Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models

Abstract

Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs' classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, N=2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the "Dotsenko-Fateev matrix model action". The latter is the "DF action" evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. ... .

arXiv comment49 pages, several diagrams and tables; conformal to version accepted for publication in JHEP

References in corpus (67)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Two and three-point functions in Liouville theory
• Two-dimensional conformal field theory and the butterfly effect
• Exactly solvable Richardson-Gaudin models for many-body quantum systems
• Universal Spectrum of 2d Conformal Field Theory in the Large c Limit
• Gate-Voltage Studies of Discrete Electronic States in Al Nanoparticles
• Virasoro Conformal Blocks and Thermality from Classical Background Fields
• Entanglement Entropy at Large Central Charge
• Quantum Hall Physics - hierarchies and CFT techniques
• Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches
• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems
• Integrability of the pairing hamiltonian
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• Black Hole Collapse in the 1/c Expansion
• Classical Conformal Blocks and Painleve VI
• Conformal Bootstrap, Universality and Gravitational Scattering
• Off-Shell Bethe Ansatz Equation and N-point Correlators in the SU(2) WZNW Theory
• Off-Shell Bethe Ansatz Equation for Gaudin Magnets and Solutions of Knizhnik-Zamolodchikov Equations
• Large N limit of the exactly solvable BCS model: analytics versus numerics
• A Numerical Approach to Virasoro Blocks and the Information Paradox
• Monodromic vs geodesic computation of Virasoro classical conformal blocks
• Higher-point conformal blocks and entanglement entropy in heavy states
• Classical conformal blocks via AdS/CFT correspondence
• From global to heavy-light: 5-point conformal blocks
• The matrix model version of AGT conjecture and CIV-DV prepotential
• Conformal Field Theory and the Exact Solution of the BCS Hamiltonian
• On the Late-Time Behavior of Virasoro Blocks and a Classification of Semiclassical Saddles
• Classical geometry from the quantum Liouville theory
• Isomonodromy, Painlevé Transcendents and Scattering off of Black Holes
• Kerr-de Sitter Greybody Factors via Isomonodromy
• Holographic interpretation of 1-point toroidal block in the semiclassical limit
• Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry
• Semi-classical Virasoro blocks: proof of exponentiation
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• Renyi Entropy for Local Quenches in 2D CFTs from Numerical Conformal Blocks
• The toroidal block and the genus expansion
• Integrable time-dependent Hamiltonians, solvable Landau-Zener models and Gaudin magnets
• Transformations of Spherical Blocks
• Many-point classical conformal blocks and geodesic networks on the hyperbolic plane
• Probing beyond ETH at large $c$
• Liouville theory and uniformization of four-punctured sphere
• Bootstrapping 2D CFTs in the Semiclassical Limit
• Classical irregular block, N=2 pure gauge theory and Mathieu equation
• Perturbative classical conformal blocks as Steiner trees on the hyperbolic disk
• Quantum Hitchin Systems via beta-deformed Matrix Models
• Semiclassical and quantum Liouville theory on the sphere
• Classical Liouville action on the sphere with three hyperbolic singularities
• Large $c$ Virasoro Blocks from Monodromy Method beyond Known Limits
• Equivalence of Geometric h<1/2 and Standard c>25 Approaches to Two-Dimensional Quantum Gravity
• Accessory parameters in confluent Heun equations and classical irregular conformal blocks
• Standard and geometric approaches to quantum Liouville theory on the pseudosphere
• New Properties of Large-$c$ Conformal Blocks from Recursion Relation
• Classical Virasoro irregular conformal block
• On the monodromy problem for the four-punctured sphere
• Classical conformal blocks
• Classical Virasoro irregular conformal block II
• Classical limit of irregular blocks and Mathieu functions
• Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations
• Matching branches of non-perturbative conformal block at its singularity divisor
• Richardson-Gaudin models and broken integrability
• Classical irregular blocks, Hill's equation and PT-symmetric periodic complex potentials
• Torus classical conformal blocks
• Integrability and Conformal Symmetry in the BCS model
• Semiclassical and quantum Liouville theory

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