Abstract
We show that the entropy of strings that wind around the Euclidean time circle is proportional to the Noether charge associated with translations along the T-dual time direction. We consider an effective target-space field theory which includes a large class of terms in the action with various modes, interactions and α′ corrections. The entropy and the Noether charge are shown to depend only on the values of fields at the boundary of space. The classical entropy, which is proportional to the inverse of Newton’s constant, is then calculated by evaluating the appropriate boundary term for various geometries with and without a horizon. We verify, in our framework, that for higher-curvature pure gravity theories, the Wald entropy of static neutral black hole solutions is equal to the entropy derived from the Gibbons-Hawking boundary term. We then proceed to discuss horizonless geometries which contain, due to the back-reaction of the strings and branes, a second boundary in addition to the asymptotic boundary. Near this “punctured” boundary, the time-time component of the metric and the derivatives of its logarithm approach zero. Assuming that there are such non-singular solutions, we identify the entropy of the strings and branes in this geometry with the entropy of the solution to all orders in α′. If the asymptotic region of an α′-corrected neutral black hole is connected through the bulk to a puncture, then the black hole entropy is equal to the entropy of the strings and branes. Later, we discuss configurations similar to the charged black p-brane solutions of Horowitz and Strominger, with the second boundary, and show that, to leading order in the α′ expansion, the classical entropy of the strings and branes is equal exactly to the Bekenstein-Hawking entropy. This result is extended to a configuration that asymptotes to AdS.
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References
J.D. Bekenstein, Black holes and the second law, Lett. Nuovo Cim. 4 (1972) 737 [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
C.G. Callan and J.M. Maldacena, D-brane approach to black hole quantum mechanics, Nucl. Phys. B 472 (1996) 591 [hep-th/9602043] [INSPIRE].
J.C. Breckenridge, R.C. Myers, A.W. Peet and C. Vafa, D-branes and spinning black holes, Phys. Lett. B 391 (1997) 93 [hep-th/9602065] [INSPIRE].
J.C. Breckenridge, D.A. Lowe, R.C. Myers, A.W. Peet, A. Strominger and C. Vafa, Macroscopic and microscopic entropy of near extremal spinning black holes, Phys. Lett. B 381 (1996) 423 [hep-th/9603078] [INSPIRE].
G.T. Horowitz, J.M. Maldacena and A. Strominger, Nonextremal black hole microstates and U duality, Phys. Lett. B 383 (1996) 151 [hep-th/9603109] [INSPIRE].
G.T. Horowitz and J. Polchinski, Selfgravitating fundamental strings, Phys. Rev. D 57 (1998) 2557 [hep-th/9707170] [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].
Y. Chen, J. Maldacena and E. Witten, On the black hole/string transition, arXiv:2109.08563 [INSPIRE].
Y. Chen and J. Maldacena, String scale black holes at large D, JHEP 01 (2022) 095 [arXiv:2106.02169] [INSPIRE].
A. Dabholkar, Tachyon condensation and black hole entropy, Phys. Rev. Lett. 88 (2002) 091301 [hep-th/0111004] [INSPIRE].
R. Brustein and Y. Zigdon, Black hole entropy sourced by string winding condensate, JHEP 10 (2021) 219 [arXiv:2107.09001] [INSPIRE].
A. Giveon, Explicit microstates at the Schwarzschild horizon, JHEP 11 (2021) 001 [arXiv:2108.04641] [INSPIRE].
R. Brustein, A. Giveon, N. Itzhaki and Y. Zigdon, A puncture in the Euclidean black hole, JHEP 04 (2022) 021 [arXiv:2112.03048] [INSPIRE].
R. Brustein and Y. Zigdon, Effective field theory for closed strings near the Hagedorn temperature, JHEP 04 (2021) 107 [arXiv:2101.07836] [INSPIRE].
M. Dine, E. Gorbatov, I.R. Klebanov and M. Krasnitz, Closed string tachyons and their implications for nonsupersymmetric strings, JHEP 07 (2004) 034 [hep-th/0303076] [INSPIRE].
W. Schulgin and J. Troost, The heterotic string at high temperature (or with strong supersymmetry breaking), JHEP 10 (2011) 047 [arXiv:1107.5316] [INSPIRE].
D.L. Jafferis and E. Schneider, Stringy ER=EPR, arXiv:2104.07233 [INSPIRE].
A. Giveon and N. Itzhaki, Stringy Information and Black Holes, JHEP 06 (2020) 117 [arXiv:1912.06538] [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].
I. Halder, D.L. Jafferis and D. Kolchmeyer, A duality in string theory on AdS3, arXiv:2208.00016 [INSPIRE].
R. Brustein and Y. Zigdon, Thermal equilibrium in string theory in the Hagedorn phase, JHEP 05 (2022) 031 [arXiv:2201.03541] [INSPIRE].
Y. Chen, Spectral form factor for free large N gauge theory and strings, JHEP 06 (2022) 137 [arXiv:2202.04741] [INSPIRE].
E.Y. Urbach, String stars in anti de Sitter space, JHEP 04 (2022) 072 [arXiv:2202.06966] [INSPIRE].
B. Balthazar, J. Chu and D. Kutasov, Winding Tachyons and Stringy Black Holes, arXiv:2204.00012 [INSPIRE].
Y. Matsuo, Fluid model of black hole/string transition, arXiv:2205.15976 [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
M. Visser, Dirty black holes: Entropy as a surface term, Phys. Rev. D 48 (1993) 5697 [hep-th/9307194] [INSPIRE].
T. Jacobson and R.C. Myers, Black hole entropy and higher curvature interactions, Phys. Rev. Lett. 70 (1993) 3684 [hep-th/9305016] [INSPIRE].
T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
A.A. Tseytlin, Mobius Infinity Subtraction and Effective Action in σ Model Approach to Closed String Theory, Phys. Lett. B 208 (1988) 221 [INSPIRE].
V.A. Kazakov and A.A. Tseytlin, On free energy of 2-D black hole in bosonic string theory, JHEP 06 (2001) 021 [hep-th/0104138] [INSPIRE].
G.T. Horowitz and A. Strominger, Black strings and P-branes, Nucl. Phys. B 360 (1991) 197 [INSPIRE].
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press (2011) [DOI].
T.H. Buscher, A Symmetry of the String Background Field Equations, Phys. Lett. B 194 (1987) 59.
T.H. Buscher, Path Integral Derivation of Quantum Duality in Nonlinear Sigma Models, Phys. Lett. B 201 (1988) 466 [INSPIRE].
T.G. Mertens, Hagedorn String Thermodynamics in Curved Spacetimes and near Black Hole Horizons, Ph.D. Thesis, Universiteit Gent (2015) [arXiv:1506.07798] [INSPIRE].
J. Polchinski, TASI lectures on D-branes, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 96): Fields, Strings, and Duality, pp. 293–356 (1996) [hep-th/9611050] [INSPIRE].
M.J. Bowick, L. Smolin and L.C.R. Wijewardhana, Role of String Excitations in the Last Stages of Black Hole Evaporation, Phys. Rev. Lett. 56 (1986) 424 [INSPIRE].
L. Susskind, Some speculations about black hole entropy in string theory, hep-th/9309145 [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].
T. Damour and G. Veneziano, Selfgravitating fundamental strings and black holes, Nucl. Phys. B 568 (2000) 93 [hep-th/9907030] [INSPIRE].
G. Papallo and H.S. Reall, On the local well-posedness of Lovelock and Horndeski theories, Phys. Rev. D 96 (2017) 044019 [arXiv:1705.04370] [INSPIRE].
R. Brustein and Y. Sherf, Causality Violations in Lovelock Theories, Phys. Rev. D 97 (2018) 084019 [arXiv:1711.05140] [INSPIRE].
R.-G. Cai, R.-K. Su and P.K.N. Yu, Thermodynamics for black strings and p-branes, Phys. Lett. A 195 (1994) 307 [INSPIRE].
J.H. Horne, G.T. Horowitz and A.R. Steif, An Equivalence between momentum and charge in string theory, Phys. Rev. Lett. 68 (1992) 568 [hep-th/9110065] [INSPIRE].
M. Visser, Dirty black holes: Thermodynamics and horizon structure, Phys. Rev. D 46 (1992) 2445 [hep-th/9203057] [INSPIRE].
S.D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
I. Bena, E.J. Martinec, S.D. Mathur and N.P. Warner, Fuzzballs and Microstate Geometries: Black-Hole Structure in String Theory, arXiv:2204.13113 [INSPIRE].
A. Dabholkar, Quantum Entanglement in String Theory, arXiv:2207.03624 [INSPIRE].
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Brustein, R., Zigdon, Y. On the entropy of strings and branes. J. High Energ. Phys. 2022, 112 (2022). https://doi.org/10.1007/JHEP10(2022)112
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DOI: https://doi.org/10.1007/JHEP10(2022)112