandour and C.B. Thorn, Nonperturbative weak coupling analysis of the quantum Liouville field theory, Annals Phys. 153 (1984) 147 [INSPIRE].
J. Polchinski, Remarks On The Quantum Liouville Theory, presented at Strings ’90 Conference, College Station U.S.A., March 12-17 1990.
A. Belavin, A.M. Polyakov and A. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
H.L. Verlinde, Conformal field theory, 2d Gravity, and Quantization Of Teichmüller Space, Nucl. Phys. B 337 (1990) 652 [INSPIRE].
L. Chekhov and V. Fock, Quantum Teichmüller space, Theor. Math. Phys. 120 (1999) 1245 [math/9908165] [INSPIRE].
J. Teschner, Quantum Liouville theory versus quantized Teichmüller spaces, Fortsch. Phys. 51 (2003) 865 [hep-th/0212243] [INSPIRE].
D. Gaiotto and E. Witten, Knot Invariants From Four-Dimensional Gauge Theory, to appear.
A. Strominger and T. Takayanagi, Correlators in time-like bulk Liouville theory, Adv. Theor. Math. Phys. 7 (2003) 369 [hep-th/0303221] [INSPIRE].
A.B. Zamolodchikov, On the three-point function in minimal Liouville gravity, hep-th/0505063 [INSPIRE].
I. Kostov and V. Petkova, Bulk correlation functions in 2 − D quantum gravity, Theor. Math. Phys. 146 (2006) 108 [hep-th/0505078] [INSPIRE].
I. Kostov and V. Petkova, Non-rational 2 − D quantum gravity. I. World sheet CFT, Nucl. Phys. B 770 (2007) 273 [hep-th/0512346] [INSPIRE].
I. Kostov and V. Petkova, Non-Rational 2D Quantum Gravity II. Target Space CFT, Nucl. Phys. B 769 (2007) 175 [hep-th/0609020] [INSPIRE].
V. Schomerus, Rolling tachyons from Liouville theory, JHEP 11 (2003) 043 [hep-th/0306026] [INSPIRE].
W. McElgin, Notes on Liouville Theory at c ≤ 1, Phys. Rev. D 77 (2008) 066009 [arXiv:0706.0365] [INSPIRE].
S. Fredenhagen and V. Schomerus, On minisuperspace models of S-branes, JHEP 12 (2003) 003 [hep-th/0308205] [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].
P.H. Ginsparg and G.W. Moore, Lectures on 2 − D gravity and 2 − D string theory, hep-th/9304011 [INSPIRE].
J. Teschner, On the Liouville three point function, Phys. Lett. B 363 (1995) 65 [hep-th/9507109] [INSPIRE].
H. Poincaré, Les Fonctions Fuchsiennes et l’Equation Δu = e u, J. Math. Pures Appl. 4 (1898) 137.
G. Gibbons, S. Hawking and M. Perry, Path Integrals and the Indefiniteness of the Gravitational Action, Nucl. Phys. B 138 (1978) 141 [INSPIRE].
V.P. Frolov, D. Fursaev and D.N. Page, Thorny spheres and black holes with strings, Phys. Rev. D 65 (2002) 104029 [hep-th/0112129] [INSPIRE].
M. Umehara and K. Yamada, Metrics of Constant Curvature 1 with Three Conical Singularities on the 2-Sphere, Illinois J. Math. 44 (2000) 72 [math/9801137].
A. Giveon, D. Kutasov and N. Seiberg, Comments on string theory on AdS 3, Adv. Theor. Math. Phys. 2 (1998) 733 [hep-th/9806194] [INSPIRE].
J. Teschner, The Minisuperspace limit of the SL(\( {2},\;\mathbb{C} \))/SU(2) WZNW model, Nucl. Phys. B 546 (1999) 369 [hep-th/9712258] [INSPIRE].
J. Teschner, Operator product expansion and factorization in the H + (3) WZNW model, Nucl. Phys. B 571 (2000) 555 [hep-th/9906215] [INSPIRE].
J. Teschner, Crossing symmetry in the H(3) + WZNW model, Phys. Lett. B 521 (2001) 127 [hep-th/0108121] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and SL(\( {2},\;\mathbb{R} \)) WZW model 1.: The Spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and the SL(\( {2},\;\mathbb{R} \)) WZW model. Part 3. Correlation functions, Phys. Rev. D 65 (2002) 106006 [hep-th/0111180] [INSPIRE].
V. Knizhnik and A. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two-Dimensions, Nucl. Phys. B 247 (1984) 83 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Physics Reviews. Vol. 104: Conformal Field Theory and Critical Phenomena in Two-dimensional Systems, Taylor & Francis, Inc., New York U.S.A. (1989).
L. Hadasz, Z. Jaskolski and M. Piatek, Analytic continuation formulae for the BPZ conformal block, Acta Phys. Polon. B 36 (2005) 845 [hep-th/0409258] [INSPIRE].
A. Zamolodchikov, Conformal Symmetry In Two-dimensions: An Explicit Recurrence Formula For The Conformal Partial Wave Amplitude, Commun. Math. Phys. 96 (1984) 419.
A. Zamolodchikov, Two-dimensional Conformal Symmetry and Critical Four-spin Correlation Functions in the Ashkin-Teller Model (in Russian), Zh. Eksp. Teor. Fiz. 90 (1986) 1808.
A. Zamolodchikov, Conformal Symmetry in Two-dimensional Space: Recursion Representation of the Conformal Block, Theor. Math. Phys. 73 (1987) 1088.
S. Deser, R. Jackiw and S. Templeton, Topologically Massive Gauge Theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [INSPIRE].
E. Witten, Fivebranes and Knots, arXiv:1101.3216 [INSPIRE].
B. Freivogel and M. Kleban, A Conformal Field Theory for Eternal Inflation, JHEP 12 (2009) 019 [arXiv:0903.2048] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
E. Whitakker and G. Watson, A Course in Modern Analysis, fourth edition, Cambridge University Press, Cambridge U.K. (1927).
E. Witten, Analytic Continuation Of Chern-Simons Theory, arXiv:1001.2933 [INSPIRE].
M. Abramovitz and I. Stegun, National Bureau of Standards Applied Mathematics Series. Vol. 55: Handbook of Mathematical Functions, tenth edition, United States Department of Commerce, Washington U.S.A. (1972).
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1108.4417
Rights and permissions
About this article
Cite this article
Harlow, D., Maltz, J. & Witten, E. Analytic continuation of Liouville theory. J. High Energ. Phys. 2011, 71 (2011). https://doi.org/10.1007/JHEP12(2011)071
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2011)071