Minimal Area Surfaces Corresponding to Wilson Loops in Gauge Theories

In this dissertation, we study the properties of Wilson loops in strongly coupled gauge theories by using one of the most important results of the AdS/CFT correspondence, namely, that the expectation value of the Wilson loop operator is related to the area of the minimal area surface ending on the Wilson loop. In particular, we concentrate on the main example of AdS/CFT correspondence which is the duality between the N = 4 Super Yang-Mills (SYM) theory in 4 dimensions and type IIB string theory in AdS5 × S5 space. In this case, the minimal area surfaces live in Anti de Sitter ( AdS) space and therefore have integrability properties we explore in this dissertation. Furthermore, N = 4 SYM is a conformal theory and therefore the expectation value of Wilson loop operators is conformally invariant, a property that we make manifest in all the results. The work consists of two main projects where we find new solutions for minimal area surfaces dual to Wilson loops and study the conformal and integrability properties of minimal area surfaces in general Euclidean AdS space. In the first project, we use the Mathieu equation, a standard example of a periodic potential, to obtain a new class of Wilson loops such that the area of the dual minimal area surface can be computed analytically in terms of eigenvalues of such equation. The area of the minimal surface is invariant under λ-deformations, a hidden symmetry that arises due to integrability. This result provides an important example to check the perturbative method developed by Dekel to compute near circular Wilson loops. It can also be reduced in a special case to an early result proposed by Toledo. There are several interesting limits of the Wilson loop solutions we found, including circular and multi-wound circular cases, as well as null-polygons studied by Alday and Maldacena, which are related to scattering amplitudes of even number of gluons. Finally, in the near null-case, we found that the potential becomes a series of separated wells each associated with two light-like segments leading to a simple perturbative method for these Wilson loops. In the second project, we consider minimal area surfaces in generic Euclidean AdSd in a similar way as we did previously in Euclidean AdS3. We give a formula for the area in terms of a generalized form of the conformal arc-length, conformal curvature and conformal torsion of the Wilson loop. This formula is explicitly reparameterization invariant, conformally invariant as well as invariant under λ-deformations and therefore should be a main starting point to study Wilson loops in conformal field theories. To obtain this result, we use the Pohlmeyer reduction method. We show how an arbitrary Wilson loop living at the boundary determines the boundary conditions for the fields appearing in the Pohlmeyer reduction. Solving the Pohlmeyer equations determines an extra set of boundary conditions and allows to set up a linear problem along the Wilson loop that determines the shape of the λ-deformed curves from the original loop. This set of boundary conditions can also be determined by the condition that all λ-deformed contours are periodic, or equivalently the vanishing of an infinite set of conserved charges derived from integrability