Classical Lie bialgebras for AdS/CFT integrability by contraction and reduction

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 \mathfrak{d}(2,1;\epsilon)𝔡(2,1;ϵ) along with standard \mathfrak{sl}(2)𝔰𝔩(2) which form supersymmetric isometries on 3D AdS space. The two major steps in this construction consist in the contraction to a 3D Poincaré superalgebra and a certain reduction to a deformation of the \mathfrak{u}(2|2)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.