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Singular Legendrian unknot links and relative Ginzburg algebras

Johan Asplund

TL;DR

This work builds a bridge between quiver data with frozen subquivers and symplectic invariants of stopped Weinstein manifolds by constructing a subcritical piece $Z_0(Q)$ and a singular Legendrian attaching link $\boldsymbol\Lambda(Q,F)$, then proving a quasi-isomorphism $CE^*(\boldsymbol\Lambda(Q,F);Z_0(Q))\simeq \mathscr G^*_n(Q,F)$ to the $n$-dimensional relative Ginzburg algebra. It shows that a canonical subalgebra corresponds to $\mathscr G^*_{n-1}(F)$ and that a strong relative smooth Calabi–Yau structure exists for the inclusion and the associated Ginzburg map, enabling derived equivalences between Perf of the CE-algebra and the partially wrapped Fukaya category via the Orlov functor. The proof combines a 1D simplicial decomposition, local Chekanov–Eliashberg calculations for simple Legendrian pieces, and a gluing/colimit framework to realize the global algebra as a relative Ginzburg algebra; explicit chain maps and homotopies establish the quasi-isomorphism and Calabi–Yau structures. This relative perspective extends known non-relative and weak-relative Calabi–Yau phenomena to stopped settings and sets the stage for pre-Calabi–Yau structures and nontrivial potentials in future work. Overall, the paper provides a concrete geometric-realization of relative Ginzburg algebras as Chekanov–Eliashberg algebras of singular Legendrians in stopped Weinstein boundaries, with strong Calabi–Yau control on both the algebra and the associated Orlov functor.

Abstract

We associate to a quiver and a subquiver $(Q,F)$ a stopped Weinstein manifold $X$ whose Legendrian attaching link is a singular Legendrian unknot link $\varLambda$. We prove that the relative Ginzburg algebra of $(Q,F)$ is quasi-isomorphic to the Chekanov--Eliashberg dg-algebra of $\varLambda$. It follows that the Chekanov--Eliashberg dg-algebra of $\varLambda$ relative to its boundary dg-subalgebra, and the Orlov functor associated to the partially wrapped Fukaya category of $X$ both admit a strong relative smooth Calabi--Yau structure.

Singular Legendrian unknot links and relative Ginzburg algebras

TL;DR

This work builds a bridge between quiver data with frozen subquivers and symplectic invariants of stopped Weinstein manifolds by constructing a subcritical piece and a singular Legendrian attaching link , then proving a quasi-isomorphism to the -dimensional relative Ginzburg algebra. It shows that a canonical subalgebra corresponds to and that a strong relative smooth Calabi–Yau structure exists for the inclusion and the associated Ginzburg map, enabling derived equivalences between Perf of the CE-algebra and the partially wrapped Fukaya category via the Orlov functor. The proof combines a 1D simplicial decomposition, local Chekanov–Eliashberg calculations for simple Legendrian pieces, and a gluing/colimit framework to realize the global algebra as a relative Ginzburg algebra; explicit chain maps and homotopies establish the quasi-isomorphism and Calabi–Yau structures. This relative perspective extends known non-relative and weak-relative Calabi–Yau phenomena to stopped settings and sets the stage for pre-Calabi–Yau structures and nontrivial potentials in future work. Overall, the paper provides a concrete geometric-realization of relative Ginzburg algebras as Chekanov–Eliashberg algebras of singular Legendrians in stopped Weinstein boundaries, with strong Calabi–Yau control on both the algebra and the associated Orlov functor.

Abstract

We associate to a quiver and a subquiver a stopped Weinstein manifold whose Legendrian attaching link is a singular Legendrian unknot link . We prove that the relative Ginzburg algebra of is quasi-isomorphic to the Chekanov--Eliashberg dg-algebra of . It follows that the Chekanov--Eliashberg dg-algebra of relative to its boundary dg-subalgebra, and the Orlov functor associated to the partially wrapped Fukaya category of both admit a strong relative smooth Calabi--Yau structure.
Paper Structure (10 sections, 7 theorems, 12 equations, 9 figures)

This paper contains 10 sections, 7 theorems, 12 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Related results
  3. Outline of the proof
  4. Future directions
  5. Relative Ginzburg algebras
  6. Stopped Weinstein manifold associated to a quiver and a frozen subquiver
  7. Weinstein manifolds and stops
  8. A simplicial decomposition
  9. Proof of the main theorem
  10. Computation of the Chekanov--Eliashberg dg-algebra

Key Result

Theorem 1.1

Let $n\geq 4$ and consider a quiver $Q$ together with a subquiver $F \subset Q$. There are quasi-isomorphisms of dg-algebras $CE^\ast(\varLambda(Q,F);Z_0(Q)) \cong \mathscr G^\ast_n(Q,F)$ and $\mathcal{B} \cong \mathscr G^\ast_{n-1}(F)$, such that the following diagram commutes

Figures (9)

  • Figure 1: Left: A quiver $Q$ and frozen subquiver $F$ marked with blue snowflakes and blue arrows. Right: Front projection of corresponding singular Legendrian unknot link $\varLambda(Q,F)$ with frozen components marked in blue.
  • Figure 2: Left: Plumbing quivers $Q$ with a set $F_0$ of frozen vertices and a set $F_1$ of frozen edges. Right: The corresponding graph $C$ with vertex set $V \cup F_0 \cup E \cup F_1$. The vertices represented by black dots $\bullet$ belong to $V$, the vertices represented by blue snowflakes belong to $F_0$, the vertices represented by red squares $\textcolor{red}{\blacksquare}$ belong to $E$ and the vertices represented by blue triangles belong to $F_1$.
  • Figure 3: Top left: A quiver with a frozen subquiver $(Q,F)$. Top right: The simplicial complex $C$ associated to $(Q,F)$. Bottom: The simplicial decomposition of the pair $(Z_0(Q),P_0(F))$.
  • Figure 4: Top left: Local depiction of a vertex $v\in E$. Top right: Two copies of $B^{2n-2}$ with $\varLambda_{v}$ which becomes the $(n-1)$-dimensional Hopf link with boundary in $\partial(B^{2n-2} \sqcup B^{2n-2})$ after a small perturbation in a Darboux chart. Bottom: A slice of $\varLambda_{v}$ in generic position after a small perturbation, where the Reeb chords $g_e$ and $g_e^\ast$ are visible.
  • Figure 5: Top left: Local depiction of a vertex $v\in F_1$. Top right: Three copies of $B^{2n-2}$ with $\varLambda_{v}$ being two $(n-1)$-dimensional Legendrian disks with boundary in $\partial(B^{2n-2} \#_{B^{2n-4}} B^{2n-2} \#_{B^{2n-4}} B^{2n-2})$ in a Darboux chart. Bottom left: The boundary of $\varLambda_1$ in the leftmost copy of $B^{2n-2}$. Bottom middle: The boundary of $\varLambda_1 \cup \varLambda_2$ in the middle blue copy of $B^{2n-2}$. Bottom right: The boundary of $\varLambda_2$ in the rightmost copy of $B^{2n-2}$.
  • ...and 4 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • Conjecture 1.5
  • Definition 2.1: $n$-dimensional Ginzburg algebra
  • Definition 2.2: $n$-dimensional relative Ginzburg algebra
  • Lemma 2.3
  • proof
  • ...and 18 more