The Vertex on a Strip
Amer Iqbal, Amir-Kian Kashani-Poor
TL;DR
The paper develops a strip-based topological vertex framework for toric Calabi-Yau geometries, enabling all sums over internal Young diagrams to be performed and allowing nontrivial external representations. By distinguishing the two local curve types, (-2,0) and (-1,-1), it derives explicit pairing factors and proves that strip amplitudes can be assembled recursively into closed forms, with clear flop and geometric-engineering interpretations. The authors establish flop invariance of Gopakumar-Vafa invariants for strip-decomposable geometries and connect the strip construction to Nekrasov's partition function for 4d N=2 gauge theories through geometric engineering. This work provides robust building blocks toward handling more general toric webs and lays groundwork for systematic GV invariant extraction from topological string amplitudes.
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.