Two and three-point functions in Liouville theory

H. Dorn; H.-J. Otto

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Two and three-point functions in Liouville theory

H. Dorn, H.-J. Otto

High Energy Physics - Theory · Submitted 23 Mar 1994 · 19 references

Pith's verdict: The pre-DOZZ construction of generic-β Liouville two- and three-point functions: the closed forms are right in hindsight, but the continuation's normalization and α↔2/α duality are never independently pinned down.

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arXiv IDhep-th/9403141 (version 3)

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CitationH. Dorn, H.-J. Otto. "Two and three-point functions in Liouville theory." arXiv:hep-th/9403141 (1994). https://pith.science/paper/TMFKIWJS

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Editor's note

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This is the Dorn–Otto paper that gave closed-form two- and three-point functions of Liouville exponentials for arbitrary real β, three years before DOZZ. In hindsight the answers are correct, and that matters: it is the paper that showed the Goulian–Li continuation could be pushed off the rational-s grid of minimal models. If you work on noncritical strings, this is where the generic-β correlators first appear in closed form, and it is worth knowing what the derivation actually establishes and what it leaves open. What is new and done well: eq. (33) for A2 is a clean closed form, the Dotsenko–Fateev reduction of A3 to the f-function products is tidy, and the quasiclassical limit (37) reproduces Seiberg's semiclassical structure. That last one is a real independent check, not a self-consistency loop, and it deserves credit. The pole-zero analysis in §5 is careful, and the observation that A3 vanishes whenever all three β sit on A2 zeros is a genuinely useful structural fact for factorization arguments. Soft spots: the derivation is a continuation, and the authors say so in §1 (a more general validity proof still lacking). Footnote 3 is the sharpest instance: sin(πs)Γ(0) is assigned a value because that choice makes §3 work. That is a convention fixed by the check it is supposed to support, and the authors are upfront about it. The §3 Liouville-equation check is a nontrivial Gamma identity and does test the β-dependence, including the relative normalization between β1=0 and β1=α, but it is homogeneous in A3, so it cannot test the overall constant. The stress-test note is right about that, though it overstates: the overall normalization is inherited from the zero-mode integration and the µ² scaling, not from (42). The point that does land is self-duality. Everything is built on b = α²/2 with α = α_-, and the α_+ = 2/α_- branch is never discussed. The exact correlator must be a single function of β; the paper never shows its formula is. §4 on φ correlators is admittedly unfinished, and the authors say they have no deeper understanding of their own procedure. §5's mass-shell conjecture is flagged as a conjecture and depends on 4-point functions that do not exist yet. Who it is for: anyone working on 2D gravity or noncritical strings who wants the pre-DOZZ derivation and the pole-zero structure. Send it out.

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Major objections (3)

[§3, Eq. (42)] The Liouville-equation check is homogeneous in A3: the left side is 4(Δ2-Δ3)^2 ∂_{β1}A3|_{β1=0} and the right side is (αμ²/2)A3(α,β2,β3), so any constant rescaling A3 -> cA3 preserves the identity. It is also evaluated at fixed α, relating A3 at β1=0 to A3 at β1=α for the same α. Consequently the check tests the β-dependence and the relative sign, but it cannot test the overall normalization or the α-dependence of A3. Since the abstract claims construction of the correlators for generic real coefficients, and the exact correlator should be a single function of β independent of which branch α_± of Eq. (6) is used, a concrete additional test is available: verify that A3 is invariant under the exchange α_- <-> α_+ (equivalently b <-> 1/b with b = α²/2), or exhibit the reflection property β -> Q-β. If the construction singles out α_- and does not manifest this duality, the authors should say so explicitly and explain why the duality is not expected to hold at this level.
[Footnote 3 and §3] Footnote 3 fixes the ambiguous product sin(πs)Γ(0) -> π(-1)^{s+1} and states that this sign choice 'will be crucial in section 3 to ensure the validity of the Liouville equation.' The section 3 check then uses the resulting A3. This makes the one nontrivial consistency check partly definitional: the convention is selected because it makes the check pass, and the check is then performed on the expression built with that convention. The authors should either derive the convention independently (for example from a limiting case where the correlator is known, or from the functional integral) or state plainly that the check validates only the relative sign and the β-dependence, not the normalization. As written, the abstract's 'strong argument in favour of the procedure' overstates the independence of the evidence.
[§5, factorization paragraph] The statement that 'by explicit inspection of all possible cases one can exclude this situation' (three zeros of A3 for β_i at mass-shell values with no coinciding poles) is asserted without exhibiting the inspection. This claim is load-bearing for the factorization of higher string S-matrix elements, which the authors present as a motivation for the whole construction. At least the representative cases, or a short table of the pole-zero configurations for the three β_i, should be shown so the reader can verify the exclusion.

Minor objections (6)

[Eq. (31)] Eq. (31) writes G2 = A2 |z1-z2|^{4Δ}, whereas the standard CFT form obtained from Eq. (11) with Δ1=Δ2=Δ and Δ3=0 is |z1-z2|^{-4Δ}, and the paper's own quasiclassical result Eq. (37) has |z1-z2|^{-4b} with 4b ≈ 4Δ. Please check the sign of the exponent in Eq. (31).
[Eqs. (13), (14), (29), (33)] A stray factor 'α' appears immediately after Γ(-s3) in Eqs. (13) and (14) and in the prefactors of Eqs. (29) and (33). It is not clear whether this is a genuine factor of α, a power, or a typographical artifact. Please clarify the notation so the prefactors can be checked against the Goulian-Li result quoted in the text.
[§4] The choice β_j = α for the Liouville-field correlators is admitted to be 'motivated purely by technical reasons,' and the resulting polynomial-in-l_ij expressions are not derived from first principles. This is honestly flagged, but the section would read more cleanly if the heuristic status of these expressions were stated at the outset rather than at the end.
[§2, one-point function] The statement that the one-point function method 'yields inconsistent results' because the dependence on the fixed unintegrated w1, w2 does not cancel is an important negative result about the scope of the continuation. It deserves a sentence explaining whether this is expected to persist for all β or only for generic β, since it bears on the generality claimed in the abstract.
[References] Reference [15] (D'Hoker and Jackiw) does not appear to be cited anywhere in the text. Please either cite it where the Liouville action and the conformal anomaly are introduced or remove it.
[§5, opening] The drastic change in analytic structure between Re(α²) > 0 and Re(α²) < 0 is a central observation of the section. It would help to state the domain of validity of the continuation at the beginning of §2, so the reader knows from the outset which regime the closed forms (14)-(17) and (33) are meant to cover.

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The paper

The paper claims

The paper constructs closed-form two- and three-point functions for Liouville exponentials with arbitrary real exponent, and checks the three-point function against the Liouville equation of motion.

The reading

Liouville theory supplies the Weyl degree of freedom of a two-dimensional metric, so its correlation functions are what one needs to dress vertex operators in noncritical string theory and in conformal models beyond the minimal series. The obstacle is that the exponential $e^{βφ}$ is only a free-field construct for special rational values of $β$; for generic real $β$ the operator is an infinite series and no direct computation was available. The authors take the Goulian–Li continuation — treat the number $s$ of cosmological-term insertions as a variable and continue it away from integers — strip it of the degenerate-kinematics restrictions used in the original derivation, and obtain explicit expressions for the three-point amplitude $A_3(β_1,β_2,β_3)$ and the two-point amplitude $A_2(β)$. The evidence offered is a functional identity: the three-point function of $e^{β_2φ}e^{β_3φ}$ with one insertion of $∂^2φ$ equals $αμ^2/2$ times the three-point function with $e^{αφ}$ in that slot, which is the Liouville equation of motion at the level of correlators. If the construction is right, one has for the first time a usable closed form for the gravitational dressing of two- and three-point functions at arbitrary real dressing charge, plus a concrete proposal for where the mass shell of a noncritically dressed tachyon sits.

What carries the argument

The engine is the continued function $f(a,b|s)$, defined for integer $s$ as $Σ_{j=0}^{s-1}Γ(a+bj)$ and extended to arbitrary complex arguments by an integral representation. It obeys four functional relations — shift in $s$, shift in $a$, shift by $b$, and a reflection $b→-b$ — which let the authors move gamma-function arguments around without ever leaving the continued domain. The poles and zeros of $e^{f}$ are then read off from those relations: poles at $a = -bj-ℓ$ and zeros at $a+bs = -bj-ℓ$ for nonnegative integers $j,ℓ$. This is what makes the otherwise formal continuation controllable, and it is what produces the pole-zero spectrum of $A_2$ and $A_3$ that the paper later uses to argue about mass shells. A second piece of machinery is the treatment of the residual Möbius volume: the two-punctured sphere has one conformal Killing vector, so the naive $β_3→0$ limit of $A_3$ diverges, and the authors divide by the explicit volume $V^{(2)}_{CKV}$ of the subgroup fixing $z_1,z_2$ to obtain a finite $A_2$.

Extended reading notes

Core claim

The paper's central claim is that the two- and three-point functions of Liouville exponentials $e^{β_jφ}$ exist in closed form for generic real $β_j$. The three-point function is reduced, by a Möbius transformation argument, to the standard conformal structure with a $β$-dependent amplitude $A_3$ given by a Dotsenko–Fateev integral over $s_3 = (Q-Σβ_j)/α$ screening insertions; evaluating that integral and continuing in $s_3$ produces $A_3$ as a product of gamma functions times three factors $F_i$ built from a continued sum $f(a,b|s)$ of log-gamma terms. The two-point function follows either by taking a $β_3→0$ limit of $A_3$ with the volume of the residual Möbius subgroup divided out, or by differentiating the two-point function with respect to $μ^2$ and matching against the once-integrated three-point function; both routes give the compact expression for $A_2(β)$ in eq. (33). The load-bearing check is eq. (42): the authors verify that $4(Δ_2-Δ_3)^2 ∂_{β_1}A_3|_{β_1=0}$ equals $αμ^2/2$ times $A_3(α,β_2,β_3)$, which is precisely the Liouville equation of motion $∂^2φ = (αμ^2/2)e^{αφ}$ promoted to a statement about three-point functions. They read this as strong evidence that the continuation procedure is the right one, since a uniqueness proof was previously available only for the rational coefficients needed to dress minimal models.

Load-bearing premise

The whole result rests on continuing a sum of log-gamma terms from integer to arbitrary complex values, and on one convention for the otherwise meaningless product $Γ(0)sin(πs)$ — a sign choice the authors say is fixed by requiring the Liouville-equation check to come out right, rather than derived independently.

Editorial extensions

If this is right

Gravitational dressing factors for two- and three-point functions become available at arbitrary real dressing charge, not only at the rational values tied to minimal models.
The pole-zero structure of $A_2(β)$ yields a candidate mass-shell condition: the tachyon zero at $β = Q/2$ gives $(k-P/2)^2 = (D-1)/12$, with the graviton case shifted to $(k-P/2)^2 = (D-25)/12$.
All zeros of $A_2$ other than $β=Q/2$ lie above $Q/2$ and correspond to wave functions that are not normalizable as $φ→+∞$, so they are excluded and the microscopic puncture sits on the boundary of the allowed set.
Because $A_3$ vanishes whenever any one of the three $β_j$ is taken from the zero table of $A_2$, and no configuration of three coinciding poles can rescue it, the three-point string $S$-matrix element vanishes for on-shell external states — the property factorization of higher $S$-matrix elements would need.
The quasiclassical limit $Q→∞$ reproduces the previously known structure of the normalized two-point function, $G_2/Z ∝ (2b-1)^{-1}(-m^2/8)^{-2b}|z_1-z_2|^{-4b}$, now in explicit form.

Reading between the lines

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

The Liouville-equation check is a one-parameter test of a two-parameter continuation: it fixes the ambiguous $\sin(πs)Γ(0)$ convention but cannot by itself confirm that the $β$-dependence away from $β_1=0$ is right, so the natural next test is a four-point function or an independent canonical free-field computation of $A_2$.
The mass-shell conjecture is stated in $D$ target-space dimensions rather than $D+1$, and the authors themselves note that the $(D+1)$-dimensional reading of the on-shell condition works cleanly only at $μ=0$; whether the conjecture survives at $μ≠0$ is decidable once the special four-point cases are computed.
If the mass-shell zeros of $A_2$ are the right on-shell condition, then the tower of excluded zeros above $β=Q/2$ is a prediction about which would-be states are absent from the noncritical spectrum — a statement that a matrix-model or continuum computation at $c_M<1$ could in principle check.
The failure of the one-point function to be well defined in this scheme, which the authors trace to the absence of an $SL(2,\mathbb{C})$-invariant vacuum, suggests that any correlator of the Liouville field itself must be defined with at least two spectator exponentials — a structural constraint that would carry over to any alternative construction of $φ$ correlators.

Original abstract

Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two and three-point correlation functions for Liouville exponentials with generic real coefficients. As a strong argument in favour of the procedure we prove the Liouville equation of motion on the level of three-point functions. The analytical structure of the correlation functions as well as some of its consequences for string theory are discussed. This includes a conjecture on the mass shell condition for excitations of noncritical strings. We also make a comment concerning the correlation functions of the Liouville field itself.

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Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two and three-point correlation functions for Liouville exponentials with generic real coefficients. As a strong argument in favour of the procedure we prove the Liouville equation of motion on the level of three-point functions. The analytical structure of the correlation functions as well as some of its consequences for string theory are discussed. This includes a conjecture on the mass shell condition for excitations of noncritical strings. We also make a comment concerning the correlation functions of the Liouville field itself.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two and three-point correlation functions for Liouville exponentials with generic real coefficients.	contribution 0.90	0.95	0.49	asserted 1.00
s1	As a strong argument in favour of the procedure we prove the Liouville equation of motion on the level of three-point functions.	result 0.81	0.95	0.25	asserted 0.97
s2	The analytical structure of the correlation functions as well as some of its consequences for string theory are discussed.	contribution 0.74	0.81	0.10	asserted 0.81
s3	This includes a conjecture on the mass shell condition for excitations of noncritical strings.	interpretation 0.51	0.82	0.10	speculative 0.96
s4	We also make a comment concerning the correlation functions of the Liouville field itself.	contribution 0.90	0.83	0.04	unclear 0.76

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References

Works cited (19)

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H.-J. Otto, G. Weigt, Phys. Lett. B159 (1985) 341; Z. f. Phys. C31 (1986) 219

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J. L. Gervais, Nucl. Phys. B391 (1993) 287

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H. Dorn, H.-J. Otto, Phys. Lett. B291 (1992) 39

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R. Myers, V. Periwal, Phys. Rev. Lett. 70 (1993) 2841 and Conformally Invariant Oﬀ-shell Strings preprint hep-th/9311083

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hep-th 2009-09 accept novelty 7.0 of 10

Loop operators in a family of 4d N=2 gauge theories are identified with Liouville loop operators on a Riemann surface, reproducing Pestun's Wilson loop formula and predicting exact 't Hooft and dyonic loop values.
A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
hep-th 2009-07 conditional novelty 7.0 of 10

The paper conjectures and partially tests a dictionary equating A_{N-1} Toda correlators with SU(N) quiver instanton partition functions, generalizing the Liouville/SU(2) correspondence.
Strings in AdS_3 and the SL(2,R) WZW Model. Part 1: The Spectrum
hep-th 2000-01 conditional novelty 7.0 of 10

The SL(2,R) WZW model's spectrum must include spectral-flowed representations, which supply the long strings of AdS3 and remove an artificial upper bound on string mass.
Smoothing the Bouncing-Geodesic Singularity in AdS/CFT
hep-th 2026-11 conditional novelty 6.0 of 10

Finite-c quantum corrections convert sharp complex-time singularity imprints of the black hole interior into smooth Gaussian profiles, while leaving short-distance, OPE-protected singularities intact.
virasoro_blocks: Software for the 2d bootstrap
hep-th 2026-10 conditional novelty 6.0 of 10

Virasoro_blocks makes Virasoro conformal blocks usable in semidefinite-program bootstraps and yields scalar-gap and moment bounds that converge toward, and are saturated by, known 2D CFTs including Liouville theory.
Semiclassical Liouville Theory on the Real Projective Plane: A Complex Interpretation of the Bootstrap
hep-th 2026-09 accept novelty 6.0 of 10

Complex saddles with a negative-definite metric, selected by analytic continuation of the zero-mode integral, reproduce the exact RP^2 one-point function of Liouville theory through one loop.
Holographic entanglement entropy with conformal boundary conditions
hep-th 2026-08 conditional novelty 6.0 of 10

In AdS3 with conformal boundary conditions, holographic entanglement entropy is still the minimal-surface area over 4G_N, and the dual Liouville plus T Tbar theory gives entropy governed by the effective central charge c_eff.
On the local conformal structure of Imaginary Liouville theory
math-ph 2026-08 conditional novelty 6.0 of 10

Imaginary Liouville theory satisfies local conformal Ward identities and BPZ equations, conditional on an unproved decay conjecture for imaginary Gaussian multiplicative chaos.
Aspects of Closed Matricial Worlds
hep-th 2026-07 conditional novelty 6.0 of 10

Higher-genus corrections to the 2D de Sitter disk path integral follow a universal structure that at large boundary length mimics topological gravity, with new explicit expressions through genus six.
The Runkel-Watts string
hep-th 2026-07 conditional novelty 6.0 of 10

The Runkel-Watts string is a solvable family of two-dimensional string theories whose all-genus amplitudes are generated by SU(2) Yang-Mills TQFT data combined with VMS quantum volumes, and it has a dual matrix-integr...
The many facets of a hyperbolic tetrahedron: open and closed triangulations of 3d gravity
hep-th 2026-04 unverdicted novelty 6.0 of 10

Open Virasoro TQFT equals fixed-length/angle 3d gravity path integrals on compact regions and yields the CTV–scalar Virasoro relation via open-closed duality.
On random matrix statistics of 3d gravity
hep-th 2025-12 unverdicted novelty 6.0 of 10

3d gravity on Σ_{g,n} × I with EOW branes equals the Virasoro minimal string random matrix model, with exact match for g=0 n=2 and inner-product formulation for negative Euler characteristic.
Black Hole Entropy from String Entanglement
hep-th 2025-09 unverdicted novelty 6.0 of 10

String entanglement entropy between folded strings in the sine-Liouville CFT, computed via worldsheet replica trick, accounts for black-hole thermal entropy, with the vertex-operator part matching exactly in the low-t...
Resurgence for large $c$ expansion in Coulomb gas formalism
hep-th 2025-08 conditional novelty 6.0 of 10

The two conformal blocks of the phi21 four-point function are each other's Stokes corrections in the large-C expansion, and their z-monodromy is governed by alien calculus in the Borel plane.
Statistics in 3d gravity from knots and links
hep-th 2025-08 conditional novelty 6.0 of 10

Fragmentation of knots and links by Wilson lines builds wormhole amplitudes that give non-perturbative Gaussian and non-Gaussian corrections to OPE statistics in 3d gravity and CFT2.
Timelike Liouville theory and AdS$_3$ gravity at finite cutoff
hep-th 2025-08 unverdicted novelty 6.0 of 10

Proposes that AdS3 gravity at finite cutoff is dual to a CFT2 coupled to timelike Liouville theory deformed by a marginal operator, with checks via semiclassical partition functions and EOM matching.
$T\bar{T}$-deformed correlators from a 2D gravity description
hep-th 2025-07 conditional novelty 6.0 of 10

Using a massive gravity formulation, the paper derives all-order leading-logarithmic T-bar-T corrections to two- and three-point CFT correlators, reproducing the known two-point result and obtaining a new closed form ...
It from ETH: Multi-interval Entanglement and Replica Wormholes from Large-$c$ BCFT Ensemble
hep-th 2025-05 conditional novelty 6.0 of 10

The RT formula for entanglement in AdS3/CFT2, including phase transitions and multi-interval vacuum entropies, is derived from an assumed large-c ensemble of (B)CFT data, under the 'It from ETH' paradigm.
On the CFT dual of superstring on AdS$_3$
hep-th 2025-04 conditional novelty 6.0 of 10

This paper verifies, at second order in conformal perturbation theory, that the proposed marginal deformation of a symmetric orbifold CFT reproduces the residues of three-point superstring correlators on AdS3 x S3 x T4.
MHV leaf amplitudes from parafermions
hep-th 2025-01 conditional novelty 6.0 of 10

A dual 2D CFT description of MHV leaf amplitudes using non-compact parafermions is shown to reproduce the known dressed Liouville realization.
Anyon Condensation in Virasoro TQFT: Wormhole Factorization
hep-th 2024-12 conditional novelty 6.0 of 10

Condensing a diagonal anyon in Virasoro TQFT factorizes wormhole partition functions and produces Liouville CFT on the two boundary surfaces.
Solving 3d Gravity with Virasoro TQFT
hep-th 2023-04 conditional novelty 6.0 of 10

3d quantum gravity with negative cosmological constant is reformulated as two copies of a Virasoro TQFT, making gravity partition functions on hyperbolic 3-manifolds algorithmically computable.
A proposal for 3d quantum gravity and its bulk factorization
hep-th 2022-10 conditional novelty 6.0 of 10

In an effective 3d gravity model, black hole entropy is reproduced as entanglement entropy of edge modes whose symmetry is the quantum semi-group SL_q^+(2,R), matching the Virasoro modular S-matrix to the quantum grou...
Holographic thermal correlators from supersymmetric instantons
hep-th 2022-06 conditional novelty 6.0 of 10

An exact, instanton-expandable formula for the thermal scalar two-point function in 4D holographic CFTs, checked against direct numerical solution and against known light-cone bootstrap results.
Exact solution of Kerr black hole perturbations via CFT$_2$ and instanton counting. Greybody factor, Quasinormal modes and Love numbers
hep-th 2021-05 conditional novelty 6.0 of 10

Explicit instanton-series connection coefficients for the confluent Heun equation yield analytic greybody factors, quasinormal mode conditions, and Love numbers for Kerr black holes.
Conformal Block Expansion in Celestial CFT
hep-th 2021-04 conditional novelty 6.0 of 10

The four-point celestial amplitude from massive scalar exchange is decomposed into 2D conformal blocks: a tower of integer-dimension scalar exchanges plus a light-ray continuum, with prefactors quadratic in OPE coeffi...
BCFT entanglement entropy at large central charge and the black hole interior
hep-th 2020-04 conditional novelty 6.0 of 10

Direct large-central-charge BCFT calculations of twist-operator correlators reproduce the holographic entanglement entropy phase transitions, including the transition where the entanglement wedge picks up part of a bl...
5D partition functions, q-Virasoro systems and integrable spin-chains
hep-th 2013-12 conditional novelty 6.0 of 10

Five-dimensional gauge theory partition functions factorize into holomorphic blocks, degenerate to three-dimensional partition functions under mass tuning, and share their special-function structure with q-Virasoro re...
Entanglement Entropy at Large Central Charge
hep-th 2013-03 conditional novelty 6.0 of 10

At large central charge, the entanglement entropy of N disjoint intervals in a 2d CFT equals a sum of single-interval entropies selected by the dominant OPE channel, reproducing the Ryu-Takayanagi minimal-surface formula.
SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls
hep-th 2011-03 conditional novelty 6.0 of 10

The partition function of 3d SL(2,R) Chern-Simons theory on a punctured surface times an interval is conjectured to equal that of a 3d N=2 theory on a duality wall, a (3+3) analog of AGT.

See all 80 Pith citations

How later papers used it

179 later papers cite it

Evidence: substantial. 178 of the citing papers share no author with it, and 22 of those carry a confident label from Pith's citation classifier.

Pith read 11 of the 255 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

2 later papers by other authors are listed below: 2 used its method or tool and 1 compared its own results with it. 1 of them appears under more than one label.

Uses its method or tool 2 of 9 papers with this label

Defined its 2-point function as the limit of the 3-point function when one insertion tends to the identity as in this paper, and used the DOZZ formula for the 3-point structure constant to compute the overall normalization.

Boundary timelike Liouville theory: bulk 1-point & boundary 2-point functions (2021) · 0.83 · paraphrased: the licence does not allow quoting
Expressed its F-matrix in terms of the quantum 6j-symbol, with the DOZZ three point function of this paper appearing in the formula.

Solving the Schwarzian via the Conformal Bootstrap (2017) · 0.87 · paraphrased: the licence does not allow quoting

Compares with it 1 of 11 papers with this label

Found that the integral giving the amplitude between its states matches exactly with the b to 0 limit of this paper's DOZZ formula.

Solving the Schwarzian via the Conformal Bootstrap (2017) · 0.93 · paraphrased: the licence does not allow quoting

How the literature uses it

161 arXiv papers cite this in 228 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 · 152 sentences without a confident label

Uses its method or tool 9 papers

In order to do this we are going to use the DOZZ formulation of Liouville theory [ref] .

Bootstrapping closed string field theory · Classical bootstrap for string vertices · 0.99
Nevertheless, we can leverage the exact solution of the worldsheet CFT ([ref]) [ref] to deduce the analytic structure of the string amplitudes viewed as analytic functions of the external momenta.

The complex Liouville string: the worldsheet · Introduction · 0.92
At each point in the base, we then perform the integration along the fiber: [equation] where in going from the second to the third line we used the Verlinde inner product [ref] between Liouville blocks [equation] and used the compact notation [ref] [equation] $𝒞$ being a choice of channel decomposition, $𝐏$ being the internal momenta, $ρ_0$ being the Cardy density, and $C_0$ being the DOZZ formula [ref] up to some normalization [ref] .

On random matrix statistics of 3d gravity · Other amplitudes · 0.93

Uses its data or values 1 paper

The structure constant $C(α,α_1,α_2)$ of the Liouville field theory was computed [ref] [ref] [ref] to be [equation] where $Υ(x)$ is defined by [equation] for $0<Re(x) <Q$ and analytically continued to the whole complex plane.

Dangerous Liouville Wave -- exactly marginal but non-conformal deformation · Quantum Liouville wave · 0.84

Compares with it 10 papers

The final result reads [equation] where $Δ_i = η_i (1-η_i)/b^2$ and $C_123$ is given by [equation] with [equation] This matches with the classical limit of the DOZZ formula [ref] .

Liouville Theory, AdS$_2$ String, and Three-Point Functions · Solving the functional equation · 0.99
By comparing to the exact correlation functions, known from [ref] , the authors of [ref] identified the saddles that contribute to the corresponding functional integrals.

Semi-Classical Analysis of the String Theory Cigar · Introduction and Overview · 0.88
Incorporating these leg factors, we finally get f(_1)f(_2)/f(^∗_3) F__1,_2,_3 F__1^∗,_2^∗,_3^∗ ∼'_0 (2_1)(2_2)(2_3)/( _1+_2+_3 -Q)(_2+_3-_1)(_3+_1-_2)(_1+_2-_3) , where [equation] This is precisely the DOZZ formula [ref] apart from the extra normalization factor [equation] Notice that this normalization factor depends on the coefficient $$ of the exponential term $∫d^2z e^2bφ$ in the Liouville action which can be shifted away by shifting $→+$ const, and is therefore clearly scheme-depedent.

Non-Perturbative Topological Strings And Conformal Blocks · Three-Point Functions · 0.91

Cites it as support 1 paper

However, the OPE in the unphysical range of the Liouville momentum formally makes sense, and indeed it was even used in the original derivation of the three-point function [ref] [ref] [ref] .

Refined Cigar and Omega-deformed Conifold · Refined cigar and $Ω$-deformation · 0.96

Cites it as background 48 papers

These partition functions were until recently [ref] unknown, with the sole exception of the W$_2$ case, i.e. the Liouville case, whose three point structure constants are given by the famous DOZZ formula [ref] .

Toda 3-Point Functions From Topological Strings · Introduction · 0.81
The spacelike structure constants $C(a_1 , a_2, a_3)$ are given by the Dorn–Otto–Zamolodchikov–Zamolodchikov (DOZZ) formula [ref] (see [ref] for more discussions).

Quantum Gravity from Timelike Liouville theory · Conformal bootstrap for Liouville · 0.83
The Virasoro primaries of the $c=25$ Liouville CFT are scalar operators $V_P$ labeled by their Liouville momenta $P≥0$, of scaling dimensions $Δ_P=2+2P^2$, subject to the normalization ⟨V_P_1(z,z̅)V_P_2(0)⟩=πδ(P_1-P_2)/|z|^2Δ_P_1 Their structure constants are given by the DOZZ formula [ref] .[Explicitly, in our normalization convention, the $c=25$ Liouville structure constants are given in (2.9) of [ref] .] The closed string asymptotic states, as insertions on the worldsheet, are given by the BRST cohomology representatives of the form V^±_ω= g_s cc̃ e^±i ωX^0 V_P=ω/2, where the superscript $+$ and $-$ corresponds to in- and out-states, respectively.

Multi-Instanton Calculus in c = 1 String Theory · Multiple ZZ-instantons from the worldsheet perspective · 0.81

Discussion

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