Open string amplitudes and large order behavior in topological string theory

We propose a formalism inspired by matrix models to compute open and closedtopological string amplitudes in the B-model on toric Calabi-Yau manifolds. Wefind closed expressions for various open string amplitudes beyond the disk, andin particular we write down the annulus amplitude in terms of theta functionson a Riemann surface. We test these ideas on local curves and local surfaces,providing in this way generating functionals for open Gromov-Witten invariantsin the spirit of mirror symmetry. In the case of local curves, we study theopen string sector near the critical point which leads to 2d gravity, and weshow that toric D-branes become FZZT branes in a double-scaling limit. We usethis connection to compute non-perturbative instanton effects due to D-branesthat control the large order behavior of topological string theory on thesebackgroundsComment: 40 pages, 5 figures, v3: further clarifications and references adde