Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Franks' dichotomy for toric manifolds, Hofer-Zehnder conjecture, and gauged linear sigma model

Shaoyun Bai, Guangbo Xu

TL;DR

The work proves a broad Franks-type dichotomy for Hamiltonian dynamics on compact toric symplectic manifolds by linking homological fixed-point counts to an eventual proliferation of periodic points. Central to the approach is the gauged linear sigma model (GLSM), which recasts pseudoholomorphic curve counts as integer vortex counts with upstairs–downstairs correspondence, enabling transversality without virtual perturbations and admitting bulk deformations. By leveraging mirror symmetry to achieve generic semisimplicity of bulk-deformed quantum cohomology and coupling this with a mod p framework, the authors establish uniform bounds on boundary depth and demonstrate linear growth of total bar length under prime iterations, yielding the Hofer–Zehnder dichotomy in the toric setting. They also develop a comprehensive algebraic and Floer-theoretic toolkit—Floer–Novikov complexes, persistence modules, reduced barcodes, and Z/p-equivariant theory—bridging symplectic topology, algebraic geometry, and dynamical consequences for toric quotients. The results illuminate deep connections between mirror symmetry and Hamiltonian dynamics and suggest potential extensions to broader symplectic quotients and complete intersections using GLSM techniques and integral Floer theories.

Abstract

We prove that for any compact toric symplectic manifold, if a Hamiltonian diffeomorphism admits more fixed points, counted homologically, than the total Betti number, then it has infinitely many simple periodic points. This provides a vast generalization of Franks' famous two or infinity dichotomy for periodic orbits of area-preserving diffeomorphisms on the two-sphere, and establishes a conjecture attributed to Hofer-Zehnder in the case of toric manifolds. The key novelty is the application of gauged linear sigma model and its bulk deformations to the study of Hamiltonian dynamics of symplectic quotients.

Franks' dichotomy for toric manifolds, Hofer-Zehnder conjecture, and gauged linear sigma model

TL;DR

The work proves a broad Franks-type dichotomy for Hamiltonian dynamics on compact toric symplectic manifolds by linking homological fixed-point counts to an eventual proliferation of periodic points. Central to the approach is the gauged linear sigma model (GLSM), which recasts pseudoholomorphic curve counts as integer vortex counts with upstairs–downstairs correspondence, enabling transversality without virtual perturbations and admitting bulk deformations. By leveraging mirror symmetry to achieve generic semisimplicity of bulk-deformed quantum cohomology and coupling this with a mod p framework, the authors establish uniform bounds on boundary depth and demonstrate linear growth of total bar length under prime iterations, yielding the Hofer–Zehnder dichotomy in the toric setting. They also develop a comprehensive algebraic and Floer-theoretic toolkit—Floer–Novikov complexes, persistence modules, reduced barcodes, and Z/p-equivariant theory—bridging symplectic topology, algebraic geometry, and dynamical consequences for toric quotients. The results illuminate deep connections between mirror symmetry and Hamiltonian dynamics and suggest potential extensions to broader symplectic quotients and complete intersections using GLSM techniques and integral Floer theories.

Abstract

We prove that for any compact toric symplectic manifold, if a Hamiltonian diffeomorphism admits more fixed points, counted homologically, than the total Betti number, then it has infinitely many simple periodic points. This provides a vast generalization of Franks' famous two or infinity dichotomy for periodic orbits of area-preserving diffeomorphisms on the two-sphere, and establishes a conjecture attributed to Hofer-Zehnder in the case of toric manifolds. The key novelty is the application of gauged linear sigma model and its bulk deformations to the study of Hamiltonian dynamics of symplectic quotients.
Paper Structure (99 sections, 84 theorems, 431 equations, 6 figures)

This paper contains 99 sections, 84 theorems, 431 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Pseudoholomorphic curves vs. vortices
  4. Pseudoholomorphic curves and the counting problem
  5. Mathematical theory of the GLSM
  6. Exact relation between upstairs and downstairs
  7. Outline of the proof
  8. Hamiltonian Floer theory via GLSM
  9. Mirror symmetry via GLSM and generic semisimplicity
  10. Boundary depth estimate and linear growth of total bar length
  11. Outlook and speculations
  12. Outline of the paper
  13. Geometric preliminaries
  14. Toric manifolds as symplectic quotients
  15. Symplectic vortex equation
  16. ...and 84 more sections

Key Result

Theorem A

Let $X$ be a compact symplectic toric manifold and $\phi: X \to X$ be a Hamiltonian diffeomorphism with isolated fixed points. If the number of fixed points of $\phi$, counted homologically, is greater than the total rank of homology of $X$, then $\phi$ has infinitely many simple periodic points.

Figures (6)

  • Figure 1:
  • Figure 2: Forgetting a weighted input.
  • Figure 3: A closed-open domain. The component with a cylindrical end is the component $C^{\rm Ham}$.
  • Figure 4: True boundaries of a 1-dimensional moduli space. The pictures represent the case when the weakly bounding cochain is zero and the insertions are all variables of the Hochschild cochains. One can draw the picture for general cases by arbitrarily inserting weakly bounding cochains on the boundary.
  • Figure 5: The moduli space of balanced treed disks with two interior inputs and one boundary inputs. This moduli space is parametrized by one variable $\rho \in [-1, 1]$.
  • ...and 1 more figures

Theorems & Definitions (191)

  • Theorem A
  • Conjecture 1.1: Hofer--Zehnder conjecture
  • Theorem B
  • Corollary 1.2
  • Theorem C
  • Theorem D
  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 181 more