How Accurate is the Quantum String Bethe Ansatz?

TL;DR

This work assesses the accuracy of the quantum string Bethe ansatz for finite-length strings in $AdS_5\times S^5$ by comparing asymptotic Bethe equations to explicit one-loop sigma-model calculations. It shows that, while the Bethe ansatz reproduces the infinite-volume spectrum exactly, finite-size corrections are exponentially small in the string length, arising from Casimir-type effects and not captured by the asymptotic framework. In a semi-classical spinning-string setup, the non-exponential part of the one-loop energy matches the Bethe prediction, whereas the remaining exponentially suppressed terms are missed, illustrating the asymptotic nature of the approach. The results motivate development of genuinely finite-volume Bethe equations or additional degrees of freedom to capture wrapping effects across coupling regimes, linking weak and strong coupling behavior.

Abstract

We compare solutions of the quantum string Bethe equations with explicit one-loop calculations in the sigma-model on AdS(5)xS(5). The Bethe ansatz exactly reproduces the spectrum of infinitely long strings. When the length is finite, we find that deviations from the exact answer arise which are exponentially small in the string length.

Figures (1)

• Figure 1: The one-loop correction to the energy as a function of $\mathcal{J}$ for $m=1$ and ${\mathcal{J}}= 1 + 0.1 N$. The blue dashed line is the exact string energy (\ref{['exacte']}), the red solid line is the Bethe-ansatz result (\ref{['BBBB']}).