Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model

TL;DR

This paper develops integrable lattice models with non-compact quantum group symmetry, including sinh-Gordon and XXZ models, using quantum dilogarithm-based R-matrices, and analyzes their spectral properties via separation of variables and Q-operators.

Contribution

It introduces a new class of integrable lattice models with non-compact quantum group symmetry and constructs explicit R-matrices and Q-operators for spectral analysis.

Findings

Constructed integrable lattice models with non-compact quantum group symmetry.

Reformulated spectral problems as finite difference equations with analytic properties.

Established connections to continuous sinh-Gordon theory and relations between massive and massless models.

Abstract

We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of U_q(sl_2)) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R-matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called Q-operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous space-time. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.