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

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.

60 pages, LaTeX2e, (v.2: minor changes, presentation of section D in Appendix improved, v3: minor last corrections, close to published version)

References in corpus (34)

• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• Separation of Variables. New Trends.
• How Algebraic Bethe Ansatz works for integrable model
• High energy QCD as a completely integrable model
• Liouville theory revisited
• Discrete Heisenberg-Weyl Group and Modular Group
• Quantum Dilogarithm
• Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))
• On structure constants and fusion rules in the $SL(2,\BC)/SU(2)$ WZNW model
• Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality
• Operator product expansion and factorization in the $H_3^{+}$-WZNW model
• Quantum Inverse Scattering Method. Selected Topics
• On the Operator Content of the Sinh-Gordon Model
• Form Factors for Integrable Lagrangian Field Theories, the Sinh-Gordon Model
• Liouville bootstrap via harmonic analysis on a noncompact quantum group
• Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables
• A lecture on the Liouville vertex operators
• High Energy Asymptotics of Multi--Colour QCD and Exactly Solvable Lattice Models
• Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains
• Noncommutative Hypergeometry
• Baxter's Q-operator for the homogeneous XXX spin chain
• Separation of variables for the quantum SL(2,R) spin chain
• R-operator, co-product and Haar-measure for the modular double of U_q(sl(2,R))
• Crossing Symmetry in the $H_3^{+}$ WZNW model
• On the Thermodynamic Bethe Ansatz Equation in Sinh-Gordon Model
• Finite temperature expectation values of local fields in the sinh-Gordon model
• Angular quantization and form-factors in massive integrable models
• CONNECTIONS of the LIOUVILLE MODEL and XXZ SPIN CHAIN
• Quasi-classical Study of Form Factors in Finite Volume
• Noncompact SL(2,R) spin chain
• Integral representation for the eigenfunctions of quantum periodic Toda chain
• Baxter Q-operators of the XXZ chain and R-matrix factorization
• Thermodynamics and conformal properties of XXZ chains with alternating spins
• Baxter equations and Deformation of Abelian Differentials

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