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Strings in AdS_3 and the SL(2,R) WZW Model. Part 3: Correlation Functions

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Juan Maldacena, Hirosi Ooguri

High Energy Physics - Theory · Submitted 20 Nov 2001 · 42 references

Pith's verdict: A solid, foundational paper on AdS3 worldsheet correlators; the 'if' half of the factorization claim is genuinely checked, the 'only if' half is generically true but oversold in the abstract.

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arXiv IDhep-th/0111180 (version 2)
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CitationHirosi Ooguri, Juan Maldacena. "Strings in AdS_3 and the SL(2,R) WZW Model. Part 3: Correlation Functions." arXiv:hep-th/0111180 (2001). https://pith.science/paper/Y6SMGZP4
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The paper

The paper claims

String four-point functions on AdS$_3$ are shown to factorize into physical intermediate states only when a spin-sum condition holds, and the authors argue the failure outside that range is the physically correct answer.

The reading

This paper asks which correlation functions of string theory on three-dimensional anti-de Sitter space can be read as honest correlation functions of the dual boundary conformal field theory, and it answers by computing them and inspecting their singularities. The authors take the worldsheet theory to be the $SL(2,R)$ WZW model, whose spectrum they fixed in the first two papers of the series, and they work with the Euclidean $SL(2,C)/SU(2)$ coset where the four-point function is known through an integral over solutions of the Knizhnik-Zamolodchikov equation. Their central result is that the four-point function of short strings with vanishing spectral-flow label factorizes into a sum of products of three-point functions, summed over physical intermediate states, only when $j_1+j_2

What carries the argument

The load-bearing device is the worldsheet factorization ansatz for the $SL(2,C)/SU(2)$ four-point function, an integral over $j=1/2+is$ of $C(j)|F_j(z,x)|^2$, where $F_j$ is the unique solution of the Knizhnik-Zamolodchikov equation with the prescribed small-$z$ behaviour and $C(j)$ is a ratio of two three-point coefficients and a two-point coefficient. The integrand has singularities at $z=x$, a point in the middle of worldsheet moduli space, with exponent $k-j_1-j_2-j_3-j_4$; the authors explain this as a worldsheet instanton that grows large only when the worldsheet cross ratio equals the target-space cross ratio. Deforming the $j$ contour from $j=1/2+is$ to $j=(k-1)/2+is$ picks up poles at $j=j_1+j_2+n$ and $j=k-j_1-j_2+n$, and it is the second family that produces the terms spoiling factorization. Splitting the $z$ integral into $|z|<1$ and $|z|>1$, and regularizing the small-$|z|$ region so that the $j$ and $z$ integrals can be exchanged, lets the authors read off the intermediate conformal weights from the powers of $x$.

Extended reading notes

Core claim

The paper establishes that the target-space four-point function of four short strings with $w=0$ admits an operator-product expansion in the standard sense, $F(x) = x^{J-J_1-J_2} C$ with $C$ built from two three-point functions divided by a two-point function, precisely when the external spins obey $j_1+j_2<k+1/2$ and $j_3+j_4<k+1/2$. The intermediate states that appear are short strings with $w=0$ in the range $1/2<j<k-1/2$, long strings with $w=1$, and two-particle states; nothing else contributes. The authors verify the coefficients match, including a factor $|2j-1+(k-2)w|$ in the short-string two-point function that is easy to lose when the worldsheet delta function is cancelled against the volume of the conformal group. When the spin-sum condition fails, residues from a second set of poles contribute powers of $x$ that correspond to no physical operator, and the paper identifies these as the source of the expected pole at $j_1+j_2+j_3+j_4-k=1$. The authors read the breakdown as a genuine feature of a boundary theory with a non-compact target space, in the same way that a quantum-mechanical operator $e^{i k x}$ has a restricted domain of validity when the potential well is finite.

Load-bearing premise

The argument depends on the contour deformation in the worldsheet spin integral being faithful to the physical amplitude: that no pole of the two-point coefficient or the conformal block crosses the contour, and that the worldsheet modulus integral can be regularized so it may be exchanged with the spin integral.

Editorial extensions

If this is right

  • The spectrum of intermediate states in the four-point function is fixed to short strings with $w=0$, long strings with $w=1$, and two-particle states, matching the spectrum proposed in the first paper of the series.
  • Three-point functions of discrete states make sense only when $j_1+j_2+j_3<k$; beyond that the analytic continuation exists but carries no boundary-theory interpretation.
  • Two short strings with $w=0$ cannot fuse into a short string with $w=1$, because the three-point coefficient carries a factor of the inverse conformal volume that vanishes once the operators are normalized to finite two-point functions.
  • The winding-violation rule for an $N$-point function in the $x$ basis, $w_i-\sum_{j\neq i} w_j \le N-2$, follows from the current-algebra highest-weight conditions alone.
  • Long-string two-point functions come with a pure phase that reads as the phase shift of a long string travelling in from the boundary, shrinking, and returning.

Reading between the lines

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

  • If the factorization condition is really a statement about the non-compactness of the boundary target space, then deforming the theory to compactify that direction should restore factorization for all spins; the paper suggests adding Ramond-Ramond fields as exactly such a deformation, which would be a sharp check.
  • The same middle-of-moduli-space singularity at $z=x$ should appear in higher-point functions whenever a subset of worldsheet moduli coincides with a subset of target-space moduli, which is a prediction that could be tested against any explicit five-point computation.
  • The failure of factorization looks structurally like the statement that a product of two operators can map the vacuum out of the Hilbert space, which suggests that the boundary theory is not well defined on higher-genus worldsheets; the authors note this for the torus in the companion paper.
  • Because the long-string two-point function is finite without the extra $|2j-1+(k-2)w|$ factor that short strings require, the distinction between short and long strings may be visible directly in the normalization of boundary correlators rather than only in their spectrum.
Original abstract

We consider correlation functions for string theory on AdS_3. We analyze their singularities and we provide a physical interpretation for them. We explain which worldsheet correlation functions have a sensible physical interpretation in terms of the boundary theory. We consider the operator product expansion of the four point function and we find that it factorizes only if a certain condition is obeyed. We explain that this is the correct physical result. We compute correlation functions involving spectral flowed operators and we derive a constraint on the amount of winding violation.

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, 4,705 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.47 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.30 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We consider correlation functions for string theory on AdS_3. We analyze their singularities and we provide a physical interpretation for them. We explain which worldsheet correlation functions have a sensible physical interpretation in terms of the boundary theory. We consider the operator product expansion of the four point function and we find that it factorizes only if a certain condition is obeyed. We explain that this is the correct physical result. We compute correlation functions involving spectral flowed operators and we derive a constraint on the amount of winding violation.

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 correlation functions for string theory on AdS_3. contribution 0.67 0.87 0.04 asserted 0.74
s1 We analyze their singularities and we provide a physical interpretation for them. contribution 0.91 0.95 0.12 asserted 1.00
s2 We explain which worldsheet correlation functions have a sensible physical interpretation in terms of the boundary theory. contribution 0.75 0.93 0.15 asserted 1.00
s3 We consider the operator product expansion of the four point function and we find that it factorizes only if a certain condition is obeyed. result 0.95 0.96 0.23 asserted 1.00
s4 We explain that this is the correct physical result. interpretation 0.94 0.74 0.17 asserted 1.00
s5 We compute correlation functions involving spectral flowed operators and we derive a constraint on the amount of winding violation. result 0.81 0.96 0.23 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
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not examined
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  • 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. If Re(j) > k−1 2 , the worldsheet for the two point function grows uniformly on S 2 toward the boundary. 11 In several respects, these instantons are similar to instantons in ordinary Yang Mills theory in four dimensions. In this latter case their action depends logarithmically on the size of the instanton (analogous to e −ρ0 in our case) and if we are in a given theta vacuum, the instanton can dissolve into the vacuum. 15
Figure 2. Here we see the change in behavior of the semiclassical geodesics when we go from the case of j3 < j1 + j2 in (a) to the case j3 > j1 + j2 in (b). It is instructive to look at the semiclassical description of this divergence. Suppose ji are large, then correlation functions can be computed by considering a particle of masses proportional to ji with trajectories that intersect the boundary at the points where the 29
Figure 3. Change in behavior of the classical worldsheet when Pji < k in (a) to the case where Pji > k in (b). In (b) the worldsheet is driven to the boundary of AdS. One might think that this is a non-local effect in the BCFT. In order to remove it it seems that we need counter-terms which are spread all over the S 2 where the BCFT is defined. We would like to propose a different interpretation. The BCFT is local and this divergence is simply due to the non-compactness of the BCFT target space. In other words, we do not remove the divergence. The origin of this divergence, which we will expla
Figure 4. The solid line indicates the integration contour for (4.10) in the j complex plane. We highlighted the location of some poles in C(j). Here all external ji are of the form ji = 1 2 + isi. There are similar poles with j1, j2 → j3, j4, there are also some other poles that will not be important for our purposes. j + j + n k - j - j + n 1 1 2 1 2 1/2 (k-1)/2 k-2+1/2 2 | j - j | - n | j - j | - n 2 - 1
Figure 5. The solid line indicates the integration contour after we analytically continue (4.10) in the external ji. Some poles of the form |j1 −j2| −n have crossed the integration contour so we should include their residues. There are similar poles with j3, j4. We separated the poles along the imaginary direction for clarity, although they are all along the real axis when ji are real. The expression (4.10) is valid if all external labels j1, · · · , j4 are close to the line j = 1 2 + is. The expression for other values of j1, · · · j4 is defined by analytic continuation. When we do this, some
Figure 6. We shifted the integration contour to j = k 2 − 1 2 + is. We picked up contributions from Poles1 and Poles2. This figure represents the case when j1+j2 < k 2 and j3 + j4 > k 2 . 37
Figure 7. Under the spectral flow, a global SL(2, C) descendant |ψi of spin J 3 0 = m among the lowest energy states in D w=0 j or C w=0 j,α turns into the lowest weight state of the discrete representation d + J with J = m + k 2 w. The figure shows the flow of D w=0 j . The resulting operator is denoted by Φw,j J,J¯ (x, z). With respect to the global SL(2, R) algebra generated by J a 0 the state |ψi is the lowest weight state of a discrete representation d + J with J = m + k 2 w, independently of whether |ψi was in d ± j or cj,α of the SL(2, R) algebra generated by J˜a

Figures reproduced from arXiv:hep-th/0111180 by the authors.

References
How later papers used it

123 later papers cite it

Evidence: substantial. 122 of the citing papers share no author with it, and 22 of those carry a confident label from Pith's citation classifier.

Pith read 27 of the 475 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 12 papers with this label
Uses its data or values 1 of 2 papers with this label
How the literature uses it

121 arXiv papers cite this in 482 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 · 324 sentences without a confident label

Contests it 1 paper
Builds on it 3 papers
  • Similarly to the $=1$ case considered in [ref] , this can be solved exactly.

    On spectrally flowed local vertex operators in AdS$_3$ · Two-point functions · 0.91

  • In the $m$-basis, the independence from the inserting points $u_a^±$ becomes clearer.] [ref] ; here we want to prove this for an arbitrary number of them.

    A note on the spectral flow operator · Spectral flow operator · 0.97

  • The bosonic version of this theory was discussed in some detail in the seminal papers [ref] ; in what follows we extend, following [ref] , some aspects of their analysis to the supersymmetric case.

    Higher spins on AdS$_{3}$ from the worldsheet · Worldsheet string theory on AdS$_3$ · 0.88

Uses its method or tool 12 papers
  • Following [ref] , we therefore regularize the $u$-integral by introducing a cut-off parameter $ε$ and divide the range of $u$ into two regions: [equation] In region I there are only operators in the intermediate channel whose $SL(2)$ part is associated with short strings with winding number $w=0$ [ref] .

    Worldsheet four-point functions in AdS(3)/CFT(2) · Moduli integration and integral over $h$ · 0.99

  • Recently, in [ref] , (the one-particle contributions of) some extremal four-point correlators have been computed on the worldsheet using a general method for $SL(2)$ correlation functions developed in [ref] .

    Worldsheet operator product expansions and p-point functions in AdS3/CFT2 · Introduction · 0.83

  • We now compute the SL(2,$ℝ)$ correlators, applying the technique developed in [ref] .

    Three-point functions in superstring theory on AdS3xS3xT4 · Three-point functions of chiral states · 1.00

Uses its data or values 2 papers
Compares with it 14 papers
  • This should be compared with the $x$ dependence of the corresponding boundary four-point function, which is $x^H-H_1-H_2$ (see e.g. (4.2) in [ref] ), where $H$ denotes the corresponding spacetime conformal weights.

    Worldsheet four-point functions in AdS(3)/CFT(2) · Moduli integration and integral over $h$ · 0.93

  • As is suggested by the results in [ref] , and unlike in the treatment of the OPE of the corresponding string amplitudes given in [ref] , it will be found that all continuous and discrete spectral flow sectors which are permitted by the selection rules have finite and non-vanishing OPE coefficients for some range of values of the \(j\) quantum number.

    Notes on the SL(2,R) CFT · Introduction · 0.86

  • Indeed, while vertex operators in the former are built upon a specific lowest-weight state by means of translations in $x$-space, a single vertex in the latter gives rise to both $𝒟^+_J$ and $𝒟^-_J$, as stated in [ref] and already stressed below Eq. [ref].

    Winding modes and the reflection symmetry in AdS$_3$ · Final remarks · 0.86

Cites it as support 1 paper
Cites it as background 74 papers
  • We also review the known facts about the unflowed four-point functions [ref] and introduce the conjecture of [ref] relating them to their spectrally flowed counterparts.

    Spectral flow and the conformal block expansion for strings in AdS$_3$ · Introduction · 0.99

  • 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

  • We will mostly follow the conventions of [ref] , and work in the bosonic SL(2,$$)-WZW model at level $k>3$ [ref] .

    On spectrally flowed local vertex operators in AdS$_3$ · Definitions and conventions · 0.83

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