Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
A. G. Bytsko, J. Teschner
TL;DR
The authors construct integrable lattice models with non-compact modular double symmetry (the modular XXZ magnet and lattice sinh-Gordon), employing representations of U_q(sl2, R) and the non-compact quantum dilogarithm to ensure self-adjoint Hamiltonians. They develop a QISM framework augmented by explicitly built Q-operators and a separation-of-variables approach, obtaining Baxter-type equations and a quantum Wronskian that characterize the spectrum beyond the conventional Bethe ansatz. The work clarifies why the algebraic Bethe ansatz fails in non-compact settings, provides a complete SOV-based spectral program, and connects the lattice theories to their continuum sinh–Gordon and Liouville counterparts via careful scaling limits and dualities (b ↔ b^{-1}). These results illuminate how massive and massless integrable structures relate in non-compact contexts and offer a concrete lattice realization of continuum sinh–Gordon with rigorous spectral data. The methods and dualities presented have potential implications for understanding non-compact quantum field theories and their discretizations, as well as for exploring lattice Liouville correspondences.
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.