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Flop Invariance of Refined Topological Vertex and Link Homologies

Masato Taki

TL;DR

The paper extends the refined, two-parameter topological string formalism to local toric Calabi–Yau geometries, analyzing flop behavior of refined amplitudes using the refined topological vertex and free-fermion techniques. It proves refined flop invariance for subdiagrams and links the invariant amplitudes to homological Hopf-link invariants via slicing invariance, proposing a closed-form superpolynomial for the Hopf link and validating it in simple cases. A key technical advance is a transpose identity for skew Schur functions that underpins the flop transformation, enabling more efficient computation of Hopf-link homologies. Collectively, the results provide refined, computable expressions for Hopf-link invariants and deepen the connection between refined topological strings and link homologies.

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.

Flop Invariance of Refined Topological Vertex and Link Homologies

TL;DR

The paper extends the refined, two-parameter topological string formalism to local toric Calabi–Yau geometries, analyzing flop behavior of refined amplitudes using the refined topological vertex and free-fermion techniques. It proves refined flop invariance for subdiagrams and links the invariant amplitudes to homological Hopf-link invariants via slicing invariance, proposing a closed-form superpolynomial for the Hopf link and validating it in simple cases. A key technical advance is a transpose identity for skew Schur functions that underpins the flop transformation, enabling more efficient computation of Hopf-link homologies. Collectively, the results provide refined, computable expressions for Hopf-link invariants and deepen the connection between refined topological strings and link homologies.

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.

Paper Structure

This paper contains 9 sections, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Flop Transition of Reined Vertex
  3. Calculation of Subdiagrams
  4. Proof of formula
  5. Flop, Slicing Invariance, and Homological Link Invariants
  6. Homological Link Invariants
  7. Homological Link Invariants from Flop
  8. Conclusion
  9. Conventions and Formulas

Figures (3)

  • Figure 1: The refined topological vertex $\mathop C\nolimits_{\lambda \mu \nu } (t,q)$
  • Figure 2: The two resolved conifolds (a) and (b) are related via the flop transition.
  • Figure 3: The superpolynomial $\bar{\mathcal{P}}$ and $\hat{Z}$ are related under the slicing invariance and the flop transition.