A Relativistic Relative of the Magnon S-Matrix

TL;DR

This paper develops a relativistic scattering theory based on q-deformation of the magnon S-matrix, revealing a finite spectrum that matches the soliton spectrum of the semi-symmetric space sine-Gordon theory, with implications for integrable models.

Contribution

It introduces a new relativistic S-matrix framework linked to affine quantum groups, extending the magnon S-matrix with a finite spectrum at roots of unity.

Findings

The constructed S-matrix aligns with the soliton spectrum of the semi-symmetric space sine-Gordon theory.

The algebraic structure involves a twisted affine loop superalgebra with extended supersymmetry.

The spectrum is finite and matches known relativistic integrable models at specific deformation parameters.

Abstract

We construct a relativistic scattering theory based on a q deformation and large string tension limit of the magnon S-matrix of the string world sheet theory in AdS_5 x S^5. The S-matrix falls naturally into a previously studied class associated to affine quantum groups, in this case for a twisted affine loop superalgebra associated to an outer automorphism of sl(2|2). This infinite algebra includes the celebrated triply extended psl(2|2) x R^3 algebra, but only two of the centres, the lightcone components of the 2-momentum, are non-vanishing. The algebra has the interpretation as an extended supersymmetry algebra including a non-trivial R-symmetry. The representation theory of this algebra has some complications in that tensor products are reducible but indecomposable, however, we find that structure meshes perfectly with the bootstrap, or fusion, equations of S-matrix theory. The bootstrap equations can then be used inductively to generate the complete S-matrix. Unlike the magnon theory, the relativistic theory only has a finite set of states and we find that - at least when the deformation parameter q is a root of unity - the spectrum matches precisely the soliton spectrum of the relativistic theory underlying the Pohlmeyer reduction of the string world sheet theory known as the semi-symmetric space sine-Gordon theory.