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

TL;DR

This paper compares gauge and string Bethe Ansatz solutions for the highest anomalous dimension operators in ${ m extbf{N}=4}$ SYM, revealing differences at strong coupling and confirming the string theory prediction of the Gubser-Klebanov-Polyakov law.

Contribution

It provides a detailed comparison of gauge and string Bethe Ansatz solutions for highest states, highlighting their differing behaviors at strong coupling and confirming the string prediction.

Findings

Gauge and string solutions exhibit different exponents in strong coupling limit.

String solution reproduces the Gubser-Klebanov-Polyakov law.

Analytic expression for level n as a function of U(1) charge.

Abstract

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.