Summing planar diagrams

TL;DR

This paper demonstrates how summing planar open string diagrams on D3-branes can be reformulated as the propagation of a closed string with a specific Hamiltonian, connecting open string diagrams to closed string backgrounds.

Contribution

It provides a novel recasting of planar open string diagrams as closed string propagation with an explicit Hamiltonian, without relying on supergravity background assumptions.

Findings

Explicit form of the hole operator P and its properties

Derivation of a supersymmetric Hamiltonian in different regimes

Closed form for scattering of closed strings from D3-branes

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 = 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.