Half-BPS Wilson Loops in AdS/CFT

Abstract

The Maldacena-Wilson loop is a versatile gauge-invariant non-local operator that can be used to probe the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. When the contour is a straight line or a circle, the Wilson loop preserves half the superconformal symmetries and can be calculated in $\mathcal{N}=4$ super Yang-Mills (SYM) in the planar limit using perturbation theory or for all values of the coupling and rank of the gauge group using supersymmetric localization. In the supergravity regime, the dual object in $AdS_5$ is the surface of minimal area that is incident on the Wilson loop on the boundary. Part I of this thesis rederives many of the classic results involving the expectation value of the circular Wilson loop on both the CFT and AdS sides of AdS/CFT. The summation of ladder diagrams in perturbation theory is carried out in detail. The necessary random matrix tools--- including Faddeev-Popov gauge fixing, the method of orthogonal polynomials and the saddle point approximation--- are developed and used to evaluate the integrals over Hermitian matrices and imaginary Hermitian matrices that represent the expectation values of the $U(N)$ and $SO(N)$ circular Wilson loops. Part I also carries out a novel confirmation of AdS/CFT by calculating the large $N$, large $\lambda$ limit of the $SO(N)$ Wilson loop in the spinor representation and replicating the result in AdS by minimizing the volume of a D5-brane wrapped on an $\mathbb{R}P^4$ subspace of $AdS_5\times \mathbb{R}P^5$ and incident on the Wilson loop. Shifting focus, Part II of the thesis explores the $AdS_2/CFT_1$ correspondence connecting the field theory characterizing fluctuations of the $AdS_2$ minimal surface to the one-dimensional defect conformal field theory living on the straight Wilson line. The four-point functions of protected composite defect primaries are calculated and used to extract operator product expansion (OPE) data for the defect CFT. The anomalous dimensions of conglomerated operators appearing in the OPE of two composite primaries are determined, some of which are checked by looking at the two-point functions of the conglomerated operators using point-splitting regularization. There is evidence of universal properties of the anomalous dimensions, as well as hints of not-yet-understood structure underlying the defect CFT, which may be illuminated in future work