The Vertex on a Strip

TL;DR

This paper develops simplified rules for calculating topological string partition functions on a class of local Calabi-Yau geometries using the topological vertex, enabling more efficient computations and applications.

Contribution

It introduces new rules for deriving closed-form topological string partition functions on strip-based Calabi-Yau geometries, extending the utility of the topological vertex method.

Findings

Derived simple rules for partition functions on strip geometries.

Analyzed topological string behavior under flops.

Confirmed Nekrasov's conjecture in general form.

Abstract

We demonstrate that for a broad class of local Calabi-Yau geometries built around a string of IP^1's - those whose toric diagrams are given by triangulations of a strip - we can derive simple rules, based on the topological vertex, for obtaining expressions for the topological string partition function in which the sums over Young tableaux have been performed. By allowing non-trivial tableaux on the external legs of the corresponding web diagrams, these strips can be used as building blocks for more general geometries. As applications of our result, we study the behavior of topological string amplitudes under flops, as well as check Nekrasov's conjecture in its most general form.