Supersymmetric extension of qKZ-Ruijsenaars correspondence

Abstract

We describe the correspondence of the Matsuo-Cherednik type between the quantum n-body Ruijsenaars-Schneider model and the quantum Knizhnik-Zamolodchikov equations related to supergroup GL(N |M ). The spectrum of the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the Z 2 -grading for a fixed value of N + M , so that N + M + 1 different qKZ systems of equations lead to the same n-body quantum problem. The obtained results can be viewed as a quantization of the previously described quantumclassical correspondence between the classical n-body Ruijsenaars-Schneider model and the supersymmetric GL(N |M ) quantum spin chains on n sites.where the number of indices j k such that j k = a is equal to M a for all a = 1, . . . , K. The dual vectors J are defined in so that J J ′ = δ J,J ′ .Then the statement of the qKZ-Ruijsenaars correspondence is as follows [19]. For any solution of the qKZ equations (1.1) Φ = J Φ J J from the weight subspace V({M a }) the function Ψ = J Φ J , Φ J = Φ J (x 1 , ..., x n ) (1.7) 4 The quantum R-matrices entering (1.2) are assumed to be unitary: R ij (x)R ji (−x) = id. 5 The set {e ab | a, b = 1...K} is the standard basis in Mat(K, C): (e ab ) ij = δ ia δ jb .

The Dynamical equations for $\mathfrak{gl}(n\vert m)$

“…Proof. The (18a) was proven in [11], so we are to prove the second. Let D a be the a th dynamical operator.…”

“…Theorem 3.1. The following covectors, constructed in [11] provide the Matsuo-Cherednik map for Dynamical equations ( 2)…”

“…There is a relation between the solutions of the (quantum) Knizhnik-Zamolodchikov equations ((q)KZ for short) and the spectral problem of quantum systems Calogero-Moser type, proposed in [3] and later extended in [4], [5]. The recent discussion [15], [11] originated in the quantum-classical correspondence [13], [12].…”

Fifth-order generalized Heisenberg supermagnetic models

“…Supersymmetry has its origins in quantum field theory and it receives a lot of attention [15]- [19]. A variety of integrable systems have been generalized into their supersymmetric equations, for instance, the Korteweg-de Vries (KdV) equation [20], the Heisenberg supermagnet (HS) model [21,22] and the NLSE [23].…”

Quantum trace formulae for the integrals of the hyperbolic Ruijsenaars-Schneider model

“…The Ruijsenaars-Schneider (RS) models [1,2] continue to provide an outstanding theoretical laboratory for the study of various aspects of Liouville integrability, both at the classical and quantum level, see, for instance, [5][6][7][8][9][10]. Also, new interesting applications of these type of models were recently found in conformal field theories [11].…”