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

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

Loop Equations and bootstrap methods in the lattice

Pure gauge theories can be formulated in terms of Wilson Loops correlators by means of the loop equation. In the large-N limit this equation closes in the expectation value of single loops. In particular, using the lattice as a regulator, it becomes a well defined equation for a discrete set of loops. In this paper we study different numerical approaches to solving this equation. Previous ideas gave good results in the strong coupling region. Here we propose an alternative method based on the observation that certain matrices $\hat{\rho}$ of Wilson loop expectation values are positive definite. They also have unit trace (\hat{\rho}\succeq 0, \mbox{tr} \hat{\rho}=1), in fact they can be defined as density matrices in the space of open loops after tracing over color indices and can be used to define an entropy associated with the loss of information due to such trace S_{WL}=-\mbox{tr}[ \hat{\rho}\ln \hat{\rho}]. The condition that such matrices are positive definite allows us to study the weak coupling region which is relevant for the continuum limit. In the exactly solvable case of two dimensions this approach gives very good results by considering just a few loops. In four dimensions it gives good results in the weak coupling region and therefore is complementary to the strong coupling expansion. We compare the results with standard Monte Carlo simulations.Comment: LaTeX, 46 pages, 17 figures. v2: References adde

Minimal area surfaces in AdS_{n+1} and Wilson loops

The AdS/CFT correspondence relates the expectation value of Wilson loops in N=4 SYM to the area of minimal surfaces in AdS_5 In this paper we consider minimal area surfaces in generic Euclidean AdS_{n+1} using the Pohlmeyer reduction in a similar way as we did previously in Euclidean AdS_3. As in that case, the main obstacle is to find the correct parameterization of the curve in terms of a conformal parameter. Once that is done, the boundary conditions for the Pohlmeyer fields are obtained in terms of conformal invariants of the curve. After solving the Pohlmeyer equations, the area can be expressed as a boundary integral involving a generalization of the conformal arc-length, curvature and torsion of the curve. Furthermore, one can introduce the \lambda-deformation symmetry of the contours by a simple change in the conformal invariants. This determines the \lambda-deformed contours in terms of the solution of a boundary linear problem. In fact the condition that all \lambda deformed contours are periodic can be used as an alternative to solving the Pohlmeyer equations and is equivalent to imposing the vanishing of an infinite set of conserved charges derived from integrability.Comment: 29 pages, LaTeX, 1 figur