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

We consider the operators with highest anomalous dimension $\Delta$ 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 $\Delta$ from weak to strong 't Hooft coupling $\lambda$ 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 $\nu$ in the leading order expansion $\Delta\sim \lambda^{\nu}$. We find $\nu = 1/2$ and $\nu = 1/4$ 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 $\lambda$. Instead, the string solution perfectly reproduces the Gubser-Klebanov-Polyakov law $\Delta = 2\sqrt{n} \lambda^{1/4}$. In particular, we provide an analytic expression for the integer level $n$ as a function of the U(1) charge in both sectors.