Topological String on Toric CY3s in Large Complex Structure Limit

TL;DR

This paper introduces a non planar topological vertex formalism to analyze the topological string partition function on toric Calabi-Yau threefolds in the large complex structure limit, connecting geometric and gauge theory methods.

Contribution

It develops a novel non planar topological vertex approach and applies it to compute the topological string partition function for specific toric CY3s in a new limit.

Findings

Derived a non planar topological vertex formalism.

Computed the topological string partition function for local elliptic curves.

Linked geometric structures with supersymmetric gauged linear sigma models.

Abstract

We develop a non planar topological vertex formalism and we use it to study the A-model partition function $\mathcal{Z}_{top}$ of topological string on the class of toric Calabi-Yau threefolds (CY3) in large complex structure limit. To that purpose, we first consider the $T^{2}\times R$ special Lagrangian fibration of generic CY3-folds and we give the realization of the class of large $\mu $ toric CY3-folds in terms of supersymmetric gauged linear sigma model with \emph{non zero} gauge invariant superpotentials $% \mathcal{W}(\Phi ) $. Then, we focus on a one complex parameter supersymmetric $U(1) $ gauged model involving six chiral superfields ${\Phi_{i}}$ with $\mathcal{W}=\mu (\prod\nolimits_{i=0}^{5}\Phi_{i}) $ and we use it to compute the function $\mathcal{Z}_{top}$ for the case of the local elliptic curve in the limit $\mu \to \infty $.