Abstract
We consider σ-models on para-complex ZT-cosets, which are analogues of those on complex homogeneous target spaces considered recently by D. Bykov. For these models, we show the existence of a gauge-invariant Lax connection whose Poisson brackets are ultralocal. Furthermore, its light-cone components commute with one another in the sense of Poisson brackets. This extends a result of O. Brodbeck and M. Zagermann obtained twenty years ago for hermitian symmetric spaces.
Highlights
We consider σ-models on para-complex ZT -cosets, which are analogues of those on complex homogeneous target spaces considered recently by D
Faced with the problem of non-ultralocality in any given integrable field theory, it is natural to seek an alternative Lax matrix for this theory which would not suffer from the presence of δ -terms in its Poisson brackets
We show that classical para-complex ZT -cosets admit an ultralocal Lax connection
Summary
We describe the particular class of ZT -cosets which we shall consider. An important role in the whole analysis is played by an element u in the Cartan subalgebra of g whose eigenvalues in the adjoint representation are integers between −T + 1 and T − 1 Let us note that the definition of the ZT -automorphism in (2.3) is such that the root vector associated with the negative of the longest root has grade 1 with respect to the ZT -gradation, namely [u, Fθ] = (1 − T )Fθ =⇒ σ(Fθ) = ωFθ. For compact real forms, the Z2-cosets constructed in the previous paragraphs correspond to Kahlerian symmetric spaces. These are the cosets considered in [22]. For T 2, and to fix the ideas, the pseudo-Riemannian manifolds such as SL(p1 + · · · + pT ) S(GL(p1) × · · · × GL(pT )) are para-complex ZT -cosets and non-symmetric whenever T > 2
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