Dualities in quantum integrable many-body systems and integrable probabilities -- I

Abstract

In this study we map the dualities observed in the framework of integrable probabilities into the dualities familiar in a realm of integrable many-body systems. The dualities between the pairs of stochastic processes involve one representative from Macdonald-Schur family, while the second representative is from stochastic higher spin six-vertex model of TASEP family. We argue that these dualities are counterparts and generalizations of the familiar quantum-quantum (QQ) dualities between pairs of integrable systems. One integrable system from QQ dual pair belongs to the family of inhomogeneous XXZ spin chains, while the second to the Calogero-Moser-Ruijsenaars-Schneider (CM-RS) family. The wave functions of the Hamiltonian system from CM-RS family are known to be related to solutions to (q)KZ equations at the inhomogeneous spin chain side. When the wave function gets substituted by the measure, bilinear in wave functions, a similar correspondence holds true. As an example, we have elaborated in some details a new duality between the discrete-time inhomogeneous multispecies TASEP model on the circle and the quantum Goldfish model from the RS family. We present the precise map of the inhomogeneous multispecies TASEP and 5-vertex model to the trigonometric and rational Goldfish models respectively, where the TASEP local jump rates get identified as the coordinates in the Goldfish model. Some comments concerning the relation of dualities in the stochastic processes with the dualities in SUSY gauge models with surface operators included are made.

Bispectrality of $AG_2$ Calogero-Moser-Sutherland system

“…The relation (1.1) for the root system A n , another function ψ, and an arbitrary coupling parameter was established recently by Kharchev and Khoroshkin in [17]. We refer to [16] and references therein for aspects of this and various other dualities in the broad realm of integrable systems.…”

Quantum nonequilibrium dynamics from Knizhnik-Zamolodchikov equations

“…A comprehensive study of dualities between various quantum and classical many-body systems, including dualities of KZ equations, with application to the calculation of integrable probabilities in the stochastic process are presented in ref. [79]. The probabilities in the stochastic process are treated within the conventional machinery of integrable models, including transfer matrices and Bethe ansatz equations.…”