Wilson loops in large-N theories

In the AdS/CFT correspondence, correlation functions of conformal fields are related to amplitudes of a quantum theory in AdS [1 – 3] (for a review, see for example [4]). Since the conformal group of the boundary and the isometry group of AdS are identical, correlation functions defined in this way are conformally symmetric. However not all quantum theories in AdS can be related to CFT at the boundary in this way. The correlation functions on the boundary must obey the axioms of CFT. For example, any CFT contains the energy-momentum tensor in the operator algebra, and one must be able to compute correlation functions including the energy-momentum tensor. There has to be a field in AdS to which it can couple to, namely the graviton. Thus a quantum theory dual to CFT is necessarily gravitational. A quantum gravity may or may not be a string theory. In some cases, however, there are operators in CFT which can directly couple to string states. Wilson loop operators are examples of such operators. So we may hope to learn about stringy aspects of the AdS/CFT correspondence by studying the Wilson loops. Also, these are fundamental gauge invariant operators in gauge theories, and we may hope to learn about gauge theories from the point of view of AdS. In this talk, I will discuss Wilson loops in the N = 4 supersymmetric Yang-Mills theory in four dimensions, based on my work with Nadav Drukker and David Gross [5]. First we review basic properties of the Wilson loops in N = 4 theory. This part of the talk is purely field theoretical. The N = 4 theory contains massless scalar fields, and they must be taken into account in constructing the Wilson loop. The presence of the scalar fields improves ultra-violet properties of the loop operators. We will then discuss