Abstract
We explain how to obtain new classical integrable field theories by assembling two affine Gaudin models into a single one. We show that the resulting affine Gaudin model depends on a parameter γ in such a way that the limit γ → 0 corresponds to the decoupling limit. Simple conditions ensuring Lorentz invariance are also presented. A first application of this method for σ-models leads to the action announced in [1] and which couples an arbitrary number N of principal chiral model fields on the same Lie group, each with a Wess-Zumino term. The affine Gaudin model descriptions of various integrable σ-models that can be used as elementary building blocks in the assembling construction are then given. This is in particular used in a second application of the method which consists in assembling N − 1 copies of the principal chiral model each with a Wess-Zumino term and one homogeneous Yang-Baxter deformation of the principal chiral model.
Highlights
Constructing classical field theories which are integrable is difficult
In this article we have shown that this language can already be put to good use at the classical level to build new classical integrable field theories with infinitely many parameters
A usual drawback of working at the Hamiltonian level is that Lorentz invariance is not explicit, making it more difficult to identify which of the integrable field theories constructed are relativistic
Summary
Constructing classical field theories which are integrable is difficult. the property of integrability is established at the Lagrangian level if one is able to find a Lax connection whose zero curvature equation is equivalent to the equations of motion derived from the given action. The Hamiltonian of the field theory should be expressible as a linear combination of the Hamiltonians of the local affine Gaudin model, just as in the finite-dimensional setting It is clear from the relation (1.8) and the common pole structure of the twist function and Gaudin Lax matrix that the zeroes of φ(z) coincide with the poles of L (z, x). In this article we shall consider the class of integrable field theories which are realisations of affine Gaudin models associated with g, whose twist functions have simple zeroes and with Hamiltonian density given by a linear combination. One can use the exact same construction as in (1.7) to define the Lax matrix of the affine Gaudin model corresponding to this new integrable field theory.
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