Structure Constants and Conformal Bootstrap in Liouville Field Theory

Abstract

An analytic expression is proposed for the three-point function of the exponential fields in the Liouville field theory on a sphere. In the classical limit it coincides with what the classical Liouville theory predicts. Using this function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and verify numerically that it satisfies the conformal bootstrap equations, i.e., that the operator algebra thus defined is associative. We consider also the Liouville reflection amplitude which follows explicitly from the structure constants.

31 pages, 2 Postscript figures. Important note about existing (but unfortunately previously unknown to us) paper which has significant overlap with this work is added

References in corpus (1)

• Two and three-point functions in Liouville theory

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• Vacuum energy of the Bukhvostov-Lipatov model
• Reflection Amplitudes of Boundary Toda Theories and Thermodynamic Bethe Ansatz
• On the time-dependent description for the decay of unstable D-branes
• Minisuperspace computation of the Mabuchi spectrum
• Boundary One-Point Functions, Scattering Theory and Vacuum Solutions in Integrable Systems
• Virasoro blocks at large exchange dimension
• Critical Non-Abelian Vortex and Holography for Little String Theory
• On two-dimensional quantum gravity and quasiclassical integrable hierarchies
• Notes on S-Matrix of Non-critical N=2 String
• Finite cut-off JT and Liouville quantum gravities on the disk at one loop
• Sine-Gordon Expectation Values of Exponential Fields With Variational Perturbation Theory
• Open minimal strings and open Gelfand-Dickey hierarchies
• Ryu-Takayanagi Formula for Multi-Boundary Black Holes from 2D Large-$c$ CFT Ensemble
• c-Recursion for multi-point superconformal blocks. NS sector
• The two-sphere partition function in two-dimensional quantum gravity at fixed area
• Scattering of low lying states in the black hole atmosphere
• Stringy horizons and generalized FZZ duality in perturbation theory
• Duality in Liouville Theory as a Reduced Symmetry
• Minimal gravity and Frobenius manifolds: bulk correlation on sphere and disk
• A Note on Z_2 Symmetries of the KZ Equation
• On the S-matrix of Liouville theory
• On generalization of 2d gravity with the simplest non-invariant states
• Singular Liouville fields and spiky strings in $\rr^{1,2}$ and $SL(2,\rr)$
• Exact Correlation Functions in the Brownian Loop Soup
• Liouville Conformal Field Theory on even-dimensional spheres
• Non-renormalization for the Liouville wave function
• Form factors of descendant operators: Resonance identities in the sinh-Gordon model
• Log-correlated Random Energy Models with extensive free energy fluctuations: pathologies caused by rare events as signatures of phase transitions
• The two-sphere partition function in two-dimensional quantum gravity
• A Causal Algebra for Liouville Exponentials
• Anyon Condensation in Virasoro TQFT: Wormhole Factorization
• Perturbed Logarithmic CFT and Integrable Models
• On the Liouville coupling constants
• Geometry of bounded critical phenomena
• The $Ad{S}_3\times S^1$ Chiral Ring
• On Krichever tau-function and Verlinde formula
• Modular bootstrap agrees with path integral in the large moduli limit
• Scaling limit of the staggered six-vertex model with $U_q(\mathfrak{sl}(2))$ invariant boundary conditions
• Gravitational action for a massive Majorana fermion in 2d quantum gravity
• The DeWitt Equation in Quantum Field Theory
• Joining-splitting interaction of non-critical string
• Semiclassical and quantum Liouville theory
• Perturbative test of exact vacuum expectation values of local fields in affine Toda theories
• Holography for bulk states in 3D quantum gravity
• High-energy asymptotics of D-brane decay amplitudes from Coulomb gas electrostatics
• On the semiclassical 3-point function in AdS_3
• Characters, Coadjoint Orbits and Duistermaat-Heckman Integrals
• Modular Integrals in Minimal Super Liouville Gravity
• Riemann-Hilbert treatment of Liouville theory on the torus: The general case
• Conformally invariant elliptic Liouville equation and its symmetry preserving discretization
• Perturbation theory in radial quantization approach and the expectation values of exponential fields in sine-Gordon model
• Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere
• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models
• Semi-Classical Analysis of the String Theory Cigar
• The imaginary Toda field theory
• Crossing invariant correlation functions at $c=1$ from isomonodromic $\tau$ functions
• The Euler anomaly and scale factors in Liouville/Toda CFTs
• Entanglement Negativity in Flat Holography
• $x-y$ swap for $(2,2p+1)$ minimal string
• New Discrete States in Two-Dimensional Supergravity
• Large and small Density Approximations to the thermodynamic Bethe Ansatz
• SLE($\kappa ,\rho$)and Boundary Coulomb Gas
• The Large-$c$ Virasoro Identity Block is a Semi-Classical Liouville Correlator
• Quantum dilaton gravity as a linear dilaton conformal field theory
• The continuation method and the real analyticity of the accessory parameters: the general elliptic case
• The chiral SYK model in three-dimensional holography
• Dual Frobenius manifolds of minimal gravity on disk
• Open-closed 3d gravity as a random ensemble
• Liouville Theory, AdS${}_2$ String, and Three-Point Functions
• Resonances in sinh- and sine-Gordon models and higher equations of motion in Liouville theory
• On non-homogeneous tachyon condensation in closed string theory
• Quantum Analytic Langlands Correspondence
• Partition Functions, Duality, and the Tube Metric
• On the tau function of the hypergeometric equation
• Semiclassical approach to form factors in the sinh-Gordon model
• On the analytical continuation of lattice Liouville theory
• Algebraic approach to form factors in the complex sinh-Gordon theory
• Scaling functions from q-deformed Virasoro characters
• On construction of correlation numbers in super Minimal Liouville Gravity in the Ramond sector
• Logarithmic operators in $c=0$ bulk CFTs
• Liouville perturbation theory for Laughlin state and Coulomb gas
• On mini-superspace limit of boundary three-point function in Liouville field theory
• Solvable spectral problems from 2d CFT and N=2 gauge theories
• ODE/IQFT correspondence for the generalized affine $\mathfrak{sl}(2)$ Gaudin model
• The two-sphere partition function from timelike Liouville theory at three-loop order
• Quantisation of Gauged SL(2,R) WZNW Theories
• Non invariant zeta-function regularization in quantum Liouville theory
• Topological defects in the Liouville field theories with different cosmological constants
• Higher order BPZ equations for Liouville conformal field theory
• Lecture notes on strings in AdS${}_3$ from the worldsheet and the AdS${}_3$/CFT${}_2$ duality
• A Family of Probability Distributions Consistent with the DOZZ Formula: Towards a Conjecture for the Law of 2D GMC
• Critical Three-Dimensional Ising Model on Spheriods from the Conformal Bootstrap
• Backbone three-point correlation function in the two-dimensional Potts model
• Remarks on A_2 Toda Field Theory
• Applications of Reflection Amplitudes in Toda-type Theories
• Liouville theory on a horizon: point particle/scalar field duality and Page-like curve
• Non-critical string, Liouville theory and geometric bootstrap hypothesis
• Proof of $A_n$ AGT conjecture at $\beta =1$
• A Bäcklund transformation for elliptic four-point conformal blocks
• Torus one-point correlation numbers in minimal Liouville gravity
• Strings and D-branes in holographic backgrounds
• Irreducible Virasoro representations in Liouville conformal field theory
• Towards $W_3$ classical blocks with semi-degenerate operators
• Wronskians, dualities and FZZT-Cardy branes
• Holographic thermal propagator from modularity
• Possible large-N fixed-points and naturalness for O(N) scalar fields
• Coarse-Grained Fixed-Point Tensor Networks and Holographic Reflected Entropy in 3D Gravity
• Highest-weight vectors and three-point functions in GKO coset decomposition
• Ring of physical states in the M(2,3) Minimal Liouville gravity