Minimal surfaces in hyperbolic space and defect CFTs

Abstract

The Wilson loop is one of the most fundamental and physically interesting operators in a gauge theory. It is gauge invariant and encodes the quark-antiquark potential. The strong-coupling expectation value of Wilson loops in large N gauge theories was also one of the motivating problems for the development of the AdS/CFT correspondence. We derive the form of the Wilson loop in a non-Abelian gauge theory and evaluate select loops in the weak coupling limit. We then perform the same calculations in the strong-coupling limit via AdS/CFT, using the mathematical formalism developed by Maldacena for evaluating the Euclidean Wilson loop on the AdS Poincare boundary. The formalism depends on finding the area of a minimal surface whose boundary is the desired Wilson loop. In the strong-coupling limit, the equations of motion of the minimal surface may be solved using the Pohlmeyer reduction, a process described by Kruczenski and others. These tools allow for the perturbative evaluation of quasi-circular Wilson loops, explicit expressions for which have been worked out to high order by those authors. We confirm the expressions given for the elliptical loop. We then apply the formalism to small deformations to the Wilson loop, which correspond to localized operators in the 1-dimensional “defect CFT” on the loop. We investigate the strong-coupling expectation value of these deformations, and describe how one might search for a novel symmetry within the Pohlmeyer reduction of the defect CFT.