  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 - (g_s N) P. 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 P 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 sigma <-> tau
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.
