Yangian Gelfand-Zetlin Bases, gl(N)-Jack Polynomials and computation of Dynamical Correlation Functions in the Spin Calogero-Sutherland Model

Abstract

We consider the gl(N)-invariant Calogero-Sutherland Models with N=1,2,3,... in a unified framework, which is the framework of Symmetric Polynomials. By the framework we mean an isomorphism between the space of states of the gl(N)-invariant Calogero-Sutherland Model and the space of Symmetric Laurent Polynomials. In this framework it becomes apparent that all gl(N)-invariant Calogero-Sutherland Models are manifestations of the same entity, which is the commuting family of Macdonald Operators. Macdonald Operators depend on two parameters $q$ and $t$. The Hamiltonian of gl(N)-invariant Calogero-Sutherland Model belongs to a degeneration of this family in the limit when both $q$ and $t$ approach the N-th elementary root of unity. This is a generalization of the well-known situation in the case of Scalar Calogero-Sutherland Model (N=1). In the limit the commuting family of Macdonald Operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the gl(N)-invariant Calogero-Sutherland Model. The limits of Macdonald Polynomials which we call gl(N)-Jack Polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The gl(N)-Jack Polynomials describe the orthogonal eigenbasis of gl(N)-invariant Calogero-Sutherland Model in exactly the same way as Jack Polynomials describe the orthogonal eigenbasis of the Scalar Model (N=1). For each known property of Macdonald Polynomials there is a corresponding property of gl(N)-Jack Polynomials. As a simplest application of these properties we compute two-point Dynamical Spin-Density and Density Correlation Functions in the gl(2)-invariant Calogero-Sutherland Model at integer values of the coupling constant.

26 pages, AMSLatex

Cited by in corpus (38)

• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity
• Matrix model from N = 2 orbifold partition function
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence
• Exact dynamical structure factor of the degenerate Haldane-Shastry model
• On spectrum of ILW hierarchy in conformal field theory II: coset CFT's
• Superconformal minimal models and admissible Jack polynomials
• Exact Dynamics of the SU(K) Haldane-Shastry Model
• Quantum Cohomology and Quantum Hydrodynamics from Supersymmetric Quiver Gauge Theories
• Spin-Ruijsenaars, q-deformed Haldane-Shastry and Macdonald polynomials
• Jack polynomials with prescribed symmetry and hole propagator of spin Calogero-Sutherland model
• Quantum Hydrodynamics from Large-n Supersymmetric Gauge Theories
• N=2 gauge theories, instanton moduli spaces and geometric representation theory
• Beta-ensembles for toric orbifold partition function
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• Rodrigues Formula for the Nonsymmetric Multivariable Hermite Polynomial
• Conformal field theory and edge excitations for the principal series of quantum Hall fluids
• Dynamical correlation functions of the spin Calogero-Sutherland model
• Bosonic and Fermionic Eigenstates for Generalized Sutherland Models
• From fermionic spin-Calogero-Sutherland models to the Haldane-Shastry chain by freezing
• Macdonald polynomials for super-partitions
• Algorithms for representations of quiver Yangian algebras
• Algebraic study on the $A_{N-1}$- and $B_N$-Calogero models with bosonic, fermionic and distinguishable particles
• Ramond singular vectors and Jack superpolynomials
• Virasoro Constraint for Uglov Matrix Model
• Wreath Macdonald polynomials as eigenstates
• Supersymmetric polynomials and algebro-combinatorial duality
• Quantum algebra in the mixed light pseudoscalar meson states
• Bethe ansatz inside Calogero-Sutherland models
• Finite temperature structure factor in the Haldane-Shastry spin chain
• Vogel's universality and the classification problem for Jacobi identities
• NSR singular vectors from Uglov polynomials
• Derivation of Green's Function of Spin Calogero-Sutherland Model by Uglov's Method
• Particle Propagator of Spin Calogero-Sutherland Model
• `Composite particles' and the eigenstates of Calogero-Sutherland and Ruijsenaars-Schneider
• AGT basis in SCFT for c=3/2 and Uglov Polynomials
• Towards mixed phase correlators in monomial matrix models