Efficient precision quantization in AdS/CFT

TL;DR

The authors develop an efficient algebraic-curve–based quantization method for the $AdS_5\times S^5$ superstring, introducing off-shell fluctuation energies to obtain the one-loop energy shift for general multi-cut (notably two-cut) solutions. They apply the formalism to the giant magnon, deriving explicit finite-size corrections and showing that their results agree with Lüscher-Klassen-Melzer formulas. The work provides compact closed-form expressions for the off-shell frequencies in terms of branch-point data and demonstrates how unit-circle, pole, branch-point, and unphysical-fluctuation contributions combine to reproduce the finite-volume spectrum. The findings offer a scalable, cross-checkable route to finite-size effects in planar AdS/CFT and connect semiclassical quantization with exact S-matrix methods. The methodology and results have broad applicability to other integrable string configurations beyond the giant magnon.

Abstract

Understanding finite-size effects is one of the key open questions in solving planar AdS/CFT. In this paper we discuss these effects in the AdS_5xS^5 string theory at one-loop in the world-sheet coupling. First we provide a very general, efficient way to compute the fluctuation frequencies, which allows to determine the energy shift for very general multi-cut solutions. Then we apply this to two-cut solutions, in particular the giant magnon and determine the finite-size corrections at subleading order. The latter are then compared to the finite-size corrections from Luscher-Klassen-Melzer formulas and found to be in perfect agreement.

Figures (5)

• Figure 1: Algebraic curve for classical superstrings on $AdS_5 \times S^5$. The macroscopic green cuts corresponds to a classical configuration. The wavy lines depict the several physical fluctuations. From left to right we have four bosonic $S^5$ fluctuations , four $AdS_5$ and eight fermionic fluctuations respectively. Any physical configuration has to cross the dashed line. We depict only the physical $|x|>1$ region.
• Figure 2: As we analytically continue a fluctuation energy $\Omega^{\tilde{2} \tilde{3}}(y)$ from a point $|y|>1$ to the interior of the unit circle we see that its mirror image becomes physical.
• Figure 3: Depiction of equation (\ref{['LinearCombiExample']}). On top: we see that for symmetric configurations we can obtain the off-sheel fluctuation frequency $\Omega^{\hat{2}\tilde{2}}=\Omega^{\tilde{3},\hat{3}}$ from the knowledge of the two $S^5$ and $AdS_5$ frequencies. On bottom: With this unphysical fluctuation at hand we can compute the fermionic fluctuation frequency $\Omega^{\hat{2}\tilde{3}}=\Omega^{\tilde{2}\tilde{3}}+\Omega^{\hat{2}\tilde{2}}$ in terms of the two bosonic fluctuations.
• Figure 4: Integration regions.
• Figure 5: When increasing the filling fraction of a cut, the fluctuation could pass through the cut and reappear uniting different sheets. The physical fluctuation $\tilde{2}\tilde{3}$ could become the unphysical one $\tilde{1}\tilde{2}$.