Quantum deformed magnon kinematics

TL;DR

This paper explores the quantum deformed symmetry algebra underlying the magnon kinematics in planar N=4 supersymmetric Yang-Mills theory, revealing its elliptic structure and relation to a discrete integrable model.

Contribution

It identifies the dispersion relation with the Casimir of a quantum deformed algebra, introducing an elliptic uniformization and linking to a lattice model with BMN length spacing.

Findings

Dispersion relation matches the Casimir of E_q(1,1)

Elliptic uniformization of rapidity space

Connection to a discrete integrable lattice model

Abstract

The dispersion relation for planar N=4 supersymmetric Yang-Mills is identified with the Casimir of a quantum deformed two-dimensional kinematical symmetry, E_q(1,1). The quantum deformed symmetry algebra is generated by the momentum, energy and boost, with deformation parameter q=e^{2\pi i/\lambda}. Representing the boost as the infinitesimal generator for translations on the rapidity space leads to an elliptic uniformization with crossing transformations implemented through translations by the elliptic half-periods. This quantum deformed algebra can be interpreted as the kinematical symmetry of a discrete integrable model with lattice spacing given by the BMN length a=2\pi/\sqrt{\lambda}. The interpretation of the boost generator as the corner transfer matrix is briefly discussed.