Planar diagrams in light-cone gauge

We consider the open string vacuum amplitude determining the interaction between a stack of N D3-branes and a single probe brane. When using light cone gauge, it is clear that the sum of planar diagrams (relevant in the large-N limit) is described by the free propagation of a closed string. A naive calculation suggests that the Hamiltonian of the closed string is of the form H = H0 − gsNP̂ . The same form of the Hamiltonian follows from considering the bosonic part of the closed string action propagating in the full D3-brane background suggesting the naive calculation captures the correct information. Further, we compute explicitly P̂ from the open string side in the bosonic sector and show that, in a certain limit, the result agrees with the closed string expectations up to extra terms due to the fact that we ignored the fermionic sector. We briefly discuss extensions of the results to the superstring and to the sum of planar diagrams in field theory. In particular we argue that the calculations seem valid whenever one can define a (σ ↔ τ) dual Hamiltonian in the world-sheet which in principle does not require the existence of a string action. This seems more generic than the existence of a string dual in the large-N limit.