We propose that the expectation value of a circular BPS-Wilson loop in N = 4 SUSYM can be calculated exactly, to all orders in a 1/N expansion and to all orders in g 2 N . Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in α ′ and to all orders in g s . We then compare this result with string theory. We find that the gauge theory calculation, for large g 2 N and to all orders in the 1/N 2 expansion does agree with the leading string theory calculation, to all orders in g s and to lowest order in α ′ . We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory.

The AdS-CFT correspondence suggests that the Wilson loop of the large N gauge theory with Nϭ4 supersymmetry in four dimensions is described by a minimal surface in AdS 5 ϫS 5 . We examine various aspects of this proposal, comparing gauge theory expectations with computations of minimal surfaces. There is a distinguished class of loops, which we call BPS loops, whose expectation values are free from ultraviolet divergence. We formulate the loop equation for such loops. To the extent that we have checked, the minimal surface in AdS 5 ϫS 5 gives a solution of the equation. We also discuss the zigzag symmetry of the loop operator. In the Nϭ4 gauge theory, we expect the zigzag symmetry to hold when the loop does not couple the scalar fields in the supermultiplet. We will show how this is realized for the minimal surface.

The partition function of N = 6 supersymmetric Chern-Simons-matter theory (known as ABJM theory) on S 3 , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super-matrix model is closely related to a matrix model describing topological Chern-Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on AdS 4 × CP 3 and gives the correct N 3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in CP 3 . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi-Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi-Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two 't Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi-Yau, and leads to an expansion around topological Chern-Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.We present the matrix model for the ABJM theory and that for CS theory on the lens space L(2, 1) = S 3 /Z 2 in the next section. The matrix model of ABJM has an underlying U (N 1 |N 2 ) symmetry while that of the lens space has U (N 1 + N 2 ) symmetry, which in both cases are broken to U (N 1 ) × U (N 2 ). It is easy to see that the expressions for them are related by analytical continuation of N 2 → −N 2 , or analogously a continuation of the 't Hooft coupling N 2 /k → −N 2 /k (which may be attributed to the negative level of the CS coupling of this group in the ABJM theory). We can then go on to study the lens space model and analytically continue to ABJM at the end.Conveniently, the lens space matrix model has been studied in the past [10,13,14,8]. The planar resolvent is known in closed form and the expressions for its periods are given as power series at special points in moduli space. We review the details of this matrix model and its solution in Sections 2 and 3.The matrix model of ABJM theory was derived by localization: it captures in a finite dimensional integral all observables of the full theory which preserve certain supercharges. At the time it was derived in [4], the only such observable (apart for the vacuum) was the 1/6 BPS Wilson loop constructed in [15,16,17] and 1/2 BPS vortex loop operators [18]. Indeed, the expectation value of the 1/6 BPS Wilson loop can be expressed as an observable in the ABJM matrix model, and by analytical continuation in the lens space model.Anothe...

The standard prescription for calculating a Wilson loop in the AdS/CFT correspondence is by a string world-sheet ending along the loop at the boundary of AdS. For a multiply wrapped Wilson loop this leads to many coincident strings, which may interact among themselves. In such cases a better description of the system is in terms of a D3-brane carrying electric flux. We find such solutions for the single straight line and the circular loop. The action agrees with the string calculation at small coupling and in addition captures all the higher genus corrections at leading order in α ′ . The resulting expression is in remarkable agreement with that found from a zero dimensional Gaussian matrix model.