Classical Lie Bialgebras for AdS/CFT Integrability by Contraction and Reduction

TL;DR

This paper derives classical Lie bialgebras relevant to AdS/CFT integrability by contracting and reducing quantum algebra structures, providing new insights into the algebraic foundations of integrable models in AdS space.

Contribution

It introduces a novel contraction and reduction procedure to obtain classical Lie bialgebras from quantum integrable structures related to AdS/CFT.

Findings

Derived classical r-matrices of rational and trigonometric types.

Constructed Lie bialgebras from quantum algebra contractions.

Applied results to representations of on-shell fields in AdS and flat space.

Abstract

Integrability of the one-dimensional Hubbard model and of the factorised scattering problem encountered on the worldsheet of AdS strings can be expressed in terms of a peculiar quantum algebra. In this article, we derive the classical limit of these algebraic integrable structures based on established results for the exceptional simple Lie superalgebra d(2,1;epsilon) along with standard sl(2) which form supersymmetric isometries on 3D AdS space. The two major steps in this construction consist in the contraction to a 3D Poincar\'e superalgebra and a certain reduction to a deformation of the u(2|2) superalgebra. We apply these steps to the integrable structure and obtain the desired Lie bialgebras with suitable classical r-matrices of rational and trigonometric kind. We illustrate our findings in terms of representations for on-shell fields on AdS and flat space.