Classical r-matrix of the su(2|2) SYM spin-chain

TL;DR

This paper derives the classical r-matrix from the quantum R-matrix of the su(2|2) SYM spin-chain, analyzing its algebraic structure, spectral dependence, and pole behavior, relevant for integrability in AdS/CFT correspondence.

Contribution

It provides a direct derivation of the classical r-matrix from the quantum R-matrix in a manifest su(1|2)-invariant formulation, including analysis of its algebraic and pole structure.

Findings

The classical r-matrix solves the classical Yang-Baxter equation.

It preserves an su(1|2) subalgebra and can be expressed via projectors.

Residue analysis at the simple pole offers insights into its algebraic properties.

Abstract

In this note we straightforwardly derive and make use of the quantum R-matrix for the su(2|2) SYM spin-chain in the manifest su(1|2)-invariant formulation, which solves the standard quantum Yang-Baxter equation, in order to obtain the correspondent (undressed) classical r-matrix from the first order expansion in the ``deformation'' parameter 2 \pi / \sqrt{\lambda}, and check that this last solves the standard classical Yang-Baxter equation. We analyze its bialgebra structure, its dependence on the spectral parameters and its pole structure. We notice that it still preserves an su(1|2) subalgebra, thereby admitting an expression in terms of a combination of projectors, which spans only a subspace of su(1|2) \otimes su(1|2). We study the residue at its simple pole at the origin, and comment on the applicability of the classical Belavin-Drinfeld type of analysis.