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

摘要

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