Correspondence between conformal field theory and Calogero-Sutherland model

Abstract

We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators $L_n$. We calculate explicitly the matrix elements of $L_n$ with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting case of the formula, we obtain a Pieri-type formula which represents a product of a power sum and a Jack symmetric function as a sum of Jack symmetric functions. Also, a similar expansion was found for the case when we differentiate the Jack symmetric functions with respect to power sums. As an application of our Jack-basis representation, a new diagrammatic interpretation is presented, why the singular vectors of the Virasoro algebra are proportional to the Jack symmetric functions with rectangular diagrams. We also propose a natural normalization of the singular vectors in the Verma module, and determine the coefficients which appear after bosonization in front of the Jack symmetric functions.

arXiv comment23 pages, references added

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Cited by in corpus (26)

• On combinatorial expansion of the conformal blocks arising from AGT conjecture
• Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model
• Conformal blocks in Virasoro and W theories: duality and the Calogero-Sutherland model
• AGT conjecture and Integrable structure of Conformal field theory for c=1
• Superconformal field theory and Jack superpolynomials
• The supersymmetric Ruijsenaars-Schneider model
• Rigged Configurations and Kashiwara Operators
• Applications of Laplace-Beltrami operator for Jack polynomials
• Deformed Calogero-Sutherland model and fractional Quantum Hall effect
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• The super-Virasoro singular vectors and Jack superpolynomials relationship revisited
• AFLT-type Selberg integrals
• Norms of logarithmic primaries of Virasoro algebra
• Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals
• Exchange operator formalism for N-body spin models with near-neighbors interactions
• Explicit formula for singular vectors of the Virasoro algebra with central charge less than 1
• Jack vertex operators and realization of Jack functions
• Ramond singular vectors and Jack superpolynomials
• Baker-Akhiezer functions and generalised Macdonald-Mehta integrals
• A bispectral q-hypergeometric basis for a class of quantum integrable models
• Generalized Gibbs Ensemble of 2D CFTs with U(1) Charge from the AGT Correspondence
• From Jack to Double Jack Polynomials via the Supersymmetric Bridge
• Symmetric functions in superspace: a compendium of results and open problems (including a SageMath worksheet)
• Calogero model with Yukawa like interaction
• Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions
• Super-Whittaker vector at c=3/2