Regime of validity of the long folded-string identification of Schwarzschild black hole microstates

The entropy of a large Schwarzschild black hole in perturbative string theory has been argued to be carried by the modes of long folded strings (instantly created folded strings) bound to the vicinity of the horizon, via two assumptions: a near-horizon $SL(2)_k/U(1)$ superconformal description with $\alpha' k=(\beta/2\pi)^2$ , and the string-bit count $N=kN_{\mathrm{IFS}}=2\pi e^{-2\Phi_{\mathrm{hor}}}$ , together giving $S=N$ . These are established only at large spacetime dimension $D$ , large $\beta/\ell_s$ and large integer level $k$ , with $\beta/\ell_s$ and $k$ tied by $\alpha' k=(\beta/2\pi)^2$ and hence not independent; the source notes that the near-horizon description (assumption 1) remains correct also at small $k$ and $\beta/\ell_s$ . Whether the identification — above all the string-bit count and the associated fractionation of the long string into $k$ little strings — survives in the rest of the $(D,\beta/\ell_s,k)$ parameter space, in particular below the black hole/string transition at $k=1$ where the $k$ -dependence of the entropy is known to change, and at small or non-integer $k$ , remains open. Later work supports the framework within its original regime but leaves this regime question untouched.

Background

Black holes in general relativity carry Bekenstein–Hawking entropy $S=A/4G_N$ , proportional to the horizon area $A$ , and a central test of any candidate quantum theory of gravity is to exhibit the microscopic states realizing this degeneracy. String theory passes this test for near-extremal charged black holes, but the neutral, non-rotating Schwarzschild black hole — the generic case — has no controlled microscopic accounting within perturbative string theory (weak string coupling, states organized as string excitations on a background).

Three ingredients define the setting. First, string theory has a fundamental length $\ell_s=\sqrt{\alpha'}$ (string tension $1/2\pi\alpha'$ ), and a closed string can wind around a compact direction with mass set by that direction's radius; near the Hagedorn temperature — the temperature above which the exponential growth of the string spectrum makes the thermal ensemble diverge — the winding mode of the thermal circle (the "thermal scalar") becomes light and can condense. Second, Horowitz and Polchinski (1998) showed that a typical highly excited self-gravitating fundamental string matches, as the coupling grows, a black hole of the same mass and entropy — the black hole/string correspondence — so highly excited strings and small black holes are plausibly two descriptions of the same physics, separated by a transition. Third, at large spacetime dimension $D$ the near-horizon region of a $D$ -dimensional Schwarzschild black hole reduces to the two-dimensional string black hole, described exactly by the $SL(2)_k/U(1)$ superconformal field theory ("cigar coset"): a Wess–Zumino–Witten model based on $SL(2,\mathbb{R})$ gauged by $U(1)$ , whose target space is a cylinder that smoothly caps off like a cigar; the level $k$ fixes the cigar radius $\sqrt{\alpha' k}$ and is tied to the Hawking temperature $1/\beta$ by $\alpha' k=(\beta/2\pi)^2$ (Chen and Maldacena 2022, sharpening the large- $D$ expansion).

Building on the long-string picture of black hole entropy (Mertens, Verschelde and Zakharov 2016) and on instantly created folded strings (IFS) — closed folded strings nucleated at an instant near the horizon by the time-like linear dilaton (a background in which the dilaton field, whose value sets the string coupling, grows linearly with time) behind it (Itzhaki 2018; Attali and Itzhaki 2019) — Giveon (2021) argued that the microstates accounting for the entropy of a large Schwarzschild black hole are the modes of long folded strings bound to the horizon. The argument rests on two assumptions: (1) the near-horizon theory is the $SL(2)_k/U(1)$ SCFT with $\alpha' k=(\beta/2\pi)^2$ ; and (2) the number of string bits in the horizon winding condensate is $N=kN_{\mathrm{IFS}}=2\pi e^{-2\Phi_{\mathrm{hor}}}$ , where $\Phi_{\mathrm{hor}}$ is the two-dimensional dilaton at the horizon and $N_{\mathrm{IFS}}$ the number of IFS. Together these give $S=N$ , reproducing the Bekenstein–Hawking entropy at large $D$ . A classical folded-string solution supports assumption (2) at large integer $k$ and exhibits fractionation: the long string behaves as a fractionated fundamental string of inverse effective tension $2\pi\alpha'/k$ that decomposes, at its folds, into $k$ little strings (the stringy degrees of freedom underlying NS5-branes, the five-dimensional solitonic extended objects sourced by the same antisymmetric tensor field as the fundamental string).

The identification is, however, established only for parametrically large $D$ , large $\beta/\ell_s$ and large $k$ . The source itself notes that the sizes of $\beta/\ell_s$ and $k$ are not independent — they are tied by $\alpha' k=(\beta/2\pi)^2$ — and that assumption (1) is correct also for small $k$ and $\beta/\ell_s$ , so what is genuinely in question outside the derived regime is chiefly the string-bit count (assumption 2) and the fractionation picture built on it. The two-dimensional theory itself is known (Giveon, Kutasov, Rabinovici and Sever 2005) to undergo a black hole/string transition at $k=1$ : for $k<1$ the high-energy states are long perturbative strings in a linear-dilaton throat rather than a black hole, and the $k$ -dependence of the entropy changes — for $1/2<k<1$ the large-energy entropy is $S=2\pi\ell_s E\sqrt{2-1/k}$ . The large- $k$ counting therefore cannot be continued below $k=1$ unchanged, and the fractionation argument, made at large integer $k$ , has no established analogue there. The source paper flags exactly this gap and closes with the acknowledgment that "a lot more awaits to be understood".

Problem Statement

Determine whether the identification of the microstates of Schwarzschild black holes in perturbative string theory with the modes of long folded strings bound to the horizon — through the near-horizon $SL(2)_k/U(1)$ superconformal description with $\alpha' k=(\beta/2\pi)^2$ (assumption 1) and the string-bit count $N=kN_{\mathrm{IFS}}=2\pi e^{-2\Phi_{\mathrm{hor}}}$ , implying $S=N$ (assumption 2) — remains valid outside the regime of parametrically large $D$ , large $\beta/\ell_s$ and large integer $k$ in which it was derived, and establish the correct form of the microstate description in the remaining parameter space. Note that $\beta/\ell_s$ and $k$ are not independent coordinates of that space: they are tied by $\alpha' k=(\beta/2\pi)^2$ , and the source states that assumption 1 remains correct also at small $k$ and $\beta/\ell_s$ , so what is open outside the derived regime is chiefly the string-bit count of assumption 2, the resulting entropy and the fractionation picture. Concretely, as posed by the source: (i) below the black hole/string transition ( $k<1$ ), where the $k$ -dependence of the entropy $S(M)$ is known to be modified, determine the corresponding microstate count and whether, and how, the fractionation of the long IFS into $k$ little strings is modified; and (ii) determine whether the identification and the string-bit count, established at large integer $k$ , extend to small or non-integer $k$ . (i) and (ii) both lie in the region left open by the derived regime, and the general question is settled only when both, together with the physical picture of the states, are resolved. Beyond these concrete questions, the source acknowledges that "a lot more awaits to be understood"; characterizing the detailed nature and dynamics of the horizon microstates and the implications of the identification for neutral black holes in string theory remains a programmatic direction rather than a determinate deliverable. The open region — below the black hole/string transition ( $k<1$ ) and at small or non-integer $k$ , at the $\beta/\ell_s$ tied to those $k$ by $\alpha' k=(\beta/2\pi)^2$ , this region being the determinate part of the remaining parameter space outside the derived regime — is where the identification and the count, together with the physical picture of the states, either hold (possibly in modified form) or fail.

References

Amit Giveon, "Explicit microstates at the Schwarzschild horizon", JHEP 11 (2021) 001. DOI: 10.1007/JHEP11(2021)001, arXiv:2108.04641, https://arxiv.org/abs/2108.04641
Amit Giveon, "On the cigar CFT and Schwarzschild horizons", Nucl. Phys. B 116367 (2023). DOI: 10.1016/j.nuclphysb.2023.116367, arXiv:2308.15775, https://arxiv.org/abs/2308.15775
Yiming Chen and J. Maldacena, "String scale black holes at large D", JHEP 01 (2022) 095. DOI: 10.1007/JHEP01(2022)095, arXiv:2106.02169, https://arxiv.org/abs/2106.02169
Yiming Chen, Juan Maldacena and Edward Witten, "On the black hole/string transition", JHEP 01 (2023) 103. DOI: 10.1007/JHEP01(2023)103, arXiv:2109.08563, https://arxiv.org/abs/2109.08563
A. Giveon, D. Kutasov, E. Rabinovici and A. Sever, "Phases of quantum gravity in AdS3 and linear dilaton backgrounds", Nucl. Phys. B 719 (2005) 3. DOI: 10.1016/j.nuclphysb.2005.04.015, arXiv:hep-th/0503121, https://arxiv.org/abs/hep-th/0503121
A. Giveon, N. Itzhaki and U. Peleg, "Instant folded strings and black fivebranes", JHEP 08 (2020) 020. DOI: 10.1007/JHEP08(2020)020, arXiv:2004.06143, https://arxiv.org/abs/2004.06143
Akikazu Hashimoto, Nissan Itzhaki and Uri Peleg, "A worldsheet description of instant folded strings", JHEP 02 (2023) 088. DOI: 10.1007/JHEP02(2023)088, arXiv:2209.04988, https://arxiv.org/abs/2209.04988
Amr Ahmadain, Alexander Frenkel, Krishnendu Ray and Ronak M. Soni, "Boundary description of microstates of the two-dimensional black hole", SciPost Phys. 16 (2024) 020. DOI: 10.21468/SciPostPhys.16.1.020, arXiv:2210.11493, https://arxiv.org/abs/2210.11493
R. Brustein, A. Giveon, N. Itzhaki and Y. Zigdon, "A puncture in the Euclidean black hole", JHEP 04 (2022) 021. DOI: 10.1007/JHEP04(2022)021, arXiv:2112.03048, https://arxiv.org/abs/2112.03048
Ram Brustein and Yoav Zigdon, "On the entropy of strings and branes", JHEP 10 (2022) 112. DOI: 10.1007/JHEP10(2022)112, arXiv:2208.07372, https://arxiv.org/abs/2208.07372
Indranil Halder and Daniel L. Jafferis, "Stretched horizon, replica trick and off-shell winding condensate, and all that", JHEP 07 (2025) 176. DOI: 10.1007/JHEP07(2025)176, arXiv:2402.00932, https://arxiv.org/abs/2402.00932
Roberto Emparan, Luis Lehner and Stefano Trezzi, "Love at the String Scale: Tidal Deformability Across the Black Hole-String Transition", arXiv:2608.23077, https://arxiv.org/abs/2608.23077
G. T. Horowitz and J. Polchinski, "Self-gravitating fundamental strings", Phys. Rev. D 57 (1998) 2557. DOI: 10.1103/PhysRevD.57.2557, arXiv:hep-th/9707170, https://arxiv.org/abs/hep-th/9707170
Thomas G. Mertens, Henri Verschelde and Valentin I. Zakharov, "The long string at the stretched horizon and the entropy of large non-extremal black holes", JHEP 02 (2016) 041. DOI: 10.1007/JHEP02(2016)041, arXiv:1505.04025, https://arxiv.org/abs/1505.04025
N. Itzhaki, "Stringy instability inside the black hole", JHEP 10 (2018) 145. DOI: 10.1007/JHEP10(2018)145, arXiv:1808.02259, https://arxiv.org/abs/1808.02259
K. Attali and N. Itzhaki, "The averaged null energy condition and the black hole interior in string theory", Nucl. Phys. B 943 (2019) 114631. DOI: 10.1016/j.nuclphysb.2019.114631, arXiv:1811.12117, https://arxiv.org/abs/1811.12117
Jackson R. Fliss and Alexander Frenkel, "Matrix Quantum Mechanics and Entanglement Entropy: A Review", Entropy 28 (2025) 58. DOI: 10.3390/e28010058, arXiv:2512.03163, https://arxiv.org/abs/2512.03163
Alexander Frenkel, PhD thesis, "Stringy physics at the seams", Stanford University (2025). https://inspirehep.net/literature/3063831
Krishnendu Ray, PhD thesis, "Études in holography", University of Oxford (2024). DOI: 10.5287/ora-r5emyqdgd, https://ora.ox.ac.uk/objects/uuid:ae99168e-d3bb-4a55-a325-d90e097aa9ed