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Bethe Ansatz solutions for highest states in ${\cal N}=4$ SYM and AdS/CFT duality

We consider the operators with highest anomalous dimension $Δ$ in the compact rank-one sectors $\mathfrak{su}(1|1)$ and $\mathfrak{su}(2)$ of ${\cal N}=4$ super Yang-Mills. We study the flow of $Δ$ from weak to strong 't Hooft coupling $λ$ by solving (i) the all-loop gauge Bethe Ansatz, (ii) the quantum string Bethe Ansatz. The two calculations are carefully compared in the strong coupling limit and exhibit different exponents $ν$ in the leading order expansion $Δ\sim λ^ν$. We find $ν= 1/2$ and $ν= 1/4$ for the gauge or string solution. This strong coupling discrepancy is not unexpected, and it provides an explicit example where the gauge Bethe Ansatz solution cannot be trusted at large $λ$. Instead, the string solution perfectly reproduces the Gubser-Klebanov-Polyakov law $Δ= 2\sqrt{n} λ^{1/4}$. In particular, we provide an analytic expression for the integer level $n$ as a function of the U(1) charge in both sectors.