Summing planar diagrams

Abstract: 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 − gsN P̂ where H0 is the free Hamiltonian and P̂ is the hole or loop insertion operator. We compute explicitly P̂ and study its properties. When the distance y to the D3-branes is much larger than the string length, y ≫ √ α′, 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, gsN ≫ 1, there is a region 1 ≪ y ≪ (gsN) 1 4