Quark-antiquark potential in AdS at one loop

TL;DR

The paper derives an exact analytical one-loop partition function for a string in AdS5×S5 whose world-sheet ends on two anti-parallel lines, showing that all fluctuations are governed by single-gap Lamé operators. Using the Gelfand-Yaglom method, it provides closed-form determinants for the fluctuation operators and regularizes infrared divergences by subtracting the straight-line self-energy, leading to an exact integral representation for the first strong-coupling constant a1. The resulting one-loop correction to the quark-antiquark potential is computed as a combination of analytic and numerically evaluated terms, yielding V^{(1)}_{q\bar{q}} ≈ 0.30492/L and a1 ≈ -1.3345953, in agreement with prior results. These results reinforce the integrable structure of fluctuations around elliptic-string solutions and illustrate how Lamé-type spectral problems can be leveraged to obtain precise quantum corrections in AdS/CFT Wilson loops.

Abstract

We derive an exact analytical expression for the one-loop partition function of a string in AdS_5xS^5 background with world-surface ending on two anti-parallel lines. All quantum fluctuations are shown to be governed by integrable, single-gap Lame' operators. The first strong coupling correction to the quark-antiquark potential, as defined in N=4 SYM, is derived as the sum of known mathematical constants and a one-dimensional integral representation. Its full numerical value can be given with arbitrary precision and confirms a previous result.