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

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 $\mathcal{N}=4$ 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.

42 pages, JHEP style LaTeX

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Cited by in corpus (11)

• A Crossing-Symmetric Phase for AdS_5 x S^5 Strings
• Single spike solutions for strings on S2 and S3
• Strong coupling limit of Bethe Ansatz equations
• The scaling function at strong coupling from the quantum string Bethe equations
• Scattering of single spikes
• Hubbard's Adventures in $\mathcal{N}=4$ SYM-land? Some non-perturbative considerations on finite length operators
• On the commuting charges for the highest dimension SU(2) operators in planar $\mathcal{N}=4$ SYM
• The AdS_5 x S^5 superstrings in the generalized light-cone gauge
• Highest states in light-cone $Ad{S}_5\times S^5$ superstring
• Analytic Bethe Ansatz Solutions for Highest States in the su(1|1) and su(2) Sectors
• AdS/CFT duality at strong coupling