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Strings in AdS_3 and the SL(2,R) WZW Model. Part 2: Euclidean Black Hole

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Juan Maldacena, Hirosi Ooguri, John Son

High Energy Physics - Theory · Submitted 19 May 2000 · 12 references

Pith's verdict: A clean, load-bearing derivation: the thermal AdS3 one-loop partition function is modular invariant and independently reproduces the Part 1 spectrum, with the discrete-state bound falling out of the pole structure.

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arXiv IDhep-th/0005183 (version 1)
Pith ID64OS4B3O, Pith's permanent record of this paper and its review
CitationHirosi Ooguri, John Son, Juan Maldacena. "Strings in AdS_3 and the SL(2,R) WZW Model. Part 2: Euclidean Black Hole." arXiv:hep-th/0005183 (2000). https://pith.science/paper/64OS4B3O
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The paper

The paper claims

A one-loop computation on thermal AdS$_3$ reproduces the string spectrum, including the upper bound on short-string spin.

The reading

The authors want an independent check on a spectrum they proposed in an earlier paper for strings moving in three-dimensional anti-de Sitter space. Their earlier proposal rested on spectral flow, a symmetry argued for classically and semiclassically; here they compute the one-loop vacuum amplitude on the Euclidean black hole background — equivalently, anti-de Sitter space with the Euclidean time direction made periodic, i.e. at finite temperature — and ask whether the single-string states can be read back out of it. The amplitude turns out to be modular invariant, and its pole and saddle-point structure encodes both families of states: long strings with a continuous energy, and short discrete states whose spin is confined to $1/2 < j < (k-1)/2$. If this holds, a proposal that previously looked like a choice of representations becomes a derived consequence of a well-defined path integral, and the extra poles and logarithmic divergences acquire a physical reading as an infinite volume of long strings and a Liouville wall cutting them off.

What carries the argument

Two mechanisms carry the argument. First, the unfolding trick: the sum over the winding number $n$ of the worldsheet embedding into the thermal circle is traded for a sum over images of the fundamental domain, so the $(0,m)$ term alone is integrated over a strip, with the $(0,0)$ term set aside as a temperature-independent cosmological-constant piece. Second, the pole structure of the integrand at $tau = r/w + ihat{beta}/(2pi w)$, where the map $theta - i t$ becomes holomorphic and $w$-to-one. Near such a point the induced potential $epsilon^2 e^{2rho}$ acts as a Liouville wall at $rho sim log epsilon$, so excising a disk of radius $epsilon$ around each pole is the same as imposing an infrared cutoff on how far a long string can wander. The finite, cutoff-dependent part of the resulting density of states is then checked against the derivative of the difference between the $SL(2,R)$ reflection phase and the Liouville phase, which the authors compute and find to agree.

Extended reading notes

Core claim

The paper evaluates the one-loop partition function of a bosonic string on $H_3/mathbb{Z}$ — thermal AdS$_3$, which is the same geometry as the Euclidean BTZ black hole — by direct functional integration in the style of Gawedzki, and shows the result is manifestly modular invariant once the winding and momentum sums over the thermal circle are included. Interpreting the answer as a spacetime free energy of a gas of strings, the authors recover the spectrum of their earlier proposal. The poles of the integrand at $tau = r/w + ihat{beta}/(2pi w)$ produce, after regularization, a logarithmically divergent piece whose coefficient is the long-string density of states with the continuous energy of Eq. (7); the saddle-point evaluation of the integral in each strip $hat{beta}/(2pi(w+1)) < tau_2 < hat{beta}/(2pi w)$ produces the discrete short-string energies of Eq. (3), together with the bound $1/2 < j < (k-1)/2$ as the statement that the saddle point must lie inside the strip. The authors state explicitly that, because this calculation does not assume the premises of the earlier paper, it can be regarded as a derivation of the spectrum.

Load-bearing premise

The load-bearing premise is that cutting a small disk around each pole in the modular parameter faithfully represents what a long string actually does at large radius; the authors themselves note that the finite part of the resulting density of states carries some dependence on the precise cutoff procedure.

Editorial extensions

If this is right

  • The spectrum proposed via spectral flow in the earlier paper becomes a consequence of the Euclidean path integral, so the continuous long-string energies and the discrete short-string energies stand on the same footing as the modular-invariant one-loop amplitude.
  • The upper bound $1/2 < j < (k-1)/2$ on the short-string spin is explained mechanically: it is the condition that the saddle point of the $tau_2$ integral lies inside the strip where a particular expansion of the integrand converges.
  • Logarithmic divergences at interior points of the modular domain, which would be alarming in a generic string amplitude, are reinterpreted as the infinite volume available to long strings, with the cutoff $epsilon$ playing the role of a radial wall.
  • The one-loop amplitude contains the $SL(2,R)$ reflection amplitude and the Liouville reflection amplitude, so a consistency check of the two against the density of states doubles as an independent derivation of the $SL(2,R)$ phase shift.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the density-of-states matching is taken as a derivation of the $SL(2,R)$ reflection phase, then the same bookkeeping should extend to the supersymmetric case; the paper notes only in passing that the tachyon divergence would disappear there, and checking that the long-string cutoff story survives with fermions would be a natural next step.
  • The identification of the regulator disk with a Liouville wall suggests a sharper statement than the paper makes: the finite part of the density of states should be regulator-independent up to terms that vanish as the wall is pushed away, which could be tested by comparing two different wall profiles.
  • Because the Euclidean black hole and thermal AdS$_3$ are the same background here, the result gives a stringy account of the thermal free energy of a black hole in three dimensions; the logarithmic long-string divergence is a concrete candidate for the volume scaling that any such account must reproduce.
  • The paper's remark that the earlier modular-invariance puzzle with $J^3$ insertions is resolved by the thermal setup hints that similar puzzles in other noncompact WZW models may also be artifacts of forgetting the winding sector.
Original abstract

We consider the one-loop partition function for Euclidean BTZ black hole backgrounds or equivalently thermal AdS_3 backgrounds which are quotients of H_3 (Euclidean AdS_3). The one-loop partition function is modular invariant and we can read off the spectrum which is consistent to that found in hep-th/0001053. We see long strings and discrete states in agreement with the expectations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,939 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We consider the one-loop partition function for Euclidean BTZ black hole backgrounds or equivalently thermal AdS_3 backgrounds which are quotients of H_3 (Euclidean AdS_3). The one-loop partition function is modular invariant and we can read off the spectrum which is consistent to that found in hep-th/0001053. We see long strings and discrete states in agreement with the expectations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider the one-loop partition function for Euclidean BTZ black hole backgrounds or equivalently thermal AdS_3 backgrounds which are quotients of H_3 (Euclidean AdS_3). contribution 0.83 0.50 0.07 asserted 0.93
s1 The one-loop partition function is modular invariant and we can read off the spectrum which is consistent to that found in hep-th/0001053. result 0.98 0.86 0.35 asserted 0.93
s2 We see long strings and discrete states in agreement with the expectations. result 0.75 0.84 0.25 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The sum over n is traded for the sum over copies of the fundamental domain. After multiplying (27) by the (b, c) ghosts and the internal CFT partition functions, we should integrate the resulting expression over the fundamental domain F0 of the modular parameter τ . The computation is much facilitated by the trick invented in [8] to trade the sum over n in (27) for the sum over copies of the fundamental domain. See
Figure 2. Poles in the τ plane, shown for w = 1 to 4. next subsections, we will see more precisely that it is indeed associated to the infinite volume in spacetime by relating ǫ to a long distance cutoff. Now we would like to calculate the short string spectrum. Since the long string spectrum gives a divergent result, while the short string spectrum gives a finite one, it might appear at first that extracting the contributions due to the short strings from a divergent expression such as (34) will be problematic. Fortunately we can get around this difficulty since the temperature dependence of…
Figure 3. Shifting the contour of integration picks up the pole residu

Figures reproduced from arXiv:hep-th/0005183 by the authors.

References
How later papers used it

114 later papers cite it

Evidence: substantial. 113 of the citing papers share no author with it, and 25 of those carry a confident label from Pith's citation classifier.

Pith read 28 of the 339 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 4 of 15 papers with this label
Uses its data or values 1 of 2 papers with this label

  • Computed the exact AdS_3 Hagedorn inverse temperature from the explicit expression for the thermal AdS_3 partition function computed in this paper.

    AdS$_3$ String Stars at Pure NSNS Flux (2023) · 0.96 · paraphrased: the licence does not allow quoting

Compares with it 1 of 10 papers with this label

  • Compared its own results for the H_3^+ partition function to eqn. (27) of this paper by setting α'=α''=0.

    Localization of strings on group manifolds (2025) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

110 arXiv papers cite this in 333 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 174 sentences without a confident label

Contests it 4 papers
  • Z^AdS, B_nm; k ℓ (τ) = β√(κ)/2π√(τ_2) exp( - (κ+ 2) β^2/4πL^2 τ_2 |m - n τ|^2 + 2π/τ_2 ( U_nm; kℓ)^2 )/|ϑ_1 ( U_nm;kℓ | τ) |^2 , where[Note that it is $τ$ and not $τ̅$ that appears inside the $ϑ$-function in [ref] correcting a typo in [ref] .

    Quantum Entanglement on Black Hole Horizons in String Theory and Holography · Horizon Orbifolds · contest 0.88

  • The contribution to the torus partition function from the thermal AdS factor is, for a fixed world-sheet modular parameter [ref] [In writing this expression we have corrected a typo in [ref] .

    Winding tachyons in BTZ · Comments on the 3 partition function · contest 0.74

  • This corrects an inaccuracy in [ref] which is of no further consequence there. ]

    Twisted Strings in Three-dimensional Black Holes · The Partition Function · contest 0.89

Builds on it 7 papers
Uses its method or tool 15 papers
  • In the remainder of this section, we will provide a solution in the form of an integral expression which we will obtain using the fact that (1) the deformed action has a concrete gauged sigma model form, (2) the folding/unfolding trick of [ref] can be utilized, and the fact that the $l_1=0$ limit is already solved in [ref] .

    Thermal partition function of $J_3 \bar{J}_3$ deformed $AdS_3$ · Statement of problem · 0.90

  • Our calculation follows the original computation done in [ref] and for some of the details we refer to the original treatment.

    Superstrings in Thermal Anti-de Sitter Space · The Single String Free Energy · 0.98

  • As shown in [ref] by adopting the technique in dealing with path integral of $H_3$ WZW model [ref] , one can exactly evaluate the partition function of the AdS sector, and the result is _AdS(β, μ;τ) =β(1-2/k)^1/2/(4π^2τ_2)^1/2 ∑_(n,m)(0,0) e^-β^2|m-nτ|^2/4πτ_2+2π(U_n,m)^2/τ_2/|ϑ_1(τ,U_n,m)|^2, where U_n,m(τ) =iβ(n τ̅-m)/2π√() (1/√(k)-i μ√()) , and the theta function is defined as ϑ_1(τ,U)=2q^1/8 sin(πU) ∏_m=1^∞(1-q^m)(1-zq^m)(1-z^-1q^m) , where $q=e^2πi τ, z=e^2πiU$.

    Hagedorn Strings and Correspondence Principle in AdS(3) · AdS$_3$ string and its thermal partition function · 0.88

Uses its data or values 2 papers
  • This torus amplitude computed in [ref] is: Z_TAdS_3( , ; ) = k-22 _2 _n,m Z e^- k ^24 _2|m-n |^2 + 2 ( I(U_n,m))^2 _2| _1(U_n,m| )|^2, where Our $U_n,m$ is the complex conjugate of that used in~ [ref] because we work in conventions where the left factor of the solution of the WZW model is holomorphic (left-moving on Lorentzian worldsheet).

    Revisiting the Thermal AdS partition function · Partition functions · 0.88

  • From the explicit expression for the partition function on thermal $AdS_3$ computed in [ref] , the exact $AdS_3$ Hagedorn inverse temperature is computed as[Nota Bene: In many places in the literature, there is an overall factor of $k$ in the $AdS_3$ metric that is not present in the Minkowski metric, so that the flat space coordinates are $√(k)$ times the $AdS_3$ coordinates in the $k→∞$ limit.

    AdS$_3$ String Stars at Pure NSNS Flux · AdS3 HP Equation · 0.96

Compares with it 10 papers
Cites it as support 2 papers
  • As shown in [ref] these additional poles are associated with the long strings which transverse the AdS space.

    Hagedorn Strings and Correspondence Principle in AdS(3) · The location of poles · 0.96

  • Further evidence that these representations give rise to a sensible string spectrum was provided through a path-integral calculation of the partition function of strings on the quotient of \(H_3\) corresponding to the Euclidean BTZ black hole [ref] .

    Notes on the SL(2,R) CFT · Introduction · 0.96

Cites it as background 66 papers
  • In this section we review the calculation of the thermal $AdS_3$ partition function in bosonic string theory with Neveu-Schwarz-Neveu-Schwarz flux [ref] , and its relation to the spectrum of string theory in $AdS_3$ [ref] .

    Superstrings in Thermal Anti-de Sitter Space · Thermal Three-dimensional Anti-de Sitter · 0.87

  • What [ref] considered is the worldsheet torus partition function for the sigma model ([ref]) analytically continued to $H_3 = SL(2,C)/SU(2)$ such that

    Thermal partition function of $J_3 \bar{J}_3$ deformed $AdS_3$ · Review of $SL(2,R)$ sigma model · 0.80

  • It is well known that the worldsheet CFT describing string theory on (Lorentzian) $AdS_3$ is given by the $SL(2,R)_k$ WZW model [ref] , where the level $k$ sets the AdS length $ℓ_AdS$ in terms of the string length $ℓ_s = √(α')$ via [equation] Similarly, the worldsheet theory describing Euclidean $AdS_3$ is the $SL(2,C)_k/SU(2)$ WZW model, and its orbifold $Z_β\SL(2,C)_k/SU(2)$ which compactifies the Euclidean time on a circle of periodicity $β$ describes thermal $AdS_3$ at inverse temperature $β$.

    AdS$_3$ String Stars at Pure NSNS Flux · AdS3 HP Equation · 0.95

Discussion

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