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 = H₀−g_sN  where H₀ is the free Hamiltonian and is the hole or loop insertion operator. We compute explicitly and study its properties. When the distance y to the D3-branes is much larger than the string length, y >> (α')^1/2, 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 σ-gauge. At strong coupling, g_sN >> 1, there is a region 1 << y << (g_sN)^¼ where H is a supersymmetric Hamiltonian describing the propagation of closed strings in AdS₅ × S⁵. 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.