Liouville Field Theory -- A decade after the revolution

Abstract

We review recent developments (up to January 2004) of the Liouville field theory and its matrix model dual. This review consists of three parts. In part I, we review the bosonic Liouville theory. After briefly reviewing the necessary background, we discuss the bulk structure constants (the DOZZ formula) and the boundary states (the FZZT brane and the ZZ brane). Various applications are also presented. In part II, we review the supersymmetric extension of the Liouville theory. We first discuss the bulk structure constants and the branes as in the bosonic Liouville theory, and then we present the matrix dual descriptions with some applications. In part III, the Liouville theory on unoriented surfaces is reviewed. After introducing the crosscap state, we discuss the matrix model dual description and the tadpole cancellation condition. This review also includes some original material such as the derivation of the conjectured dual action for the N = 2 Liouville theory from other known dualities and the comparison of the Liouville crosscap state with the c = 0 unoriented matrix model. This is based on my master's thesis submitted to Department of Physics, Faculty of Science, University of Tokyo on January 2004.

264 pages, 31 figures, based on master's thesis defended 31/01/2004, v2,3,4,5,6,7: references added, typos corrected, abridged version published in IJMPA

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• Branes, Rings and Matrix Models in Minimal (Super)string Theory
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• A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model
• Modular Bootstrap for Boundary N=2 Liouville Theory
• Matrix Cosmology
• Higher Equations of Motion in Liouville Field Theory
• Flux Backgrounds in 2D String Theory
• Integrable Model of Boundary Interaction: The Paperclip
• New Results on the "$a$-theorem" in Four Dimensional Supersymmetric Field Theory
• One-Point Functions of N=2 Super-Liouville Theory with Boundary
• A Matrix Model for AdS2
• On the S-matrix of Type 0 String Theory
• Little String Theory and Heterotic/Type II Duality
• Type 0 Strings in a 2-d Black Hole
• Matrix Quantum Mechanics and Two-dimensional String Theory in Non-trivial Backgrounds
• D-brane Dynamics in the c=1 Matrix Model
• Topological Matrix Models, Liouville Matrix Model and c=1 String Theory
• On the Worldsheet Derivation of Large N Dualities for the Superstring
• Boundary Ground Ring in 2D String Theory
• Matrix Models, Integrable Structures, and T-duality of Type 0 String Theory
• (m,n) ZZ branes and the c=1 matrix model
• Time Dependent Solution in Open Bosonic String Field Theory
• Classical Liouville action on the sphere with three hyperbolic singularities
• Non-Perturbative String Equations for Type 0A
• The Cap in the Hat: Unoriented 2D Strings and Matrix(-Vector) Models
• Correlation Functions and Vertex Operators of Liouville Theory
• The decay of quantum D-branes
• The tetrahedron graph in Liouville theory on the pseudosphere
• AdS Solutions of 2D Type 0A
• Renormalization Group Approach to $c=1$ Matrix Model on a circle and D-brane Decay
• Boundary Action of N=2 Super-Liouville Theory
• Note on D-Brane Effective Action in the Linear Dilaton Background
• Rolling Tachyon with Electromagnetic Field in Linear Dilaton Background
• Nonsinglet Sector of $c=1$ Matrix Model and 2D Black Hole
• Classical and Quantum Branes in c=1 String Theory and Quantum Hall Effect
• Recent progress on Liouville Field Theory
• Zero Mode Problem of Liouville Field Theory

Cited by in corpus (224)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• The Pomeron and Gauge/String Duality
• Exact half-BPS Type IIB interface solutions I: Local solution and supersymmetric Janus
• Analytic Continuation of Liouville Theory
• Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
• TFT construction of RCFT correlators III: Simple currents
• Solvable Models of Quantum Black Holes: A Review on Jackiw-Teitelboim Gravity
• Phases of Quantum Gravity in AdS3 and Linear Dilaton Backgrounds
• The Annular Report on Non-Critical String Theory
• Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory
• Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
• H(3)+ correlators from Liouville theory
• TFT construction of RCFT correlators IV: Structure constants and correlation functions
• Static Patch Solipsism: Conformal Symmetry of the de Sitter Worldline
• Thermodynamics of Black Holes in Two (and Higher) Dimensions
• M5-branes, toric diagrams and gauge theory duality
• SL(2,R)/U(1) Supercoset and Elliptic Genera of Non-compact Calabi-Yau Manifolds
• 3d & 5d gauge theory partition functions as q-deformed CFT correlators
• Biorthogonal Quantum Systems
• New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory
• Closure of the Operator Product Expansion in the Non-Unitary Bootstrap
• Boundary States for the Rolling D-branes in NS5 Background
• D-branes in N=2 Liouville and its mirror
• The Virasoro Minimal String
• The Black Hole S-Matrix from Quantum Mechanics
• Liouville Quantum Gravity on the complex tori
• Duality and KPZ in Liouville Quantum Gravity
• The uses of the refined matrix model recursion
• Exact Bremsstrahlung and Effective Couplings
• Non-compact String Backgrounds and Non-rational CFT
• Conformal field theory on the plane
• Liouville Brownian motion
• Two perspectives of the 2D unit area quantum sphere and their equivalence
• Chiral Liouville Gravity
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• Notes on Liouville Theory at c<=1
• Rolling Down the Throat in NS5-brane Background: The Case of Electrified D-Brane
• Towards physically motivated proofs of the Poincare' and geometrization conjectures
• KPZ formula derived from Liouville heat kernel
• Conifold Type Singularities, N=2 Liouville and SL(2;R)/U(1) Theories
• The unitary conformal field theory behind 2D Asymptotic Safety
• Continuum limit of fishnet graphs and AdS sigma model
• ZZ-Branes of N=2 Super-Liouville Theory
• Notes on Matrix Models
• Non-Perturbative Topological Strings And Conformal Blocks
• Liouville heat kernel: regularity and bounds
• Bootstrapping 2D CFTs in the Semiclassical Limit
• Non-BPS D-brane Near NS5-branes
• Conformal bootstrap to Rényi entropy in 2D Liouville and super-Liouville CFTs
• SYK model with quadratic perturbations: the route to a non-Fermi-liquid
• On boundary correlators in Liouville theory on AdS${}_2$
• Liouville Conformal Field Theories in Higher Dimensions
• Quantum quench in matrix models: Dynamical phase transitions, Selective equilibration and the Generalized Gibbs Ensemble
• 3d gravity from Virasoro TQFT: Holography, wormholes and knots
• Noncritical M-Theory in 2+1 Dimensions as a Nonrelativistic Fermi Liquid
• Schramm Loewner Evolution and Liouville Quantum Gravity
• On AGT description of N=2 SCFT with N_f=4
• Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus
• Superstrings on AdS_2 and Superconformal Matrix Quantum Mechanics
• Quantum Gravity from Timelike Liouville theory
• Supercurrent, Supervirial and Superimprovement
• The Stoyanovsky-Ribault-Teschner Map and String Scattering Amplitudes
• On the fusion matrix of the N=1 Neveu-Schwarz blocks
• Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies
• On Perturbation theory improved by Strong coupling expansion
• Supersymmetric Liouville theory in AdS${}_2$ and AdS/CFT
• Braiding and fusion properties of the Neveu-Schwarz super-conformal blocks
• Renormalization of Critical Gaussian Multiplicative Chaos and KPZ formula
• An action for the exact string black hole
• Bootstrapping closed string field theory
• Integrability of boundary Liouville conformal field theory
• A brief review of the 2d/4d correspondences
• From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality
• On Spectral Flow Symmetry and Knizhnik-Zamolodchikov Equation
• Equivalence of Liouville measure and Gaussian free field
• The String Theory on AdS_3 as a Marginal Deformation of a Linear Dilaton Background
• The Nothing at the Beginning of the Universe Made Precise
• Infinitely many inequivalent field theories from one Lagrangian
• Comments on 2D Type IIA String and Matrix Model
• D-Brane Propagation in Two-Dimensional Black Hole Geometries
• A note on entanglement entropy for topological interfaces in RCFTs
• Extended Eigenstate Thermalization and the role of FZZT branes in the Schwarzian theory
• Extracting classical Lyapunov exponent from one-dimensional quantum mechanics
• On the heat kernel and the Dirichlet form of Liouville Brownian Motion
• 2D Quantum Gravity at One Loop with Liouville and Mabuchi Actions
• Riemann-Hilbert treatment of Liouville theory on the torus
• Liouville theory without an action
• Geodesic distances in Liouville quantum gravity
• The complex Liouville string: the matrix integral
• Quantum Cosmology Near Two Dimensions
• Accessory parameters for Liouville theory on the torus
• Non-abelian Toda theory on AdS${}_2$ and AdS${}_2$/CFT${}_2^{1/2}$ duality
• D-branes and complex curves in c=1 string theory
• Black holes and thermodynamics - The first half century
• String Perturbation Theory on the Schwinger-Keldysh Time Contour
• Microstates of a $2d$ Black Hole in string theory
• Fluctuations in non-conformal holographic plasma at criticality
• c < 1 String from Two Dimensional Black Holes
• Elliptic recursion for 4-point superconformal blocks and bootstrap in N=1 SLFT
• Classical and Quantum Integrability of 2D Dilaton Gravities in Euclidean space
• Hairpin-Branes and Tachyon-Paperclips in Holographic Backgrounds
• Long strings condensation and FZZT branes
• Supersymmetric Biorthogonal Quantum Systems
• ADM reduction of Einstein action and black hole entropy
• Liouville D-branes in Two-Dimensional Strings and Open String Field Theory
• Integrability of Liouville theory: proof of the DOZZ Formula
• The Hitchhiker's Guide to 4d $\mathcal{N}=2$ Superconformal Field Theories
• Spectral dimension of Liouville quantum gravity
• $Ad{S}_2$ holography is (non-)trivial for (non-)constant dilaton
• Liouville Quantum Gravity and KPZ
• Matrix Model and Time-like Linear Dilaton Matter
• Liouville's Imaginary Shadow
• FZZ Algebra
• A Two-Dimensional String Cosmology
• $\mathcal{N}=1$ Liouville SCFT in Four Dimensions
• Thermodynamics of Noncritical M-Theory and the Topological A-Model
• Comment on three-point function in AdS(3)/CFT(2)
• On static solutions in 2D dilaton gravity with scalar matter
• de Sitter Harmonies: Cosmological Spacetimes as Resonances
• Unitarity Meets Channel-Duality for Rolling / Decaying D-Branes
• The Torus One-Point Diagram in Two-Dimensional String Cosmology
• Hawking Effect in 2-D String Theory
• Stochastic Ricci Flow on Compact Surfaces
• Another derivation of the geometrical KPZ relations
• AFLT-type Selberg integrals
• A Worldsheet Description of Instant Folded Strings
• Liouville gravity from Einstein gravity
• FZZ formula of boundary Liouville CFT via conformal welding
• Cationic vacancies as defects in honeycomb lattices with modular symmetries
• Nesting statistics in the O(n) loop model on random planar maps
• Matrix Quantum Mechanics on $S^1/{\mathbb{Z}}_2$
• Zamolodchikov relations and Liouville hierarchy in SL(2,R)_k WZNW model
• Discrete Liouville equation and Teichmüller theory
• Proposal for the Open String Tachyon Effective Action in the Linear Dilaton Background
• Gauge Invariant Computable Quantities In Timelike Liouville Theory
• Mabuchi spectrum from the minisuperspace
• Quantization of 2D Hořava gravity: non-projectable case
• Toda theory in AdS${}_2$ and $\mathcal{W}{A}_n$-algebra structure of boundary correlators
• Duality in 2-dimensional dilaton gravity
• New scenarios for classical and quantum mechanical systems with position dependent mass
• Semiclassical limit of the FZZT Liouville theory
• Discrete States in Light-Like Linear Dilaton Background
• Boundary Liouville Theory: Hamiltonian Description and Quantization
• Strings on AdS_2 and the High-Energy Limit of Noncritical M-Theory
• Renormalizability of Liouville Quantum Gravity at the Seiberg bound
• Conformal field theories on deformed spheres, anomalies, and supersymmetry
• 2D quantum gravity partition function on the fluctuating sphere
• Smoothness of correlation functions in Liouville Conformal Field Theory
• On the solution of Liouville equation
• Langlands duality in Liouville-H_3^+ WZNW correspondence
• c=1 from c<1: Bulk and boundary correlators
• Quantized cosmological constant in 1+1 dimensional quantum gravity with coupled scalar matter
• BRST cohomology of timelike Liouville theory
• Falling D0-Branes in 2D Superstring Theory
• Refined Cigar and Omega-deformed Conifold
• Electrostatics approach to closed string pair production from a decaying D-brane
• 3D $\mathcal{N}=1$ supergravity from Virasoro TQFT: Gravitational partition function and Out-of-time-order correlator
• Crosscap states in N=2 Liouville theory
• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Deriving the non-perturbative gravitational dual of quantum Liouville theory from BCFT operator algebra
• Can One Understand Black Hole Entropy without Knowing Much about Quantum Gravity?
• Half-BPS Geometries and Thermodynamics of Free Fermions
• Black Hole - String Transition and Rolling D-brane
• ZZ boundary states and fragmented AdS2 spaces
• Odd Dimensional Nonlocal Liouville Conformal Field Theories
• Chameleonic equivalence postulate and wave function collapse
• The ultrarelativistic limit of 2D dilaton gravity and its energy momentum tensor
• Exact M-Theory Solutions, Integrable Systems, and Superalgebras
• Notes on S-Matrix of Non-critical N=2 String
• FZZT Brane Relations in the Presence of Boundary Magnetic Fields
• c-Recursion for multi-point superconformal blocks. NS sector
• Critical Non-Abelian Vortex and Holography for Little String Theory
• Baby universes in 2d and 4d theories of quantum gravity
• A Note on Z_2 Symmetries of the KZ Equation
• Chern-Simons theory, exactly solvable models and free fermions at finite temperature
• Is cosmological constant screened in Liouville gravity with matter?
• Disk one-point function for non-rational conformal theories
• On the S-matrix of Liouville theory
• Pohlmeyer reduction for superstrings in AdS space
• Universal Dynamics in Non-Orientable CFT${}_2$
• Orientifolds in N=2 Liouville Theory and its Mirror
• Non-renormalization for the Liouville wave function
• Noncritical M-theory: Three Dimensions
• Large Strebel graphs and $(3,2)$ Liouville CFT
• Duality between 1+1 dimensional Maxwell-Dilaton gravity and Liouville field theory
• Quantized Laplacian growth, III: On conformal field theories of Laplacian growth
• Renormalization of lattice field theories with infinite-range wavelets
• D-dualized D-brane
• Conformally invariant elliptic Liouville equation and its symmetry preserving discretization
• Gravitational action for a massive Majorana fermion in 2d quantum gravity
• Quantum gravity and the cosmological constant: lessons from two-dimensional dilaton gravity
• Three-Point Function in Perturbed Liouville Gravity
• Riemann-Hilbert treatment of Liouville theory on the torus: The general case
• Refined topological amplitudes in N=1 flux compactification
• Liouville Brownian motion at criticality
• Liouville measure as a multiplicative cascade via level sets of the Gaussian free field
• On the Liouville 2D dilaton gravity models with sinh-Gordon matter
• 2d Sigma Models on Non-compact Calabi-Yau and $\mathcal{N}=2$ Liouville Theory
• Non-compact Mirror Bundles and (0,2) Liouville Theories
• Conformal Dimensions On Causal Random Geometry
• An SLE approach to four dimensional black hole microstate entropy
• Much ado about nothing: a treatise on empty and not-so-empty spacetimes
• On a relation between Liouville field theory and a two component scalar field theory passing through the random walk
• Warping effects in strongly perturbed metrics
• Noncritical M-theory and the Gross-Neveu model in 2+1 dimensions
• Eternal Inflation with Liouville Cosmology
• Gauged Linear Sigma Models for Noncompact Calabi-Yau Varieties
• Critical Non-Abelian Vortex String and 2D N=2 Black Hole
• (0,2) Chiral Liouville Field Theory
• Diffusion of Brownian particles and Liouville field theory
• Schrödinger-like Dilaton Gravity
• Anisotropic scale invariant cosmology
• Boundary timelike Liouville theory: bulk 1-point & boundary 2-point functions
• Liouville perturbation theory for Laughlin state and Coulomb gas
• Vacuum States in 2D Tachyon Effective Action
• A Family of Probability Distributions Consistent with the DOZZ Formula: Towards a Conjecture for the Law of 2D GMC
• If you want to cross singularity, wrap it!
• Higher Equations of Motion in N=2 Superconformal Liouville Field Theory
• Ensemble averaging in JT gravity from entanglement in Matrix Quantum Mechanics
• The spatio-temporal statistical structure of the turbulent dissipation field and its stochastic representation as a Gaussian Multiplicative Chaos
• Cylinder quantum field theories at small coupling
• Are Near $Ad{S}_2$ Spacetimes Dual to $Ad{S}_2$ Black Holes?
• Pseudo-hermitian Chebyshev differential matrix and non-Hermitian Liouville quantum mechanics
• The symmetric branes of the H4 WZW model