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Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex

John Chae

TL;DR

The paper addresses the problem of computing and understanding link invariants for links in $\mathbb{R} \mathbb{P}^3$ via open/closed string duality. It leverages large $N$ duality and a geometric transition to a toric Calabi-Yau geometry, enabling the use of (refined) topological vertex techniques to calculate colored unknot and Hopf link invariants. The key contributions are explicit computations yielding power-series invariants in Kahler parameters with positive $q$-expansions, the proposal that these are graded Poincaré polynomials of a link homology theory for $\mathbb{R} \mathbb{P}^3$, and a comparative analysis with the $S^3$ case that reveals novel mixed-term structure and $t$-dependence in the RP$^3$ setting. The work advances the categorification program for RP$^3$ links and supports a physical interpretation in terms of BPS state counting within open/closed string duality.

Abstract

We provide the first computations of colored unknots and Hopf link in $\mathbb{R}\mathbb{P}^3$ using both the topological vertex and its refinement. Our approach utilizes the toric Calabi-Yau threefold arising from the geometric transition of the cotangent bundle of $\mathbb{R}\mathbb{P}^3$ under the large $N$ duality. We find that the link invariants are series in the Kahler parameters of the toric Calabi-Yau manifold and the $q$-expansions of the rational functions of the series have positivity property. We conjecture that they are Poincare series of an infinite dimensional link homology theory for links in $\mathbb{R}\mathbb{P}^3$. We compare our results with that of the $S^3$ and speculate the consequences of the series nature of the invariants.

Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex

TL;DR

The paper addresses the problem of computing and understanding link invariants for links in via open/closed string duality. It leverages large duality and a geometric transition to a toric Calabi-Yau geometry, enabling the use of (refined) topological vertex techniques to calculate colored unknot and Hopf link invariants. The key contributions are explicit computations yielding power-series invariants in Kahler parameters with positive -expansions, the proposal that these are graded Poincaré polynomials of a link homology theory for , and a comparative analysis with the case that reveals novel mixed-term structure and -dependence in the RP setting. The work advances the categorification program for RP links and supports a physical interpretation in terms of BPS state counting within open/closed string duality.

Abstract

We provide the first computations of colored unknots and Hopf link in using both the topological vertex and its refinement. Our approach utilizes the toric Calabi-Yau threefold arising from the geometric transition of the cotangent bundle of under the large duality. We find that the link invariants are series in the Kahler parameters of the toric Calabi-Yau manifold and the -expansions of the rational functions of the series have positivity property. We conjecture that they are Poincare series of an infinite dimensional link homology theory for links in . We compare our results with that of the and speculate the consequences of the series nature of the invariants.
Paper Structure (17 sections, 44 equations, 3 figures)

This paper contains 17 sections, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Large $N$ Duality for $\mathbb{R} \mathbb{P}^3$
  3. Geometric Transition
  4. Link Invariant
  5. Setup
  6. Unknot
  7. Hopf Link
  8. Refined Link Invariant
  9. Setup
  10. Unknot
  11. Hopf Link
  12. Comparison with results of $S^3$
  13. Conventions
  14. 2d partitions
  15. 3d partitions
  16. ...and 2 more sections

Figures (3)

  • Figure 1: A toric graph for a Lagrangian brane in the local $\mathbb{C} P^1 \times \mathbb{C} P^1$ geometry corresponding to a colored ($\alpha$) unknot in $\mathbb{R} P^3$. The trivial partitions along the three noncompact edges are suppressed.
  • Figure 2: A toric graph for a pair of Lagrangian branes in the local $\mathbb{C} P^1 \times \mathbb{C} P^1$ geometry corresponding to a colored ($\alpha ,\gamma$) Hopf link in $\mathbb{R} P^3$. The trivial partitions along the two noncompact edges are suppressed.
  • Figure :

Theorems & Definitions (3)

  • Conjecture 3.1
  • Conjecture 4.1
  • Conjecture 4.2