Abstract
We discuss the worldsheet sigma-model whose target space is the d+1 dimensional Euclidean Schwarzschild black hole. We argue that in the limit where the Hawking temperature of the black hole, T, approaches the Hagedorn temperature, TH, it can be described in terms of a generalized version of the Horowitz-Polchinski effective theory. For d ≥ 6, where the Horowitz-Polchinski EFT [1, 2] does not have suitable solutions, the modified effective Lagrangian allows one to study the black hole CFT in an expansion in powers of d − 6 and TH − T. At T = TH, the sigma model is non-trivial for all d > 6. It exhibits an enhanced SU(2) symmetry, and is described by a non-abelian Thirring model with a radially dependent coupling. The resulting picture connects naturally to the results of [3,4,5], that relate Schwarzschild black holes in flat spacetime at large d to the two dimensional black hole. We also discuss an analogous open string system, in which the black hole is replaced by a system of two separated D-branes connected by a throat. In this system, the asymptotic separation of the branes plays the role of the inverse temperature. At the critical separation, the system is described by a Kondo-type model, which again exhibits an enhanced SU(2) symmetry. At large d, the brane system gives rise to the hairpin brane [6].
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References
G.T. Horowitz and J. Polchinski, Selfgravitating fundamental strings, Phys. Rev. D 57 (1998) 2557 [hep-th/9707170] [INSPIRE].
Y. Chen, J. Maldacena and E. Witten, On the black hole/string transition, JHEP 01 (2023) 103 [arXiv:2109.08563] [INSPIRE].
J. Soda, Hierarchical dimensional reduction and gluing geometries, Prog. Theor. Phys. 89 (1993) 1303 [INSPIRE].
R. Emparan, D. Grumiller and K. Tanabe, Large-D gravity and low-D strings, Phys. Rev. Lett. 110 (2013) 251102 [arXiv:1303.1995] [INSPIRE].
Y. Chen and J. Maldacena, String scale black holes at large D, JHEP 01 (2022) 095 [arXiv:2106.02169] [INSPIRE].
S.L. Lukyanov, E.S. Vitchev and A.B. Zamolodchikov, Integrable model of boundary interaction: The Paperclip, Nucl. Phys. B 683 (2004) 423 [hep-th/0312168] [INSPIRE].
B. Balthazar, J. Chu and D. Kutasov, Winding Tachyons and Stringy Black Holes, arXiv:2204.00012 [INSPIRE].
G.T. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D 55 (1997) 6189 [hep-th/9612146] [INSPIRE].
B. Sathiapalan, Vortices on the String World Sheet and Constraints on Toral Compactification, Phys. Rev. D 35 (1987) 3277 [INSPIRE].
Y.I. Kogan, Vortices on the World Sheet and String’s Critical Dynamics, JETP Lett. 45 (1987) 709 [INSPIRE].
K.H. O’Brien and C.I. Tan, Modular Invariance of Thermopartition Function and Global Phase Structure of Heterotic String, Phys. Rev. D 36 (1987) 1184 [INSPIRE].
J.J. Atick and E. Witten, The Hagedorn Transition and the Number of Degrees of Freedom of String Theory, Nucl. Phys. B 310 (1988) 291 [INSPIRE].
D. Kutasov, Accelerating branes and the string/black hole transition, hep-th/0509170 [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].
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. Murthy, Notes on noncritical superstrings in various dimensions, JHEP 11 (2003) 056 [hep-th/0305197] [INSPIRE].
V.A. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, Sine-Liouville/Cigar Duality, unpublished notes.
A. Giveon and D. Kutasov, Little string theory in a double scaling limit, JHEP 10 (1999) 034 [hep-th/9909110] [INSPIRE].
K. Hori and A. Kapustin, Duality of the fermionic 2-D black hole and N = 2 liouville theory as mirror symmetry, JHEP 08 (2001) 045 [hep-th/0104202] [INSPIRE].
D. Tong, Mirror mirror on the wall: On 2-D black holes and Liouville theory, JHEP 04 (2003) 031 [hep-th/0303151] [INSPIRE].
C.G. Callan and J.M. Maldacena, Brane death and dynamics from the Born-Infeld action, Nucl. Phys. B 513 (1998) 198 [hep-th/9708147] [INSPIRE].
I. Affleck, Conformal field theory approach to the Kondo effect, Acta Phys. Polon. B 26 (1995) 1869 [cond-mat/9512099] [INSPIRE].
D. Kutasov, D-brane dynamics near NS5-branes, hep-th/0405058 [INSPIRE].
Y. Nakayama, Y. Sugawara and H. Takayanagi, Boundary states for the rolling D-branes in NS5 background, JHEP 07 (2004) 020 [hep-th/0406173] [INSPIRE].
D.A. Sahakyan, Comments on D-brane dynamics near NS5-branes, JHEP 10 (2004) 008 [hep-th/0408070] [INSPIRE].
Y. Nakayama, K.L. Panigrahi, S.-J. Rey and H. Takayanagi, Rolling down the throat in NS5-brane background: The case of electrified D-brane, JHEP 01 (2005) 052 [hep-th/0412038] [INSPIRE].
R. Brustein and Y. Zigdon, Effective field theory for closed strings near the Hagedorn temperature, JHEP 04 (2021) 107 [arXiv:2101.07836] [INSPIRE].
R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].
O. Aharony and D. Kutasov, Holographic Duals of Long Open Strings, Phys. Rev. D 78 (2008) 026005 [arXiv:0803.3547] [INSPIRE].
A. Giveon, N. Itzhaki and U. Peleg, Instant Folded Strings and Black Fivebranes, JHEP 08 (2020) 020 [arXiv:2004.06143] [INSPIRE].
D.L. Jafferis and E. Schneider, Stringy ER = EPR, JHEP 10 (2022) 195 [arXiv:2104.07233] [INSPIRE].
J. Polchinski, String theory. Volume 1: An introduction to the bosonic string, Cambridge University Press (2007) [https://doi.org/10.1017/CBO9780511816079] [INSPIRE].
Acknowledgments
This work was supported in part by DOE grant DE-SC0009924 and BSF grant 2018068. The work of DK was also supported in part by National Science Foundation grant PHY-1607611 at the Aspen Center for Physics. DK thanks the Weizmann Institute, Tel Aviv University, and the Aspen Center for Physics for hospitality during part of this work.
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Balthazar, B., Chu, J. & Kutasov, D. On small black holes in string theory. J. High Energ. Phys. 2024, 116 (2024). https://doi.org/10.1007/JHEP03(2024)116
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DOI: https://doi.org/10.1007/JHEP03(2024)116