Strong coupling expansion of $1\over 2$ BPS Wilson loop in SYM theory and 2-loop Green-Schwarz string in AdS$_5 \times $S$^5$

TL;DR

The paper investigates the two-loop Green-Schwarz string corrections near the AdS$_2$ minimal surface corresponding to the 1/2 BPS circular Wilson loop in ${ m N}=4$ SYM. It shows that a UV logarithmic divergence arises at two loops, requiring a subtraction prescription to meaningfully compare with the gauge-theory strong-coupling expansion. By adopting a dimensional-regularization scheme that preserves 2d structure and a supersymmetry-consistent choice of propagator constants, the finite part of the two-loop correction can be made to match the gauge-theory coefficient for the subleading $T^{-1}$ term, highlighting the crucial role of regularization and counterterms in AdS/CFT at strong coupling. The work underscores scheme- and measure-dependence in higher-loop string computations and suggests studying additional observables to achieve robust, cross-checked tests of the duality. Extensions to ABJM theory are also discussed as a broader context for similar strong-coupling analyses.

Abstract

The exact localization result for the expectation value of the $1\over 2$ BPS circular Wilson loop in ${\cal N}=4$ SYM theory is given in the planar limit by the famous Bessel function expression: $\langle W\rangle = {2N\over \sqrt λ} I_1 ( \sqrt λ)$. Expanded in large $λ$ and expressed in terms of the AdS$_5 \times $S$^5$ string tension $T= {\sqrt λ\over 2π}$ this gives $\langle W\rangle = {\sqrt T\over 2πg_s} e^{2πT} (1- {3\over 16 π} T^{-1} + ...)$.The exponential is matched by the value of the action of the string with the AdS$_2$ world volume while the prefactor comes from the 1-loop GS string correction. Here we address the question of how the subleading $T^{-1}$ term could be reproduced by the 2-loop correction in the corresponding partition function of the AdS$_5 \times $S$^5$ GS string expanded near the AdS$_2$ minimal surface. We find that the string correction contains a non-zero UV logarithmic divergence implying that comparison with the SYM result requires a particular subtraction prescription. We discuss implications of this conclusion for checking the AdS/CFT duality at strong coupling.