Holographic Wilson Loops in Higher Representations of Supersymmetric Yang-Mills Theory

In this thesis, we study Wilson loops within the AdS/CFT correspondence. We focus on circular half-BPS Wilson loops in N=4 SYM, which are particularly tractable due to their high degree of supersymmetry and their well-defined holographic dual descriptions in terms of either fundamental strings or D-branes in AdS5xS5. On the gauge theory side, the expectation values of such Wilson loops can be computed using supersymmetric localization, which reduces the problem to a matrix model. On the string theory side, the dual computation involves evaluating the area of a string worldsheet or the volume of a D-brane worldvolume, with the Wilson loop specifying the boundary conditions for the embedding. While Wilson loops in the fundamental representation are well understood on both sides of the AdS/CFT correspondence, this thesis aims to develop new tools for studying Wilson loops in higher representations of the gauge group, specifically the rank-k symmetric and antisymmetric representations. This is motivated by a known mismatch that arises at the one-loop level between gauge theory and string theory results for these higher representations. We introduce a novel formalism based on Fredholm determinants of integral operators with a Hermite kernel. We show that the generating functions for the rank-k symmetric and antisymmetric Wilson loops can be expressed as Fredholm determinants that correspond to functions satisfying a closed system of integro-differential equations that are exact in both N and the ’t Hooft coupling λ.