Family Floer Cohomology Overview

• Family Floer Cohomology is a framework that constructs Floer invariants through equivariant techniques on symplectic manifolds with rich algebraic structures.
• It uses filtration by ideals and persistence modules to capture the birth and death of quantum cohomology classes under varying Hamiltonian actions.
• Explicit algebraic presentations in toric Fano and Calabi–Yau settings enable practical computations and deeper insights into mirror symmetry.

Family Floer cohomology is a framework for constructing and analyzing Floer-theoretic invariants in families, particularly on symplectic manifolds with additional algebraic structure such as toric or semiprojective toric varieties equipped with $\mathbb{C}^*$-actions. The interplay between Floer theory, the algebraic geometry of (quantum) cohomology, and the action of symmetry groups like $\mathbb{C}^*$ or $S^1$ yields a rich structure of filtrations, localizations, and explicit presentations of (quantum and symplectic) cohomology, both in equivariant and non-equivariant settings. Such constructions not only produce new polynomials and invariants on ordinary and quantum cohomology that reflect the underlying Floer-theoretic geometry but also encode “family” behaviors—such as persistence modules—in the presence of Hamiltonian group actions. The case of Fano and Calabi–Yau (CY) toric manifolds features particularly explicit algebraic descriptions, which are crucial for both computation and conceptual understanding.

1. Floer Theory on Symplectic $\mathbb{C}^*$-Manifolds

Consider a non-compact symplectic manifold $(Y, \omega)$ admitting one or several commuting Hamiltonian $\mathbb{C}^*$-actions. For a given action $\psi$ (with $S^1$-Hamiltonian part and moment map $H$), one defines Floer complexes $CF^*(H_\lambda)$ associated to Hamiltonians $H_\lambda$ that are linear with fixed “slope” $\lambda$ outside a compact set. The symplectic cohomology

$SH^*(Y) = \varinjlim_\lambda HF^*(H_\lambda)$

is constructed as a direct limit over such Hamiltonians.

A central object is the “rotation class” $Q_\psi \in QH^*(Y)$, defined via the action-induced Floer continuation maps and counted through trajectories equivariant with respect to the $\mathbb{C}^*$-action. There is an algebra isomorphism

$$c^* : QH^*(Y) \longrightarrow SH^*(Y) \cong QH^*(Y)_{Q_{\psi}^{-1}}$$

where localization by $Q_\psi$ inverts the effect of the action at the level of (quantum/symplectic) cohomology. For multiple actions, composition laws such as

$Q_{v'} * Q_{v''} = T^{a(v',v'')} Q_{v'+v''}$

track the minimal values of their respective moment maps and the Novikov weight.

This Floer-theoretic machinery underpins the subsequent algebraic and filtration structures.

2. Construction of Filtrations by Ideals and Persistence Modules

The $\mathbb{C}^*$-action induces a canonical filtration on quantum cohomology by ideals. For a chosen Hamiltonian slope parameter $p$, the filtration is defined by

$$F^p := \bigcap_{\lambda \geq p} \ker(c^*_\lambda : QH^*(Y) \to HF^*(H_\lambda))$$

where $c^*_\lambda$ are the continuation-induced maps at varying slopes.

This gives rise to a descending sequence of ideals

$F^0 \supset F^{p_1} \supset F^{p_2} \supset \dots$

that reflect the “energy thresholds” at which quantum cohomology classes become detectable by Floer theory. The graded pieces, or the dimensions of $F^p/F^{p'}$, can be organized into a persistence module (Editor's term), whose structure encodes the birth and death of Floer-theoretic invariants as a function of the action filtration. This directly connects to periodic barcodes in persistence homology, providing a Floer-theoretic analog of Morse-theoretic stratifications.

3. Equivariant Floer Theory and Hilbert–Poincaré Invariants

In the $S^1$-equivariant context, one works with the modules

$E^-QH^*(Y) \simeq H^*(Y)[[u]]$

where $u$ is the equivariant parameter. Here, filtrations by ideals are induced on the equivariant quantum (and symplectic) cohomology as well, and one studies completions such as $E^+QH^*(Y), E^{\infty}QH^*(Y)$. The graded slices, after passing to $u$-adic completions and considering $u=0$, define classes

$$P_j^p = (F_j^p \cap u^j E^-QH^*(Y)) / (u F_{j-1}^p \cap u^j E^-QH^*(Y))$$

whose generating function (Hilbert–Poincaré polynomial)

$f_p(t) = \sum_{j \ge 0} \dim_K P_j^p t^j$

provides a refined, graded measure of the filtered invariants at level $p$. These polynomials generalize classical invariants and capture nontrivial “quantum” information.

4. Explicit Algebraic Presentations in the Fano and Calabi–Yau Toric Case

For (semi)projective toric manifolds—including Fano and CY—the quantum cohomology ring $QH^*(Y)$ and the symplectic cohomology $SH^*(Y)$ admit concrete presentations. In the Fano (monotone) case:

• $QH^*(Y) \cong K[x_1, \dots, x_r]/\mathcal{J}$,
• $\mathcal{J}$ is the ideal of linear relations and quantum Stanley–Reisner relations.

Rotation classes $Q_v$ parameterize one-parameter subgroups and form a group in $QH^*(Y)$. Symplectic cohomology is a localization:

$SH^*(Y, v) \cong QH^*(Y)\left[(Q_v)^{-1}\right].$

In the CY case, $QH^*(Y)$ is “classical” as a vector space but the rotation classes admit higher Novikov corrections:

$$Q_{e_i} = PD[D_i] + T^{>0} (\text{lower deg terms}),$$

modifying the presentation. In both Fano and CY settings, $SH^*(Y,v)$ is identified with the Jacobian ring of the Landau–Ginzburg superpotential,

$W = \sum T^{-\lambda_i} z^{e_i}.$

These explicit models are critical for calculations in symplectic topology and mirror symmetry.

5. Interpretation via Family Floer Cohomology

The filtration structures are naturally interpreted in the framework of Family Floer Cohomology, as they track the behavior and detectability of cohomology classes parameterized by the $\mathbb{C}^*$-actions and Hamiltonian deformations:

• For families of Hamiltonians or parameterized Lagrangians, the filtration parameter $p$ records critical thresholds at which classes survive or vanish, mirroring the idea of continuity and persistence across a family (persistence module).
• Equivariant Hilbert–Poincaré polynomials serve as algebraic proxies for the behavior of equivariant family Floer invariants as the family parameter (or equivariant scale) varies.
• In the toric setting, the explicit algebraic presentations reflect how Floer-theoretic invariants transform under variation of the $\mathbb{C}^*$-action, tracing out a family-level deformation of quantum and symplectic cohomology.

This approach systematically encodes the evolution of Floer-theoretic data across families and symmetries, precisely the phenomenon central to Family Floer Cohomology.

6. Broader Significance and Connections

These methods not only produce computational tools but also provide conceptual insights:

• The filtration and persistence module structures mirror (and extend) those arising in Morse–Bott theory, but distinctly capture non-topological quantum features.
• In the paper of mirror symmetry, identification of $SH^*(Y)$ with Jacobian rings and the presentation of $QH^*(Y)$ provide the necessary algebraic underpinnings to relate symplectic invariants with functions and sheaves on the mirror.
• The approach applies to a wide class of symplectic manifolds, particularly conical symplectic resolutions and quiver varieties, where filtrations by ideals reflect the geometry of group actions.

This synthesis situates Family Floer Cohomology as not only a generalization but a natural organizing principle for Floer invariants in geometric representation theory and algebraic geometry.

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Summary Table: Floer-Theoretic Constructions and Algebraic Invariants for $\mathbb{C}^*$-Actions

Construction	Description	Setting
$Q_\psi$ (Rotation Class)	Quantum cohomology element via Floer theory	Any $\mathbb{C}^*$-action
Filtration $F^p$	Ideals in $QH^*(Y)$ by Floer detectability	All settings
Persistence module/barcode	Birth–death data of Floer invariants	Family context
$SH^*(Y,v)$	Localized symplectic cohomology	Fano, CY toric manifolds
Hilbert–Poincaré polynomial	Generating function for graded filtered slices	Equivariant context
Explicit algebraic ring	Presentations for $QH^*(Y),SH^*(Y)$	Toric, Fano, CY cases

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The framework described provides a unifying algebraic and Floer-theoretic language for organizing and calculating invariants in families, especially when toric or group symmetries are present. The explicit correspondence between filtrations, persistence modules, and algebraic presentations of cohomology deepens the conceptual and computational reach of Family Floer Cohomology in modern symplectic topology and related fields (Ritter et al., 15 Jan 2025).