Abstract
Integrable σ-models, such as the principal chiral model, {mathbb{Z}}_T -coset models for Tin {mathbb{Z}}_{ge 2} and their various integrable deformations, are examples of non-ultralocal integrable field theories described by r/s-systems with twist function. In this general setting, and when the Lie algebra mathfrak{g} underlying the r/s-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space σ-model. In the present context, the local charges are attached to certain ‘regular’ zeros of the twist function and have increasing degrees related to the exponents of the untwisted affine Kac-Moody algebra widehat{mathfrak{g}} associated with mathfrak{g} . The Hamiltonian flows of these charges are shown to generate an infinite hierarchy of compatible integrable equations.
Highlights
One of the hallmarks of integrability in a (1 + 1)-dimensional field theory is the existence of an infinite number of conserved charges
When the Lie algebra g underlying the r/s-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space σ-model
This property can be attributed to the existence of a Lax connection, depending on some auxiliary complex spectral parameter λ, whose zero curvature equation for all λ is equivalent to the equations of motion of the field theory
Summary
One of the hallmarks of integrability in a (1 + 1)-dimensional field theory is the existence of an infinite number of conserved charges. The principal chiral model and (semi-)symmetric space σ-models all have a double pole in their twist function which splits up into a pair of simple poles when their Yang-Baxter type deformation is switched on It was shown in [18, 21], based on earlier work [31,32,33,34,35,36,37,38] (see more recent related results [39,40,41]), that the charges extracted from the leading order in the expansion of the monodromy at this pair of simple poles satisfy all the relations of a Poisson algebra Uq(g), the semiclassical counterpart of the quantum group Uq(g) with q = q.
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