On Yangian and Long Representations of the Centrally Extended su(2|2) Superalgebra

TL;DR

This paper investigates the Yangian and long representations of the centrally extended su(2|2) superalgebra, revealing the non-existence of certain S-matrices and constructing a new scattering matrix for long-short representations.

Contribution

It demonstrates the non-existence of compatible S-matrices for evaluation representations of long modules and constructs a new invariant S-matrix for long-short representations.

Findings

No compatible S-matrices for evaluation representations of long modules.

Constructed a new invariant S-matrix for long-short representations.

Revealed different invariance properties under non-evaluation representations.

Abstract

The centrally extended su(2|2) superalgebra is an asymptotic symmetry of the light-cone string sigma model on AdS5 x S5. We consider an evaluation representation of the conventional Yangian built over a particular 16-dimensional long representation of the centrally extended su(2|2). Interestingly, we find that S-matrices compatible with this evaluation representation do not exist. On the other hand, by requiring centrally extended su(2|2) invariance and explicitly solving the Yang-Baxter equation, we find a scattering matrix for long-short representations of the Lie superalgebra. We notice that this S-matrix is invariant under a different representation of non-evaluation type, induced from the tensor product of short representations. Our findings concern the conventional Yangian only, and are not applied to possible algebraic extensions of the latter.