Flop Invariance of Refined Topological Vertex and Link Homologies

TL;DR

This paper explores the invariance of refined topological string amplitudes under flop transitions in toric Calabi-Yau manifolds and derives a simplified formula for homological link invariants, enhancing understanding of topological string theory and link homologies.

Contribution

It demonstrates flop invariance of refined topological vertex amplitudes and provides a new, simplified formula for homological link invariants of the Hopf link.

Findings

Refined topological string amplitudes are invariant under flop transitions.

A simple formula for homological sl(N) invariants of the Hopf link is derived.

The new invariant expression refines the Chern-Simons Hopf link invariant.

Abstract

It has been proposed recently that the topological A-model string theory on local toric Calabi-Yau manifolds has a two parameter extension. Amplitudes of the two parameter topological strings can be computed using a diagrammatic method called the refined topological vertex. In this paper we study properties of the refined amplitudes under the flop transition of toric Calabi-Yau three-folds. We also discuss that the slicing invariance and the flop transition imply a simple formula for the homological sl(N) invariants of the Hopf link. The new expression for the invariants gives a simple refinement of the Hopf link invariant of Chern-Simons theory.