Higher rank Wilson loops from a matrix model
Sean A. Hartnoll, S. Prem Kumar
TL;DR
The paper uses a Gaussian Hermitian matrix integral to compute circular Wilson loops in rank-$k$ symmetric and antisymmetric representations of ${\mathcal N}=4$ SYM at large ${N}$, obtaining exact results in the ’t Hooft coupling ${\lambda}$ and connecting to AdS/CFT via D3 and D5 brane dictionaries. Wilson loops are expressed as contour integrals of generating functions ${F_A(t)}$ and ${F_S(t)}$, and a large-${N}$ saddle-point analysis with a Wigner semicircle density yields strong- and weak-coupling limits that reproduce known supergravity results and predict corrections ${\sim e^{-\sqrt{\lambda}}}$. For antisymmetric representations a single saddle matches the D5-brane calculation, while for symmetric representations two competing saddles reproduce the D3-brane result in the appropriate limit, with a rich structure including branch cuts and potential second-sheet contributions. The work provides a concrete, controllable bridge between matrix-model computations and bulk holography for higher SU(N) representations, and it outlines directions for finite-${N}$ extensions and further refinements of the holographic dictionary.
Abstract
We compute the circular Wilson loop of N=4 SYM theory at large N in the rank k symmetric and antisymmetric tensor representations. Using a quadratic Hermitian matrix model we obtain expressions for all values of the 't Hooft coupling. At large and small couplings we give explicit formulae and reproduce supergravity results from both D3 and D5 branes within a systematic framework.