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Symplectomorphisms and spherical objects in the conifold smoothing

Ailsa Keating, Ivan Smith

TL;DR

The paper investigates the open Calabi–Yau threefolds arising from the conifold and its smoothing/resolution, proving that the compactly supported symplectic mapping class group of the smoothing $X$ contains a split copy of the free group on countably many generators and that spherical objects in the derived category $D(Y)$ (the small resolution side) are fully classified in the affine $A_1$ setting. Central to the approach is homological mirror symmetry: the authors relate symplectic autoequivalences of the wrapped Fukaya category $\EuScript{W}(X)$ to $R$-linear autoequivalences of a subcategory $\EuScript{D}\subset D(Y)$, leveraging the stability-condition structure on $D(Y)$ (Toda) to deduce infinite generation on the symplectic side. Conversely, they use Nielsen–Thurston dynamics and a localization framework to constrain autoequivalences on the algebraic side, ultimately showing that spherical objects in $D(W) $ correspond to matching-spheres in $X$ and to $\mathcal{O}_C(-1)$ in $D(Y)$. The results illuminate the interplay between 3-fold affine $A_1$ geometry, symplectic topology, and derived-category dynamics, with implications for categorical entropy and stability-condition geometry in mirror symmetry. The findings provide a novel example where the symplectic mapping class group is of infinite type and establish a complete spherical-object classification mediated by mirror symmetry, underscoring the robustness of HMS techniques in higher dimensions.

Abstract

Let $X$ denote the `conifold smoothing', the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$, or equivalently the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the `conifold resolution', by which we mean the complement of a smooth divisor in $\mathcal{O}(-1) \oplus \mathcal{O}(-1) \to \mathbb{P}^1$. We prove that the compactly supported symplectic mapping class group of $X$ splits off a copy of an infinite rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category $D(Y)$ (the three-dimensional `affine $A_1$-case'). Our results build on work of Chan-Pomerleano-Ueda and Toda, and both theorems make essential use of working on the `other side' of the mirror.

Symplectomorphisms and spherical objects in the conifold smoothing

TL;DR

The paper investigates the open Calabi–Yau threefolds arising from the conifold and its smoothing/resolution, proving that the compactly supported symplectic mapping class group of the smoothing contains a split copy of the free group on countably many generators and that spherical objects in the derived category (the small resolution side) are fully classified in the affine setting. Central to the approach is homological mirror symmetry: the authors relate symplectic autoequivalences of the wrapped Fukaya category to -linear autoequivalences of a subcategory , leveraging the stability-condition structure on (Toda) to deduce infinite generation on the symplectic side. Conversely, they use Nielsen–Thurston dynamics and a localization framework to constrain autoequivalences on the algebraic side, ultimately showing that spherical objects in correspond to matching-spheres in and to in . The results illuminate the interplay between 3-fold affine geometry, symplectic topology, and derived-category dynamics, with implications for categorical entropy and stability-condition geometry in mirror symmetry. The findings provide a novel example where the symplectic mapping class group is of infinite type and establish a complete spherical-object classification mediated by mirror symmetry, underscoring the robustness of HMS techniques in higher dimensions.

Abstract

Let denote the `conifold smoothing', the symplectic Weinstein manifold which is the complement of a smooth conic in , or equivalently the plumbing of two copies of along a Hopf link. Let denote the `conifold resolution', by which we mean the complement of a smooth divisor in . We prove that the compactly supported symplectic mapping class group of splits off a copy of an infinite rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category (the three-dimensional `affine -case'). Our results build on work of Chan-Pomerleano-Ueda and Toda, and both theorems make essential use of working on the `other side' of the mirror.
Paper Structure (31 sections, 58 theorems, 51 equations, 5 figures)

This paper contains 31 sections, 58 theorems, 51 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Context
  4. Remarks on the proofs
  5. Wrapped Fukaya categories of Liouville manifolds
  6. Preliminary wrapped Fukaya categories
  7. Actions of symplectomorphisms
  8. Gradings and orientations
  9. The conifold smoothing
  10. Descriptions of the conifold smoothing $X$
  11. Exact Lagrangian submanifolds
  12. Gradings
  13. Symplectomorphisms of $X$: first considerations
  14. The $\operatorname{PBr}_3$ action
  15. $K$-theoretic actions of symplectomorphism groups
  16. ...and 16 more sections

Key Result

Theorem 1.1

There is a split injection $\mathbb{Z}^{\ast \infty} \to \pi_0\operatorname{Symp}_{c}(X,\omega)$, where $\mathbb{Z}^{\ast \infty}$ denotes the free group on countably infinitely many generators. In particular, the symplectic mapping class group $\pi_0\operatorname{Symp}_{c}(X,\omega)$ is infinitely

Figures (5)

  • Figure 1: The toric compactification of $X$; blow up the thickened black edges on the cube for $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$. This slices off wedges of the corresponding moment polytope. Two of the four $\mathbb{F}_1$-boundary components of the result have been shaded.
  • Figure 2: Lagrangian matching spheres $S_i$ (with matching paths $\gamma_i$) and Lagrangian discs $L_i$ (see Theorem \ref{['thm:hms-equivalence']}).
  • Figure 3: For a choice of $\gamma$, visualisation of the associated full right-handed twist $\rho$, as an isotopy of $\mathbb{C}^\ast$.
  • Figure 4: The generators for $\operatorname{PBr}_3$
  • Figure 5: Matching spheres in $(X, \omega_{a,b})$ and flux values for the vanishing tori (with respect to the upper half plane). $S_1$ is in grey in each diagram to visualise intersection points.

Theorems & Definitions (130)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 120 more