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On classical \(q\)-deformations of integrable \(\sigma\)-models. (English) Zbl 1342.81182

Summary: A procedure is developed for constructing deformations of integrable \(\sigma\)-models which are themselves classically integrable. When applied to the principal chiral model on any compact Lie group \(F\), one recovers the Yang-Baxter \(\sigma\)-model introduced a few years ago by C. Klimčík. In the case of the symmetric space \(\sigma\)-model on \(F/G\) we obtain a new one-parameter family of integrable \(\sigma\)-models. The actions of these models correspond to a deformation of the target space geometry and include a torsion term. An interesting feature of the construction is the \(q\)-deformation of the symmetry corresponding to left multiplication in the original models, which becomes replaced by a classical \(q\)-deformed Poisson-Hopf algebra. Another noteworthy aspect of the deformation in the coset \(\sigma\)-model case is that it interpolates between a compact and a non-compact symmetric space. This is exemplified in the case of the \(\mathrm{SU}(2)/U(1)\) coset \(\sigma\)-model which interpolates all the way to the \(\mathrm{SU}(1, 1)/U(1)\) coset \(\sigma\)-model.

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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