Conformal blocks in Virasoro and W theories: duality and the Calogero-Sutherland model

Abstract

We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero-Sutherland Hamiltonian with non-trivial braiding properties. A generalized duality property relates the two types of second order degenerate fields. By studying this duality we found that the excited states of the Calogero-Sutherland Hamiltonian are characterized by two partitions, or in the case of WA_{k-1} theories by k partitions. By extending the conformal field theories under consideration by a u(1) field, we find that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero-Sutherland Hamiltonian. When the action of the Calogero-Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonisation, these integrals of motion can be expressed as a sum of two, or in general k, bosonic Calogero-Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states

37 pages, 2 figures; minor corrections, few typos corrected and some references added

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• A slow review of the AGT correspondence
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• Wilson Loops and Chiral Correlators on Squashed Sphere
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• Selberg Integral and SU(N) AGT Conjecture
• Conformal blocks of W_N Minimal Models and AGT correspondence
• Superconformal field theory and Jack superpolynomials
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• AGT, N-Burge partitions and W_N minimal models
• Holomorphic field realization of SH${}^c$ and quantum geometry of quiver gauge theories
• Virasoro constraint for Nekrasov instanton partition function
• Entanglement subspaces, trial wavefunctions, and special Hamiltonians in the fractional quantum Hall effect
• Integrable QFT and Longo-Witten endomorphisms
• Quantum toroidal algebras and solvable structures in gauge/string theory
• Correlation functions with fusion-channel multiplicity in W3 Toda field theory
• Effective Edge State Dynamics in the Fractional Quantum Hall Effect
• Truncated Calogero-Sutherland models
• Rigged Configurations and Kashiwara Operators
• Quantum Cohomology and Quantum Hydrodynamics from Supersymmetric Quiver Gauge Theories
• Yangian associated with 2D $\mathcal{N}=1$ SCFT
• Wilson Loops and Chiral Correlators on Squashed Spheres
• Deformed Calogero-Sutherland model and fractional Quantum Hall effect
• On Elliptic Algebras and Large-n Supersymmetric Gauge Theories
• Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential (TED) K-theory
• q-Vertex Operator from 5D Nekrasov Function
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Second level semi-degenerate fields in W3 Toda theory: matrix element and differential equation
• Rigid Fuchsian systems in 2-dimensional conformal field theories
• The non-chiral intermediate Heisenberg ferromagnet equation
• Correspondence between bulk entanglement and boundary excitation spectra in 2d gapped topological phases
• FQHE and $t{t}^{\ast }$ geometry
• Notes on 3-point functions of A_{N-1} Toda theory and AGT-W relation for SU(N) quiver
• Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence
• $n$-th parafermion ${\mathcal{W}}_N$ characters from $U(N)$ instanton counting on ${\mathbb{C}}^2/{\mathbb{Z}}_n$
• Cubic constraints for the resolvents of the ABJM matrix model and its cousins
• On the particle entanglement spectrum of the Laughlin states
• Generalized Gibbs Ensemble of 2D CFTs with U(1) Charge from the AGT Correspondence
• An SLE approach to four dimensional black hole microstate entropy
• Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions
• Fermionization, Triangularization and Integrability