On highest-energy state in the su(1|1) sector of N=4 super Yang-Mills theory

TL;DR

This work identifies the highest-energy state in the ${\mathfrak{su}}(1|1)$ sector of ${\cal N}=4$ SYM as the purely-fermionic operator ${\rm tr}(\psi^L)$ and computes its conformal dimension $\Delta(\lambda)$ across weak and strong coupling using the asymptotic gauge-theory Bethe ansatz. In the finite-$L$ regime, the state is maximized for $M=L$, with momentum distribution $p_k=\frac{2\pi k}{L}+\mathcal{O}(g^2)$, yielding $\Delta/L=\tfrac{3}{2}+2g^2-4g^4+\mathcal{O}(g^6)$; in the $L\to\infty$ limit, a Bethe root density $\rho(u)$ satisfies a linear integral equation, and the leading weak-coupling result matches the finite-$L$ expansion. At strong coupling, a consistent $p_k$ distribution gives $\Delta/L= c_1\sqrt{\lambda}+c_2+\mathcal{O}(\lambda^{-1/2})$ with $c_1=\tfrac{3\sqrt{3}}{2\pi^2}$ and $c_2\approx1.18$, indicating $\Delta$ scales as $\sqrt{\lambda}$; the authors also propose and partially analyze a potential analog using the string Bethe ansatz (AFS-type) and discuss a reduced ${\mathfrak{su}}(1|1)$ string model to illuminate the qualitative behavior. Overall, the paper demonstrates how the maximal-energy ${\mathfrak{su}}(1|1)$ state exhibits a smooth weak-to-strong coupling interpolation and reveals subtleties in extending gauge-theory results to the full string description.

Abstract

We consider the highest-energy state in the su(1|1) sector of N=4 super Yang-Mills theory containing operators of the form tr(Z^{L-M} ψ^M) where Z is a complex scalar and ψis a component of gaugino. We show that this state corresponds to the operator tr(ψ^L) and can be viewed as an analogue of the antiferromagnetic state in the su(2) sector. We find perturbative expansions of the energy of this state in both weak and strong 't Hooft coupling regimes using asymptotic gauge theory Bethe ansatz equations. We also discuss a possible analog of this state in the conjectured string Bethe ansatz equations.