  A conjecture for computing all genus topological closed string amplitudes on
toric local Calabi-Yau threefolds, by interpreting the associated 5-brane web
as a Feynman diagram, is given. A propagator and a three point vertex is
defined which allows us to write down the amplitude associated with 5-brane
web. We verify the conjecture that this amplitude is equal to the closed string
partition function by computing integer invariants for resolved conifold and
certain curves of low degree in local del Pezzo surfaces, local Hirzebruch
surfaces and their various blowups.
