Summing planar diagrams

We consider the sum of planar diagrams for open strings propagating on N D3-branes and show that it can be recast as the propagation of a closed string with a Hamiltonian H = H_0 - g_s N P where H_0 is the free Hamiltonian and P is the hole or loop insertion operator. We compute explicitly P and study its properties. When the distance y to the D3-branes is much larger than the string length, y >> l_s, small holes dominate and H becomes a supersymmetric Hamiltonian describing the propagation of a closed string in the full D3-brane supergravity background in a particular gauge that we call sigma-gauge. At strong coupling, g_s N >> 1, there is a region 1 << y << (g_sN)^(1/4) where H is a supersymmetric Hamiltonian describing the propagation of closed strings in AdS_5xS^5. We emphasize that both results follow from the open string planar diagrams without any reference to the existence of a D3-brane supergravity background. A by-product of our analysis is a closed form for the scattering of a generic closed string state from a D3-brane. Finally, we briefly discuss how this method could be applied to a field theory and describe a way to rewrite the planar Feynman diagrams as the propagation of a string with a non-local Hamiltonian by identifying the shape of the string with the trajectory of the particle.Comment: 40 pages, 9 figures, LaTeX. v2: references added v3: Appendices added expanding some calculation

Matching the circular Wilson loop with dual open string solution at 1-loop in strong coupling

We compute the 1-loop correction to the effective action for the string solution in AdS_5 x S^5 dual to the circular Wilson loop. More generically, the method we use can be applied whenever the two dimensional spectral problem factorizes, to regularize and define the fluctuation determinants in terms of solutions of one-dimensional differential equations. A such it can be applied to non-homogeneous solutions both for open and closed strings and to various boundary conditions. In the case of the circular Wilson loop, we obtain, for the 1-loop partition function a result which up to a factor of two matches the expectation from the exact gauge theory computation. The discrepancy can be attributed to an overall constant in the string partition function coming from the normalization of zero modes, which we have not fixed.Comment: 32 pages; v2: typos corrected, acknowledgments added; v3: minor corrections, references adde

Wilson loops and Riemann theta functions II

In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS3). If the Wilson loop is given by a boundary curve X(s) we define, using the integrable properties of the system, a family of curves X(lambda,s) depending on a complex parameter lambda known as the spectral parameter. This family has remarkable properties. As a function of lambda, X(lambda,s) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, X(lambda,s) has an essential singularity at the origin lambda=0. The coefficients of the expansion of X(lambda,s) around lambda=0, when appropriately integrated along the curve give the area of the corresponding minimal area surface. Furthermore we show that the same construction allows the computation of certain surfaces with one or more boundaries corresponding to Wilson loop correlators. We extend the area formula for that case and give some concrete examples. As the main example we consider a surface ending on two concentric circles and show how the boundary circles can be deformed by introducing extra cuts in the spectral curve.Comment: LaTeX, 45 pages, 10 figures. v2: typos corrected, references adde

Euclidean Wilson loops and Minimal Area Surfaces in Minkowski AdS3

The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal area surfaces in AdS5xS5 space. If the Wilson loop is Euclidean and confined to a plane (t,x) then the dual surface is Euclidean and lives in Minkowski AdS3. In this paper we study such minimal area surfaces generalizing previous results obtained in the Euclidean case. Since the surfaces we consider have the topology of a disk, the holonomy of the flat current vanishes which is equivalent to the condition that a certain boundary Schroedinger equation has all its solutions anti-periodic. If the potential for that Schroedinger equation is found then reconstructing the surface and finding the area become simpler. In particular we write a formula for the Area in terms of the Schwarzian derivative of the contour. Finally an infinite parameter family of analytical solutions using Riemann Theta functions is described. In this case, both the area and the shape of the surface are given analytically and used to check the previous results.Comment: 45 pages, 4 figures, LaTe