Complete 1-loop test of AdS/CFT

TL;DR

The paper develops a general framework to compute and interpret finite-size (1/L) corrections in nested Bethe Ansatz for high-rank and supersymmetric systems, tying these corrections to 1-loop worldsheet fluctuations in AdS/CFT. It introduces twists and two complementary derivations (transfer-matrix and bosonic-duality based) to derive a unified integral equation encoding the leading finite-size corrections, and shows its equivalence to the semiclassical 1-loop shift around arbitrary classical string solutions. For the AdS/CFT BS equations with Hernandez-Lopez phase, the authors demonstrate that the corrected integral equation reproduces the 1-loop spectrum via a sum over physical fluctuations, including careful mode-number labeling and unit-circle contributions that generate the HL phase. The work connects the algebraic-curve finite-gap description with Bethe-ansatz quantization, providing strong evidence that Beisert-Staudacher equations correctly capture the semi-classical quantization of strings on AdS_5×S^5 at this order. Together, these results establish a robust, general mechanism by which finite-size effects encode quantum fluctuations across a broad class of integrable models.

Abstract

We analyze nested Bethe ansatz (NBA) and the corresponding finite size corrections. We find an integral equation which describes these corrections in a closed form. As an application we considered the conjectured Beisert-Staudacher (BS) equations with the Hernandez-Lopez dressing factor where the finite size corrections should reproduce generic one (worldsheet) loop computations around any classical superstring motion in the AdS_5 x S^5 background with exponential precision in the large angular momentum of the string states. Indeed, we show that our integral equation can be interpreted as a sum over all physical fluctuations and thus prove the complete 1-loop consistency of the BS equations. In other words we demonstrate that any local conserved charge (including the AdS Energy) computed from the BS equations is indeed given at 1-loop by the sum of charges of fluctuations up to exponentially suppressed contributions. Contrary to all previous studies of finite size corrections, which were limited to simple configurations inside rank one subsectors, our treatment is completely general.