Open–Closed Maps in Symplectic Topology

• Open–closed maps are morphisms connecting open string invariants (Fukaya categories) with closed string data (quantum/symplectic cohomology) in symplectic topology.
• They are constructed by counting and integrating over pseudoholomorphic disks, ensuring compatibility with Gromov–Witten axioms and field theory structures.
• These maps support split-generation, mirror symmetry verifications, and extensions to spectral, relative, and operadic frameworks.

Open–closed maps are fundamental morphisms in symplectic topology and topological field theory which mediate between categorical invariants associated to "open string" data (e.g., the Fukaya category or its Hochschild/cyclic (co)homology) and "closed string" invariants (quantum or symplectic cohomology, Gromov–Witten theory, or more refined homotopical or motivic targets). These maps encode the interaction and duality between the open and closed sectors at both geometric and algebraic levels, underpinning a range of exact results and split-generation theorems in Floer theory, as well as providing the interface for mirror symmetry and field-theoretic structures.

1. Fundamental Definition and Frameworks

Given a symplectic manifold $(X,\omega)$ (possibly noncompact or relative to a divisor), the open–closed map is a chain-level or spectral map

$\mathrm{OC} : HH_*\big(\Fuk(X)\big) \to QH^*(X)$

or in S¹-equivariant/cyclic/topological refinements,

$\mathcal{OC}^{S^1} : HC^-_*\big(\Fuk(X)\big) \to QH^*(X)[[u]]$

mapping Hochschild (resp. negative cyclic) homology of the (wrapped or relative/big) Fukaya category to quantum or symplectic cohomology. In the context of Liouville domains equipped with additional tangential structures and stable homotopy orientations, this generalizes to a spectral morphism

$$\mathcal{OC} : \pi_*\,THH^R\big(\mathcal{F}(X;\Psi)\big) \to \Omega_{*+d}^{E_\Psi,\mathrm{OC}}(X;\phi)$$

targeting twisted bordism or generalized cohomology determined by the Thom spectrum $R$ associated to a tangential structure $\Psi = (\Theta\to\Phi)$ (Porcelli et al., 25 Sep 2025).

The map is constructed geometrically by counting parameterized moduli spaces of pseudoholomorphic disks in $X$ (with or without boundary), whose boundary conditions and insertions correspond to open (Lagrangian, brane) and closed (bulk) sectors, with extra decorations or constraints encoding local systems, bulk deformations, and higher genus effects (Giterman et al., 7 Sep 2025, Sheridan, 3 Nov 2025, Hirschi et al., 8 Jan 2025).

2. Algebraic and Homotopical Construction

In the standard (non-spectral) $A_\infty$ setting, the open–closed map arises from a chain map built via moduli of marked holomorphic disks:

• For a curved $A_\infty$-category with objects given by Lagrangians (equipped with local systems or bounding cochains), and morphisms given by Floer complexes or de Rham models, the chain-level open–closed map is

$OC: CC_*(C) \to H^*(X;R)$

defined explicitly by evaluating forms or cochains at boundary and interior marked points and integrating over the moduli space of stable disks (Giterman et al., 7 Sep 2025, Sheridan, 3 Nov 2025).

Operadic approaches encode the entire spectrum of open–closed and closed–open operations using the language of (dg-)operads and multicategories, notably through the operadic cubical associahedron and the machinery of operadic Floer theory. These constructions impose the $A_\infty$-structure and guarantee compatibility with Gromov–Witten and sewing axioms (Chen, 2024).

In spectral and stable homotopy theoretic settings, the map is constructed at the level of ring spectra and Thom spectra, with twisting by higher local system data:

• The essential mechanism involves a twisting construction on flow categories, a realization of the trace map $BGL_1(R) \subseteq K(R) \to R$ (Schlichtkrull's trace), and an $\eta$-correction depending on the stable Hopf map $\eta\in\pi_1^{st}$ (Porcelli et al., 25 Sep 2025).

3. Structural Properties and Theorems

The main algebraic structures and theorems around open–closed maps include:

• Module structure: The open–closed map is a $QH^*(X)$-module homomorphism. In the spectral setting, multiplicativity is realized via the cap product and the multiplicative structure in the Thom spectrum.
• Naturality: The map is compatible with symplectic embeddings and Hamiltonian isotopies of Lagrangians and their branes (including local systems) (Porcelli et al., 25 Sep 2025).
• Eigenvalue and block decomposition: In monotone and semisimple settings, both domain (split Fukaya category) and target ($QH^*(X)$, $SH^*(X)$) admit decompositions indexed by eigenvalues; $\mathrm{OC}$ respects this decomposition (Ritter et al., 2012, Hugtenburg, 2022).
• Split-generation and generation criteria: Abouzaid’s criterion asserts that if the unit of $QH^*(X)$ lies in the image of OC from the Hochschild homology of a subcategory, then that subcategory split-generates the Fukaya category—extended to semisimple blocks and to the big/relative category (Sheridan, 3 Nov 2025).
• Cardy condition: The open–closed and closed–open maps fit into a commutative diagram relating Hochschild (co)homologies and quantum cohomology, as realized in the Cardy formula (Sheridan, 3 Nov 2025).
• Torsion corrections: In the spectral version with nontrivial local systems, the difference between the class $[(L,\xi)]$ and $[L]$ is a universal two-torsion element in $R^0(L)^\times$, arising from the action of $\eta$ on the local system—vanishing upon inverting $2$ (Porcelli et al., 25 Sep 2025).

4. Refined and Equivariant Structures

Cyclic and $S^1$-equivariant maps: Taking into account cyclic or $S^1$-equivariant structures refines the map to

$\mathcal{OC}^{S^1} : HC^-_*(\Fuk(X)) \to QH^*(X)[[u]]$

which respects natural $u$-connections corresponding to Dubrovin's connection on quantum cohomology. This map intertwines the categorical TEP-structure with the enumerative $TEP$-structure, and in the semisimple case, the Givental–Teleman $R$-matrix appears as the difference between categorical and quantum splittings (Hugtenburg, 2022).

Finite group actions and Gysin comparisons: Over fields of characteristic $p$, the cyclic open–closed map connects with $\mathbb{Z}/p$-equivariant analogues via operadic and homotopy-theoretic Gysin comparisons, leading to new connections with quantum Steenrod operations and $p$-adic enumerative invariants (Chen, 2024).

5. Applications in Symplectic Topology and Mirror Symmetry

Open–closed maps are central in:

• Split-generation and non-formality proofs: For instance, injectivity of the map implies split-generation for monotone and Lagrangian-orbifold summands (Ritter et al., 2012, Tonkonog, 2015).
• Open–closed mirror symmetry: The isomorphism between the Hochschild (co)homology of the Fukaya category and quantum cohomology as realized via open–closed maps verifies mirror symmetry predictions, and matches enumerative invariants via Costello's machinery (Hirschi et al., 8 Jan 2025).
• Obstruction theory and open gravitational descendants: Open–closed maps detect obstructions and support the definition of open gravitational descendants via coupling with Chern and $\psi$-classes, with Gromov–Witten–type axioms holding at the chain level (Giterman et al., 7 Sep 2025).
• Refinements in the presence of spectral local systems: Two-torsion $\eta$ corrections in the spectral setting provide a universal measure for the dependence of bordism classes on local systems (Porcelli et al., 25 Sep 2025).

6. Extensions: Spectral, Relative, and Quantum Map Variants

Multiple generalizations of open–closed maps address different geometric settings:

• Spectral and bordism-theoretic: The spectral version encodes universal torsion corrections linked to the Hopf map and is essential for understanding the higher-categorical structure of open–closed dualities (Porcelli et al., 25 Sep 2025).
• Relative/big Fukaya category: The construction extends to varieties relative to divisors, using bounding cochains and filtrations; chain-level open–closed maps are adapted to account for bulk and relative data, ensuring compatibility with compactness and gluing (Sheridan, 3 Nov 2025).
• Quantum maps and mirror symmetry: Open–closed maps support the construction of open–closed mirror maps whose Taylor expansions (in mirror coordinates) have integral coefficients, underpinning the arithmetic of mirror symmetry (Zhou, 2010).

7. Axioms, Operadic and Field Theory Structures

Open–closed maps are governed by:

• Gromov–Witten–type axioms: These include the fundamental class axiom, divisor axiom, energy–zero axiom, degree and splitting axioms mirroring those satisfied by closed Gromov–Witten invariants (Giterman et al., 7 Sep 2025).
• Sewing and field theory axioms: In the open–closed Deligne–Mumford field theory framework, composition, gluing, and compatibilities of the moduli spaces are axiomatized at the level of symmetric monoidal functors from the open–closed category (with prescribed curvature and truncation properties) to complexes over the Novikov ring (Hirschi et al., 8 Jan 2025).
• Operadic formulations: The full system of chain-level open–closed and closed–open operations is operadically encoded, allowing for the systematic construction and comparison of equivariant, cyclic, and field-theoretic enhancements (Chen, 2024).

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Key references: (Porcelli et al., 25 Sep 2025, Giterman et al., 7 Sep 2025, Sheridan, 3 Nov 2025, Hirschi et al., 8 Jan 2025, Ritter et al., 2012, Chen, 2024, Hugtenburg, 2022, Tonkonog, 2015, Zhou, 2010).