Complete Spectrum of Long Operators in N=4 SYM at One Loop

TL;DR

Beisert, Kazakov, Sakai, and Zarembo construct the complete spectral curve for arbitrary local one‑loop operators in N=4 SYM in the thermodynamic limit, including fermions and derivatives, and demonstrate exact agreement with the Frolov–Tseytlin limit of the AdS5×S5 string curve. They introduce stacks—bound states of Bethe roots across flavors—and develop duality transformations that connect Beauty and Beast Bethe equations, unifying the integrable description. The thermodynamic analysis reveals that stacks form strings and decouple the su(4) and su(2,2) sectors at leading order, while fermions provide subleading corrections and essential supersymmetric structure. The resulting algebraic curve reproduces the string theory curve in the FT limit and suggests a supersymmetric Landau–Lifshitz interpretation, offering a robust framework for comparing gauge and string theories and guiding higher‑loop extensions.

Abstract

We construct the complete spectral curve for an arbitrary local operator, including fermions and covariant derivatives, of one-loop N=4 gauge theory in the thermodynamic limit. This curve perfectly reproduces the Frolov-Tseytlin limit of the full spectral curve of classical strings on AdS_5xS^5 derived in hep-th/0502226. To complete the comparison we introduce stacks, novel bound states of roots of different flavors which arise in the thermodynamic limit of the corresponding Bethe ansatz equations. We furthermore show the equivalence of various types of Bethe equations for the underlying su(2,2|4) superalgebra, in particular of the type "Beauty" and "Beast".

Figures (9)

• Figure 1: "Beauty" Beisert:2003yb Dynkin diagram of $\mathfrak{su}(2,2|4)$.
• Figure 2: From Beauty to Beast
• Figure 3: Interchange of $\mathfrak{su}(4)$ and $\mathfrak{su}(2,2)$
• Figure 4: Duality transformation for $\mathfrak{su}(1,1|2)$
• Figure 5: A stack of $l-k=3$ Bethe roots $u^{(j)}$, $j=k,\ldots ,l-1$. The distance between the roots $u^{(j+1)}-u^{(j)}$ is $\mathcal{O}(1)$ while their position $u^{(j)}$ is $\mathcal{O}(L)$.