Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cusped hyperbolic Lagrangians as mirrors to lines in three-space

Sebastian Haney

TL;DR

The paper constructs a singular Lagrangian lift of a 4-valent tropical vertex inside $T^*T^3$, whose smooth part is the minimally twisted five-component chain link complement, and then forms an immersed Lagrangian via a doubling trick. It develops an immersed Wrapped Floer theory to define a wrapped Fukaya category for such objects, analyzes obstructions, gradings, and unobstructedness, and shows that for generic lines in $P^3$, there exists a local system making the wrapped Floer object mirror to a line in the mirror. A tropical Lagrangian lift framework is used to relate the Floer-theoretic data to very affine lines, with the Grassmann–Plücker relations encoding the necessary compatibility conditions. The work also develops desingularizations and smoothings of the singular lift, as well as a framework for immersions that may inform future mirror symmetry statements for lines in Calabi–Yau threefolds, including the quintic mirror.

Abstract

We construct a Lagrangian in the cotangent bundle of a 3-torus whose projection to the fiber is a neighborhood of a tropical curve with a single 4-valent vertex. This Lagrangian has an isolated conical singular point, and its smooth locus is diffeomorphic to the minimally-twisted five component chain link complement, a cusped hyperbolic 3-manifold. From this singular Lagrangian, we construct an immersed Lagrangian, and determine when it is unobstructed in the wrapped Fukaya category. We show that for a generic line in projective 3-space, there is a local system on this immersed Lagrangian such that the resulting object of the wrapped Fukaya category is homologically mirror to an object of the derived category supported on the line. In the course of the proof, we construct a version of the wrapped Fukaya category with objects supported on Lagrangian immersions, which may be of independent interest.

Cusped hyperbolic Lagrangians as mirrors to lines in three-space

TL;DR

The paper constructs a singular Lagrangian lift of a 4-valent tropical vertex inside , whose smooth part is the minimally twisted five-component chain link complement, and then forms an immersed Lagrangian via a doubling trick. It develops an immersed Wrapped Floer theory to define a wrapped Fukaya category for such objects, analyzes obstructions, gradings, and unobstructedness, and shows that for generic lines in , there exists a local system making the wrapped Floer object mirror to a line in the mirror. A tropical Lagrangian lift framework is used to relate the Floer-theoretic data to very affine lines, with the Grassmann–Plücker relations encoding the necessary compatibility conditions. The work also develops desingularizations and smoothings of the singular lift, as well as a framework for immersions that may inform future mirror symmetry statements for lines in Calabi–Yau threefolds, including the quintic mirror.

Abstract

We construct a Lagrangian in the cotangent bundle of a 3-torus whose projection to the fiber is a neighborhood of a tropical curve with a single 4-valent vertex. This Lagrangian has an isolated conical singular point, and its smooth locus is diffeomorphic to the minimally-twisted five component chain link complement, a cusped hyperbolic 3-manifold. From this singular Lagrangian, we construct an immersed Lagrangian, and determine when it is unobstructed in the wrapped Fukaya category. We show that for a generic line in projective 3-space, there is a local system on this immersed Lagrangian such that the resulting object of the wrapped Fukaya category is homologically mirror to an object of the derived category supported on the line. In the course of the proof, we construct a version of the wrapped Fukaya category with objects supported on Lagrangian immersions, which may be of independent interest.
Paper Structure (20 sections, 46 theorems, 112 equations, 13 figures)

This paper contains 20 sections, 46 theorems, 112 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Immersed wrapped Floer theory
  3. Lagrangian immersions
  4. Obstructions
  5. The wrapped Fukaya category
  6. Background on SYZ mirror symmetry
  7. Lagrangian torus fibrations
  8. The Lagrangian pair of pants
  9. Singular Lagrangian lift of a 4-valent vertex
  10. The minimally twisted five-component chain link
  11. Construction
  12. First homology
  13. Weinstein neighborhoods and smoothings
  14. Lagrangian neighborhood theorems
  15. Desingularizations and lifts of smooth tropical curves
  16. ...and 5 more sections

Key Result

Theorem 1.1

The $4$-valent tropical curve $V$ in $\mathbb{R}^3$ depicted in Figure intro-fig has a Lagrangian lift $L_{\mathrm{sing}}$ with one singular point whose neighborhood is homeomorphic to the cone over a $2$-torus. Away from the singular point, $L_{\mathrm{sing}}$ is diffeomorphic to a hyperbolic link

Figures (13)

  • Figure 1: The $4$-valent tropical curve $V$ (top) and the minimally twisted five-component chain link (bottom).
  • Figure 2: A stable disk with $3$ positive punctures of Type 1 and $2$ negative punctures of Type 2.
  • Figure 3: The tropical pair of pants (left) and its coamoeba (right).
  • Figure 4: The boundary of the three holomorphic strips contributing to the Floer differential on $CW^*(L_{\mathrm{pants}},T^2)$, projected to the amoeba (left), and the coamoeba (right).
  • Figure 5: The circle of symmetry (left) and its image in the quotient orbifold (right).
  • ...and 8 more figures

Theorems & Definitions (117)

  • Theorem 1.1: Paraphrasing of Theorem \ref{['singularlift']}
  • Remark 1.1
  • Corollary 1.1
  • proof
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • ...and 107 more