All-genus calculation of Wilson loops using D-branes
Nadav Drukker, Bartomeu Fiol
TL;DR
We introduce a D3-brane with electric flux in $AdS_5\times S^5$ to compute Wilson loops in $\mathcal{N}=4$ SYM, offering an alternative to the fundamental-string description for large $k$. The D3-brane worldvolume takes the form $AdS_2\times S^2$, yielding an on-shell action that reproduces the leading $k\sqrt{\lambda}$ behavior and, via boundary terms, all-genus corrections that align with the Gaussian matrix model predictions. For the circular loop the D3-brane results match the matrix-model calculations exactly in the relevant large-$N$, large-$\lambda$ limit, validating the approach and illuminating the structure of non-planar contributions. The framework extends to 't Hooft loops via S-duality and points to broader applications, including other loop configurations and defect CFT descriptions, with explicit validity bounds on the parameter $\kappa$ to avoid backreaction.
Abstract
The standard prescription for calculating a Wilson loop in the AdS/CFT correspondence is by a string world-sheet ending along the loop at the boundary of AdS. For a multiply wrapped Wilson loop this leads to many coincident strings, which may interact among themselves. In such cases a better description of the system is in terms of a D3-brane carrying electric flux. We find such solutions for the single straight line and the circular loop. The action agrees with the string calculation at small coupling and in addition captures all the higher genus corrections at leading order in alpha'. The resulting expression is in remarkable agreement with that found from a zero dimensional Gaussian matrix model.