Classical and quantum algebras of non-local charges in \(\sigma\) models. (English) Zbl 0536.58013

In field theories the notion of complete integrability involves the existence of an infinite number of commuting local conserved charges. Some models possess also nonlocal conserved charges which do not commute among themselves. Thus nonabelian dyamical symmetry algebras arise. The paper is devoted to the investigation of such algebras both in classical and quantum cases in the framework of nonlinear \(\sigma\)-models. For a large class of classical integrable models the Poisson bracket \(\{\) \(T(\lambda)\otimes T(\mu)\}\) of monodromy matrices is ultralocal and can be expressed using r-matrix. The classical Yang-Baxter equation on the r- matrix guarantees usual properties of the Poisson bracket. In the case of classical chiral \(\sigma\)-models it turns out that there exists no definition of the Poisson bracket satisfying the antisymmetry property and the Jacobi identity. The problems found in classical theory are absent in the quantum case. In the case of the quantum 0(N) nonlinear \(\sigma\)-model the conserved quantum monodromy operator is determined which generates the quantum nonlocal charges. The expression for [\(T(\lambda)\otimes T(\mu)]\) is derived consistent with all the properties of a Lie algebra. The quantum R-matrix is found. It is shown that the quantum nonlocal charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.
Reviewer: I.Ja.Dorfman


37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81T08 Constructive quantum field theory
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