Topological string on toric CY3s in large complex structure limit

We develop a non-planar topological vertex formalism and we use it to study the A-model partition function Ztop of topological string on the class of toric Calabi-Yau threefolds (CY3) in large complex structure limit. To that purpose, we first consider the T2×R special Lagrangian fibration of generic CY3-folds and we give the realization of the class of large μ toric CY3-folds in terms of supersymmetric gauged linear sigma model with non-zero gauge invariant superpotentials W(Φ). Then, we focus on a one complex parameter supersymmetric U(1) gauged model involving six chiral superfields {Φi} with W=μ(∏i=05Φi) and we use it to compute the function Ztop for the case of the local elliptic curve in the limit μ→∞.