  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.
