A Matrix Model for the Topological String I: Deriving the Matrix Model

We construct a matrix model that reproduces the topological string partition function on arbitrary toric Calabi-Yau threefolds. This demonstrates, in accord with the BKMP remodeling the B-model conjecture, that Gromov-Witten invariants of any toric Calabi-Yau threefold can be computed in terms of the spectral invariants of a spectral curve. Moreover, it proves that the generating function of Gromov-Witten invariants is a tau function for an integrable hierarchy. In a follow-up paper, we will explicitly construct the spectral curve of our matrix model and argue that it equals the mirror curve of the toric Calabi-Yau manifold.