Abstract
We study the semi-classical limit of the reflection coefficient for the SL(2, ℝ)k /U(1) CFT. For large k, the CFT describes a string in a Euclidean black hole of 2-dimensional dilaton-gravity, whose target space is a cigar with an asymptotically linear dilaton. This sigma-model description is weakly coupled in the large k limit, and we investigate the saddle-point expansion of the functional integral that computes the reflection coefficient. As in the semi-classical limit of Liouville CFT studied in [1], we find that one must complexify the functional integral and sum over complex saddles to reproduce the limit of the exact reflection coefficient. Unlike Liouville, the SL(2, ℝ)k /U(1) CFT admits bound states that manifest as poles of the reflection coefficient. To reproduce them in the semi-classical limit, we find that one must sum over configurations that hit the black hole singularity, but nevertheless contribute to the saddle-point expansion with finite action.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B 371 (1992) 269 [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS3 and SL(2, R) WZW model. Part 1. The spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
J.M. Maldacena, H. Ooguri and J. Son, Strings in AdS3 and the SL(2, R) WZW model. Part 2. Euclidean black hole, J. Math. Phys. 42 (2001) 2961 [hep-th/0005183] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS3 and the SL(2, R) WZW model. Part 3. Correlation functions, Phys. Rev. D 65 (2002) 106006 [hep-th/0111180] [INSPIRE].
K. Gawędzki, Noncompact WZW conformal field theories, in NATO advanced study institute: new symmetry principles in quantum field theory, (1991), pg. 0247 [hep-th/9110076] [INSPIRE].
J. Teschner, On structure constants and fusion rules in the SL(2, C)/SU(2) WZNW model, Nucl. Phys. B 546 (1999) 390 [hep-th/9712256] [INSPIRE].
J. Teschner, Operator product expansion and factorization in the \( {H}_3^{+} \)-WZNW model, Nucl. Phys. B 571 (2000) 555 [hep-th/9906215] [INSPIRE].
A. Hanany, N. Prezas and J. Troost, The partition function of the two-dimensional black hole conformal field theory, JHEP 04 (2002) 014 [hep-th/0202129] [INSPIRE].
A. Giveon and D. Kutasov, Comments on double scaled little string theory, JHEP 01 (2000) 023 [hep-th/9911039] [INSPIRE].
E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
E. Witten, Analytic continuation of Chern-Simons theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].
R. Balian, G. Parisi and A. Voros, Quartic oscillator, in Feynman path integrals, (1978), pg. 337.
A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. I.H.P. Phys. Théor. 39 (1983) 211.
A. Cherman, D. Dorigoni and M. Ünsal, Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles, JHEP 10 (2015) 056 [arXiv:1403.1277] [INSPIRE].
A. Behtash, G.V. Dunne, T. Schäfer, T. Sulejmanpasic and M. Ünsal, Complexified path integrals, exact saddles and supersymmetry, Phys. Rev. Lett. 116 (2016) 011601 [arXiv:1510.00978] [INSPIRE].
A. Behtash, G.V. Dunne, T. Schäfer, T. Sulejmanpasic and M. Ünsal, Toward Picard-Lefschetz theory of path integrals, complex saddles and resurgence, Ann. Math. Sci. Appl. 02 (2017) 95 [arXiv:1510.03435] [INSPIRE].
T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai, Nonperturbative contributions from complexified solutions in CPN−1 models, Phys. Rev. D 94 (2016) 105002 [arXiv:1607.04205] [INSPIRE].
A. Behtash, G.V. Dunne, T. Schaefer, T. Sulejmanpasic and M. Ünsal, Critical points at infinity, non-gaussian saddles, and bions, JHEP 06 (2018) 068 [arXiv:1803.11533] [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].
N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl. 102 (1990) 319 [INSPIRE].
O. Aharony, A. Giveon and D. Kutasov, LSZ in LST, Nucl. Phys. B 691 (2004) 3 [hep-th/0404016] [INSPIRE].
J.B. Hartle and S.W. Hawking, Path integral derivation of black hole radiance, Phys. Rev. D 13 (1976) 2188 [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, Stringy horizons II, JHEP 10 (2016) 157 [arXiv:1603.05822] [INSPIRE].
M.V. Berry, Infinitely many Stokes smoothings in the gamma function, Proc. Roy. Soc. 434 (1991) 465.
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].
D. Kutasov, Accelerating branes and the string/black hole transition, hep-th/0509170 [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, Stringy horizons, JHEP 06 (2015) 064 [arXiv:1502.03633] [INSPIRE].
A. Giveon and N. Itzhaki, String theory at the tip of the cigar, JHEP 09 (2013) 079 [arXiv:1305.4799] [INSPIRE].
G. Giribet, Scattering of low lying states in the black hole atmosphere, Phys. Rev. D 94 (2016) 026008 [Addendum ibid. 94 (2016) 049902] [arXiv:1606.06919] [INSPIRE].
D.L. Jafferis and E. Schneider, Stringy ER=EPR, arXiv:2104.07233 [INSPIRE].
A. Giveon and N. Itzhaki, Stringy black hole interiors, JHEP 11 (2019) 014 [arXiv:1908.05000] [INSPIRE].
R. Ben-Israel, A. Giveon, N. Itzhaki and L. Liram, On the black hole interior in string theory, JHEP 05 (2017) 094 [arXiv:1702.03583] [INSPIRE].
E. Witten, Open strings on the Rindler horizon, JHEP 01 (2019) 126 [arXiv:1810.11912] [INSPIRE].
L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE].
M. Grinberg and J. Maldacena, Proper time to the black hole singularity from thermal one-point functions, JHEP 03 (2021) 131 [arXiv:2011.01004] [INSPIRE].
V. Fateev, A. Zamolodchikov and Al. Zamolodchikov, unpublished.
V. Kazakov, I.K. Kostov and D. Kutasov, A matrix model for the two-dimensional black hole, Nucl. Phys. B 622 (2002) 141 [hep-th/0101011] [INSPIRE].
S. Ribault and J. Teschner, \( {H}_3^{+} \)-WZNW correlators from Liouville theory, JHEP 06 (2005) 014 [hep-th/0502048] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2004.05223
Rights and permissions
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.
About this article
Cite this article
Jafferis, D.L., Schneider, E. Semi-classical analysis of the string theory cigar. J. High Energ. Phys. 2021, 120 (2021). https://doi.org/10.1007/JHEP12(2021)120
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)120