An SU(1|1)-Invariant S-Matrix with Dynamic Representations

TL;DR

This work studies a $\mathfrak{u}(1|1)$-invariant S-matrix arising in a long-range spin chain with ${\mathfrak{su}}(2|1)$ symmetry, viewed through a representation-theory lens. By formulating the asymptotic Bethe Ansatz and constructing the invariant S-matrix from the $\mathfrak{u}(1|1)$ algebra, it connects the long-range chain to the well-known ${\mathfrak{su}}(1|2)$ sector of ${\cal N}=4$ SYM via an $x^{\pm}$ parametrization and explicit scattering phases. The paper also relates these structures to XXZ-like deformations and quantum-algebra deformed spin chains, showing how the same $x^{\pm}$ framework underpins both long-range and nearest-neighbor integrable systems and satisfying the Yang-Baxter equation. These results clarify the origin of the Beisert S-matrix in this context and suggest broader applicability to AdS/CFT integrability and the full ${\cal N}=4$ SYM spin chain. Overall, the study advances understanding of momentum-dependent representations and invariant scattering in integrable gauge-theory spin chains.

Abstract

The spin chains originating from large-N conformal gauge theories are of a special kind: The Hamiltonian is not invariant under the symmetry algebra, it is rather a part of it. This leads to interesting properties within the asymptotic Bethe ansatz. Here we study an S-matrix with u(1|1) symmetry which arises in a long-range spin chain with fundamental spins of su(2|1).