Abstract
Lorentzian continuation of the Sine-Liouville model describes non-homogeneous rolling closed string tachyon. Via T-duality, this relates to the gauged H 3+ Wess-Zumino-Witten model at subcritical level. This model is exactly solvable. We give a closed formula for the 3-point correlation functions for the model at level k within the range 0 < k < 2, which relates to the analogous quantity for k > 2 in a similar way as how the Harlow-Maltz-Witten 3-point function of timelike Liouville field theory relates to the analytic continuation of the Dorn-Otto-Zamolodchikov-Zamolodchikov structure constants: we find that the ratio between both 3-point functions can be written in terms of quotients of Jacobi’s θ-functions, while their product exhibits remarkable cancellations and eventually factorizes. Our formula is consistent with previous proposals made in the literature.
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References
A. Sen, Rolling tachyon, JHEP 04 (2002) 048 [hep-th/0203211] [INSPIRE].
A. Sen, Tachyon matter, JHEP 07 (2002) 065 [hep-th/0203265] [INSPIRE].
A. Sen, Tachyons in string theory, Annales Henri Poincaré 4 (2003) S31 [INSPIRE].
A. Sen, Rolling tachyon boundary state, conserved charges and two-dimensional string theory, JHEP 05 (2004) 076 [hep-th/0402157] [INSPIRE].
A. Adams, J. Polchinski and E. Silverstein, Don’t panic! Closed string tachyons in ALE space-times, JHEP 10 (2001) 029 [hep-th/0108075] [INSPIRE].
M. Headrick, S. Minwalla and T. Takayanagi, Closed string tachyon condensation: an overview, Class. Quant. Grav. 21 (2004) S1539 [hep-th/0405064] [INSPIRE].
E.J. Martinec, Defects, decay and dissipated states, hep-th/0210231 [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].
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].
F. Larsen, A. Naqvi and S. Terashima, Rolling tachyons and decaying branes, JHEP 02 (2003) 039 [hep-th/0212248] [INSPIRE].
N.R. Constable and F. Larsen, The rolling tachyon as a matrix model, JHEP 06 (2003) 017 [hep-th/0305177] [INSPIRE].
J.L. Karczmarek and A. Strominger, Closed string tachyon condensation at c = 1, JHEP 05 (2004) 062 [hep-th/0403169] [INSPIRE].
J.L. Karczmarek, A. Maloney and A. Strominger, Hartle-Hawking vacuum for c = 1 tachyon condensation, JHEP 12 (2004) 027 [hep-th/0405092] [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, JHEP 11 (2003) 043 [hep-th/0306026] [INSPIRE].
D.Z. Freedman, M. Headrick and A. Lawrence, On closed string tachyon dynamics, Phys. Rev. D 73 (2006) 066015 [hep-th/0510126] [INSPIRE].
S. Ribault and R. Santachiara, Liouville theory with a central charge less than one, JHEP 08 (2015) 109 [arXiv:1503.02067] [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (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].
V. Schomerus and P. Suchanek, Liouville’s imaginary shadow, JHEP 12 (2012) 020 [arXiv:1210.1856] [INSPIRE].
M. Picco, R. Santachiara, J. Viti and G. Delfino, Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator, Nucl. Phys. B 875 (2013) 719 [arXiv:1304.6511] [INSPIRE].
W. McElgin, Notes on Liouville theory at c ≤ 1, Phys. Rev. D 77 (2008) 066009 [arXiv:0706.0365] [INSPIRE].
V. Schomerus and P. Suchanek, Liouville’s imaginary shadow, JHEP 12 (2012) 020 [arXiv:1210.1856] [INSPIRE].
M. Picco, R. Santachiara, J. Viti and G. Delfino, Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator, Nucl. Phys. B 875 (2013) 719 [arXiv:1304.6511] [INSPIRE].
Y. Hikida and T. Takayanagi, On solvable time-dependent model and rolling closed string tachyon, Phys. Rev. D 70 (2004) 126013 [hep-th/0408124] [INSPIRE].
V. Fateev, A.B. 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].
Y. Hikida and V. Schomerus, The FZZ-duality conjecture: a proof, JHEP 03 (2009) 095 [arXiv:0805.3931] [INSPIRE].
K. Hori and A. Kapustin, Duality of the fermionic 2D black hole and N = 2 Liouville theory as mirror symmetry, JHEP 08 (2001) 045 [hep-th/0104202] [INSPIRE].
A.B. Zamolodchikov, Three-point function in the minimal Liouville gravity, Theor. Math. Phys. 142 (2005) 183 [Teor. Mat. Fiz. 142 (2005) 218] [hep-th/0505063] [INSPIRE].
I.K. Kostov and V.B. Petkova, Bulk correlation functions in 2D quantum gravity, Theor. Math. Phys. 146 (2006) 108 [Teor. Mat. Fiz. 146 (2006) 132] [hep-th/0505078] [INSPIRE].
I.K. Kostov and V.B. Petkova, Non-rational 2D 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].
K. Becker and M. Becker, Interactions in the SL(2, R)/U(1) black hole background, Nucl. Phys. B 418 (1994) 206 [hep-th/9310046] [INSPIRE].
G. Giribet and C.A. Núñez, Aspects of the free field description of string theory on AdS 3, JHEP 06 (2000) 033 [hep-th/0006070] [INSPIRE].
G. Giribet and C.A. Núñez, Correlators in AdS 3 string theory, JHEP 06 (2001) 010 [hep-th/0105200] [INSPIRE].
G. Giribet and Y. Nakayama, The Stoyanovsky-Ribault-Teschner map and string scattering amplitudes, Int. J. Mod. Phys. A 21 (2006) 4003 [hep-th/0505203] [INSPIRE].
S. Ribault, private communication.
S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Four point correlation functions and the operator algebra in the two-dimensional conformal invariant theories with the central charge c < 1, Nucl. Phys. B 251 (1985) 691 [INSPIRE].
T. Fukuda and K. Hosomichi, Three point functions in sine-Liouville theory, JHEP 09 (2001) 003 [hep-th/0105217] [INSPIRE].
G. Giribet, Stringy horizons and generalized FZZ duality in perturbation theory, JHEP 02 (2017) 069 [arXiv:1611.03945] [INSPIRE].
G. Giribet and A. Ranjbar, Screening stringy horizons, Eur. Phys. J. C 75 (2015) 490 [arXiv:1504.05044] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and the SL(2, R) WZW model Part 3. Correlation functions, Phys. Rev. D 65 (2002) 106006 [hep-th/0111180] [INSPIRE].
K. Hosomichi, K. Okuyama and Y. Satoh, Free field approach to string theory on AdS 3, Nucl. Phys. B 598 (2001) 451 [hep-th/0009107] [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 structure constants and fusion rules in the SL(2, C)/SU(2) WZNW model, Nucl. Phys. B 546 (1999) 390 [hep-th/9712256] [INSPIRE].
V.S. Dotsenko, Three point correlation functions of the minimal conformal theories coupled to 2D gravity, Mod. Phys. Lett. A 6 (1991) 3601 [INSPIRE].
M. Goulian and M. Li, Correlation functions in Liouville theory, Phys. Rev. Lett. 66 (1991) 2051 [INSPIRE].
P. Di Francesco and D. Kutasov, World sheet and space-time physics in two-dimensional (super)string theory, Nucl. Phys. B 375 (1992) 119 [hep-th/9109005] [INSPIRE].
G. Giribet and L. Nicolás, Comment on three-point function in AdS 3 /CFT 2, J. Math. Phys. 50 (2009) 042304 [arXiv:0812.2732] [INSPIRE].
A. Dabholkar and A. Pakman, Exact chiral ring of AdS 3 /CFT 2, Adv. Theor. Math. Phys. 13 (2009) 409 [hep-th/0703022] [INSPIRE].
G. Giribet, A. Pakman and L. Rastelli, Spectral flow in AdS 3 /CFT 2, JHEP 06 (2008) 013 [arXiv:0712.3046] [INSPIRE].
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Giribet, G., Rado, L. On non-homogeneous tachyon condensation in closed string theory. J. High Energ. Phys. 2017, 15 (2017). https://doi.org/10.1007/JHEP08(2017)015
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DOI: https://doi.org/10.1007/JHEP08(2017)015