Double excitations in the AdS(5)/CFT(4) integrable system and the Lagrange operator

TL;DR

This paper explores the full spectrum of excitations in the AdS(5)/CFT(4) integrable system, emphasizing the role of double excitations and their implications for the structure of the on-shell Lagrangian in N=4 super Yang-Mills theory.

Contribution

It introduces the concept of double excitations for all three raising operators in the internal SU(4) symmetry and analyzes their impact on the spectrum and structure constants.

Findings

Double excitations can fully accommodate the spectrum of excitations.

Creation amplitudes for double excitations have a universal form.

The on-shell Lagrangian can be represented by four fermions on a spin chain of length two.

Abstract

It is argued that the integrable model for the planar spectrum of the AdS/CFT correspondence can accommodate for the full spectrum of excitations $D^{\alpha \dot \alpha}, \phi^{[IJ]}, \psi^I, \bar \psi_I, F^{\alpha \beta}, \tilde F^{\dot \alpha \dot \beta}$ (with $I,J \in 1 \ldots 4$) if double excitations are allowed for all three raising operators of the internal $SU(4)$ symmetry. We present a tree-level analysis of related creation amplitudes in the nested Bethe ansatz as well as in the original level-1 picture in which excitations of various flavours scatter by a true $S$-matrix. In the latter case, the creation amplitudes for all double excitations we encounter take a perfectly universal form. Building on these ideas we work out Bethe solutions and states relevant in the mixing problem concerning the on-shell Lagrangian of ${\cal N} = 4$ super Yang-Mills theory. Owing to the very existence of double excitations, the chiral Yang-Mills field strength tensor can be represented by the four fermions $\{\psi^{31}, \psi^{32}, \psi^{41}, \psi^{42}\}$ moving on a spin chain of length two. Our analysis remains restricted to leading order in the coupling, where the conformal eigenstate corresponding to the on-shell Lagrangian only comprises the pure Yang-Mills action. It should eventually be possible to augment our analysis to higher loop orders by incorporating coupling corrections in the relevant ingredients from the Bethe ansatz. Finally, it was recently realised how structure constants for operators containing the hitherto hidden half of the excitations can be computed by the hexagon formalism. We use this for a first test of our conjecture for the on-shell Lagrangian, namely that its three-point function with two half-BPS operators of equal length ought to vanish.