Non-Stationary Ruijsenaars Functions for $\kappa =t^{-1/N}$ and Intertwining Operators of Ding-Iohara-Miki Algebra

Abstract

We construct the non-stationary Ruijsenaars functions (affine analogue of the Macdonald functions) in the special case $\kappa =t^{-1/N}$, using the intertwining operators of the Ding-Iohara-Miki algebra (DIM algebra) associated with $N$-fold Fock tensor spaces. By the $S$-duality of the intertwiners, another expression is obtained for the non-stationary Ruijsenaars functions with $\kappa =t^{-1/N}$, which can be regarded as a natural elliptic lift of the asymptotic Macdonald functions to the multivariate elliptic hypergeometric series. We also investigate some properties of the vertex operator of the DIM algebra appearing in the present algebraic framework; an integral operator which commutes with the elliptic Ruijsenaars operator, and the degeneration of the vertex operators to the Virasoro primary fields in the conformal limit $q\to 1$.

References in corpus (28)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• The Refined Topological Vertex
• On combinatorial expansion of the conformal blocks arising from AGT conjecture
• A Quantum Deformation of the Virasoro Algebra and the Macdonald Symmetric Functions
• Refined BPS state counting from Nekrasov's formula and Macdonald functions
• Collective fields, Calogero-Sutherland model and generalized matrix models
• Quantum Algebraic Approach to Refined Topological Vertex
• Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra
• Finalizing the proof of AGT relations with the help of the generalized Jack polynomials
• Notes on Ding-Iohara algebra and AGT conjecture
• Changing the preferred direction of the refined topological vertex
• Kernel function and quantum algebras
• A Commutative Family of Integral Transformations and Basic Hypergeometric Series. I. Eigenfunctions
• Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence
• Higgsed network calculus
• A direct approach to the bispectral problem for the Ruijsenaars-Macdonald q-difference operators
• The Integrals of Motion for the Deformed Virasoro Algebra
• Hypergeometric theta functions and elliptic Macdonald polynomials
• A Conjecture about Raising Operators for Macdonald Polynomials
• T[U(N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences
• A Nekrasov-Okounkov formula for Macdonald polynomials
• Existence and Orthogonality of Generalized Jack Polynomials and Its $q$-Deformation
• Basic Properties of Non-Stationary Ruijsenaars Functions
• The Integrals of Motion for the Deformed W-Algebra $W_{qt}(s{l}_N^{)}$ II: Proof of the commutation relations
• Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case
• Kac determinant and singular vector of the level N representation of Ding-Iohara-Miki algebra
• Series Solutions of the Non-Stationary Heun Equation
• A Commutative Family of Integral Transformations and Basic Hypergeometric Series. II. Eigenfunctions and Quasi-Eigenfunctions

Cited by in corpus (4)

• Basic Properties of Non-Stationary Ruijsenaars Functions
• MacMahon KZ equation for Ding-Iohara-Miki algebra
• Quasi-Hopf twist and elliptic Nekrasov factor
• Quantum Toroidal Comodule Algebra of Type $A_{n-1}$ and Integrals of Motion