The magnon kinematics of the AdS/CFT correspondence

TL;DR

This work identifies the origin of magnon kinematics in the AdS/CFT framework by introducing a central abelian Hopf subalgebra ${\cal Z}$ with a non-trivial coproduct and antipode that govern length fluctuations in the planar ${\cal N}=4$ theory. Intertwiner constraints on ${\cal Z}$ fix magnon irreps to a BMN-like dispersion and reveal an elliptic structure on the rapidity plane, with the coupling entering through Fermat-curve parameters on ${\rm Spec}({\cal Z})$; crossing can be defined algebraically via the antipode acting on the central subalgebra. The authors argue for a larger Hopf symmetry ${\cal A}$, likely a quantum affine algebra at a root of unity, whose enlarged center yields ${\cal Z}$, and they illustrate this with cyclic two-dimensional irreps of ${\cal U}_{q}(\widehat{SL(2)})$ at $q^{4}=1$. In the semiclassical regime, magnons correspond to sine-Gordon solitons, reproducing the strong-coupling dispersion $E(p)\simeq (\sqrt{\lambda}/\pi)\sin(p/2)$ and linking the gauge and string pictures through a common central-subalgebra structure on the rapidity/central-geometry plane.

Abstract

The planar dilatation operator of N=4 supersymmetric Yang-Mills is the hamiltonian of an integrable spin chain whose length is allowed to fluctuate. We will identify the dynamics of length fluctuations of planar N=4 Yang-Mills with the existence of an abelian Hopf algebra Z symmetry with non-trivial co-multiplication and antipode. The intertwiner conditions for this Hopf algebra will restrict the allowed magnon irreps to those leading to the magnon dispersion relation. We will discuss magnon kinematics and crossing symmetry on the spectrum of Z. We also consider general features of the underlying Hopf algebra with Z as central Hopf subalgebra, and discuss the giant magnon semiclassical regime.

Figures (1)

• Figure 1: The algebraic curve $y^2=z(z-z^-)(z-z^+)$.