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Orbifold Hamiltonian Floer theory for global quotients

Cheuk Yu Mak, Sobhan Seyfaddini, Ivan Smith

TL;DR

This work develops bulk-deformed orbifold Hamiltonian Floer theory for global quotient orbifolds Y=[X/Γ], introducing HF(H_Y; 𝔟) and spectral invariants c^𝔟(x,−) via a robust global chart framework. Central innovations include global Kuranishi charts for orbifold moduli spaces, an ordered marked flow category enriched by global-chart data, and a zig-zag technique connecting boundary strata to fibre-product charts, enabling coherent perturbations and chain complexes. The authors establish an additive isomorphism with orbifold quantum cohomology, construct a bulk-deformed pair-of-pants product, and prove spectral invariants satisfy key properties, all in a presentation-independent, intrinsic manner. These results extend the Chen–Ruan theory to a Hamiltonian Floer context with bulk insertions, and provide a foundation for applications to dynamics on symmetric product orbifolds and beyond.

Abstract

We construct bulk-deformed orbifold Hamiltonian Floer theory for a global quotient orbifold, that is the quotient of a smooth closed symplectic manifold by a finite group acting faithfully via symplectomorphisms. The moduli spaces define an `ordered marked Flow category', which we equip with a coherent presentation via derived orbifolds. The global charts for orbifold Floer cylinders are built from moduli spaces of holomorphic curves in a quotient of projective space by a free action of the given finite group.

Orbifold Hamiltonian Floer theory for global quotients

TL;DR

This work develops bulk-deformed orbifold Hamiltonian Floer theory for global quotient orbifolds Y=[X/Γ], introducing HF(H_Y; 𝔟) and spectral invariants c^𝔟(x,−) via a robust global chart framework. Central innovations include global Kuranishi charts for orbifold moduli spaces, an ordered marked flow category enriched by global-chart data, and a zig-zag technique connecting boundary strata to fibre-product charts, enabling coherent perturbations and chain complexes. The authors establish an additive isomorphism with orbifold quantum cohomology, construct a bulk-deformed pair-of-pants product, and prove spectral invariants satisfy key properties, all in a presentation-independent, intrinsic manner. These results extend the Chen–Ruan theory to a Hamiltonian Floer context with bulk insertions, and provide a foundation for applications to dynamics on symmetric product orbifolds and beyond.

Abstract

We construct bulk-deformed orbifold Hamiltonian Floer theory for a global quotient orbifold, that is the quotient of a smooth closed symplectic manifold by a finite group acting faithfully via symplectomorphisms. The moduli spaces define an `ordered marked Flow category', which we equip with a coherent presentation via derived orbifolds. The global charts for orbifold Floer cylinders are built from moduli spaces of holomorphic curves in a quotient of projective space by a free action of the given finite group.

Paper Structure

This paper contains 32 sections, 54 theorems, 138 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Orbifold Quantum cohomology and Morse cohomology
  3. Orbifold cohomology
  4. Representable orbifold maps and admissible covers
  5. Pseudoholomorphic representable orbifold stable maps
  6. Orbifold quantum product
  7. Orbifold Morse cohomology
  8. Revisiting the quantum product using global charts
  9. Global Kuranishi charts
  10. Global charts for moduli spaces of stable maps
  11. Moduli of framed curves
  12. A polarizing line bundle and another $\Gamma$ representation
  13. Global chart
  14. Flow category generalities
  15. Ordered marked flow category
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $Y = [X / \Gamma]$ be a global quotient orbifold, and where $PD$ stands for the Poincaré dual. Then for each $H_Y \in C^{\infty}(Y \times S^1;\mathbb{R})$ that is non-degenerate (in the sense that $1$ is not an eigenvalue of the linearised return map for every $1$-periodic orbit of $H_Y$) and each $\omega_Y$-compatible almost complex structure $J_Y$,

Figures (4)

  • Figure 1: Boundary strata and boundary charts
  • Figure 2: Fibre product structure at the boundary
  • Figure 3: The differential counts zeroes of the interpolating section in the homotopy cube (cf. Figure \ref{['fig:externalcoherent']}). Black dots represent zeroes.
  • Figure 4: From left to right

Theorems & Definitions (127)

  • Theorem 1.1
  • Theorem 1.3
  • Example 2.2
  • Example 2.3
  • Theorem 2.4: OrbifoldBook, Theorem 2.45
  • Proposition 2.9: Chen-RuanAV
  • Proposition 2.12
  • proof
  • Lemma 2.13: Bao-Lawson, Bao-Lawson
  • Definition 3.3
  • ...and 117 more