Spectral Viterbo Restriction Functor

• Spectral Viterbo restriction functor is a filtered enhancement of the classical Viterbo map, refining symplectic cohomology via action filtrations and persistence modules.
• It translates geometric conditions, such as the absence of short Reeb chords, into categorical and algebraic consequences within wrapped Fukaya and Legendrian homological frameworks.
• The construction employs filtered Floer complexes and A∞-functors, enabling rigorous contradiction arguments that confirm Reeb chord existence and homological epimorphism results.

Searching arXiv for the cited papers to ground the article in the current literature. The spectral Viterbo restriction functor is a filtered enhancement of the Viterbo restriction map, or Viterbo transfer, associated to an exact symplectic embedding of Liouville domains. In the formulation developed in "Lengths of Reeb chords and Viterbo restriction" (Broćić et al., 15 Jun 2026), it is a map

$\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$

defined on filtered symplectic cohomology and refined by the action filtration to persistence-module level. In parallel, work on wrapped Fukaya categories treats Viterbo restriction as an $A_\infty$-functor between ambient and subdomain categories, and also relates it to graph correspondence functors and homological epimorphisms (Sylvan, 2019, Gao, 2020). Across these settings, the construction serves as a bridge from geometric input—especially the absence of short Reeb chords—to algebraic or categorical consequences.

1. Filtered symplectic-cohomological construction

For an exact Liouville embedding $\iota: K \hookrightarrow \Omega$, the Viterbo restriction map is presented as a filtered algebra homomorphism

$\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$

where $c$ and $b$ are slopes, more precisely action thresholds, not in the respective spectra (Broćić et al., 15 Jun 2026). In this formulation, the map is linear and compatible with the product, BV operator, units, and the PSS map. It also respects the decomposition of symplectic cohomology into free homotopy classes.

The filtered character of the construction is essential. Rather than passing immediately to unfiltered symplectic cohomology, the map is defined on filtered groups $SH_c$ and $SH_b$, and therefore induces maps of persistence modules. The action filtration is not an auxiliary bookkeeping device here; it is the mechanism by which the restriction acquires its spectral refinement. This gives a spectral-invariant viewpoint in which one tracks how distinguished classes, including the unit, behave as the action threshold increases.

The analytic construction described in the appendix of (Broćić et al., 15 Jun 2026) proceeds through a canonical identification of subquotients of Floer complexes using action filtrations and the no-escape lemma. Compatibility with the additional algebraic structures is verified by means of Hamiltonian connections, curvature estimates, and limits over auxiliary data such as perturbations and mollifications. In this sense, the restriction map is already a functorial object before any passage to direct limits.

2. Reeb-chord geometry and exact embeddings

The geometric input begins with a Legendrian submanifold $\Lambda \subset \partial \Omega$. If there does not exist a Reeb chord with endpoints on $\Lambda$ of period up to $A_\infty$0, then a geometric construction produces an exact embedding

$A_\infty$1

where $A_\infty$2 is a disk of area $A_\infty$3, and $A_\infty$4 is a small cotangent disk bundle (Broćić et al., 15 Jun 2026). This is the point at which a statement about Reeb dynamics is converted into a statement about Liouville embeddings.

That embedding provides the domain $A_\infty$5 to which the filtered Viterbo restriction is applied. For suitable $A_\infty$6, one obtains

$A_\infty$7

The parameter $A_\infty$8 therefore simultaneously controls the Reeb-chord exclusion window and the action range in which restriction is available.

The paper uses this mechanism to derive Reeb-chord existence results by contradiction. In particular, it shows that if $A_\infty$9 is covered by a product of spheres, for example the $\iota: K \hookrightarrow \Omega$0-torus, then all compact Legendrians $\iota: K \hookrightarrow \Omega$1 admit a Reeb chord for every choice of contact form on $\iota: K \hookrightarrow \Omega$2 (Broćić et al., 15 Jun 2026). The obstruction employed in that case is based on inverting degree-$\iota: K \hookrightarrow \Omega$3 classes in cohomology and is described as similar to the notion of string point invertibility introduced by Egor Shelukhin.

3. $\iota: K \hookrightarrow \Omega$4-invertibility, ideals, and the contradiction mechanism

The core obstruction theory is formulated in terms of $\iota: K \hookrightarrow \Omega$5-invertibility. A Liouville manifold $\iota: K \hookrightarrow \Omega$6 is called $\iota: K \hookrightarrow \Omega$7-invertible if the only $\iota: K \hookrightarrow \Omega$8-invariant ideal of $\iota: K \hookrightarrow \Omega$9 containing the image of $\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$0 under the PSS map is $\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$1 itself (Broćić et al., 15 Jun 2026). The relevant ideal $\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$2 is the smallest $\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$3-invariant ideal containing the degree-$\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$4 image of the PSS map under all operations, including product and BV.

At the filtered level, the construction is made explicit by

$\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$5

$\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$6

and

$\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$7

The associated $\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$8-invertibility capacity is

$\mathfrak{R} : SH_c(\Omega) \to SH_b(K),$9

The contradiction mechanism uses the specific geometry of

$c$0

For this $c$1, $c$2, so $c$3. Because the restriction map is functorial and compatible with the algebraic operations, the same vanishing propagates through the $c$4-invariant ideal generated by the degree-$c$5 classes. Hence, if the unit is generated from those classes under the relevant operations, then $c$6.

The obstruction appears because the filtered Künneth theorem identifies $c$7 with $c$8, and the image of the unit corresponds to the unit in $c$9, which is non-zero by string topology (Broćić et al., 15 Jun 2026). The assumption of no Reeb chords up to length $b$0 therefore forces a vanishing statement incompatible with the cotangent-bundle calculation. The only resolution is that the assumed Reeb-chord exclusion cannot hold.

4. Persistence, spectral models, and Legendrian-homological realizations

The term "spectral" appears in a second, closely related sense in "Lagrangian correspondences and the generalized Viterbo restriction functor" (Gao, 2020). There, both wrapped Floer complexes and linearized Legendrian complexes are filtered by action, and the comparison is established at the chain and homology levels through explicit chain-homotopy equivalences

$b$1

In this framework, the restriction functor admits a spectral realization through linearized Legendrian homology and SFT-type transfer maps.

This paper studies the classical Viterbo restriction functor for a Liouville subdomain $b$2, defined on the strongly exact subcategory $b$3 by

$b$4

and extends it to general exact Lagrangians by introducing object-wise Maurer–Cartan deformations. The extended functor takes the form

$b$5

where the deformed $b$6-operations are

$b$7

The required bounding cochains arise from pseudoholomorphic caps and can be described on the Legendrian side by linearized Legendrian complexes generated by Reeb chords together with Morse critical points. The paper proves that the $b$8-homomorphism realized by restriction coincides, up to homotopy, with the linearized SFT transfer map after passage through the spectral models. It also proves that the graph correspondence functor associated to the graph

$b$9

agrees with the extended restriction functor and specializes to the classical Viterbo restriction functor in the strongly exact case (Gao, 2020).

5. Wrapped Fukaya categorical forms of Viterbo restriction

In "Orlov and Viterbo functors in partially wrapped Fukaya categories" (Sylvan, 2019), the Viterbo transfer map is formulated as a functor

$SH_c$0

where $SH_c$1 denotes the partially wrapped Fukaya category. The construction is carried out in the sectorial framework of Ganatra–Pardon–Shende, using a Viterbo sector inside $SH_c$2 with two stops and Orlov functors associated to those stops.

The central theorem states that if $SH_c$3 is a Liouville subdomain and both $SH_c$4 and the completion of $SH_c$5 satisfy stop removal, then $SH_c$6 is a homological epimorphism (Sylvan, 2019). In particular, its image split-generates $SH_c$7. The statement is expressed module-theoretically by the quasi-isomorphism

$SH_c$8

Under stronger hypotheses, Ganatra–Pardon–Shende prove a localization result. The comparison recorded in (Sylvan, 2019) is:

Geometric setup	Result for the Viterbo functor	Reference
$SH_c$9, and $SH_b$0 all Weinstein	Localization (up to summands)	GPS
$SH_b$1 Weinstein, complement not necessarily Weinstein	Homological epimorphism	(Sylvan, 2019)
General Liouville subdomains, no stop removal hypothesis	No general statement	(Sylvan, 2019)

This categorical version and the filtered symplectic-cohomological version are structurally parallel. In both, restriction is not merely a map between invariants; it preserves additional algebraic structure and serves to transport obstructions from ambient spaces to subdomains.

6. Scope, terminology, and recurrent points of confusion

Within the cited literature, the phrase "spectral Viterbo restriction functor" refers to more than one enhancement of the classical transfer construction. In (Broćić et al., 15 Jun 2026), it denotes a filtered version of symplectic cohomological restriction built from action windows and interpreted as a map of persistence modules. In (Gao, 2020), it also denotes a realization through linearized Legendrian homology, where the transfer is represented on a filtered spectral model and then compared with wrapped Floer theory. These are different implementations of the same general principle rather than competing definitions.

A second recurrent source of confusion is the relation between restriction and localization. The categorical literature distinguishes sharply between a homological epimorphism and a genuine localization. The former is the conclusion of (Sylvan, 2019) under stop-removal hypotheses, whereas localization requires the stronger Weinstein-complement assumptions recorded there. The functor therefore need not be a localization in general, even when it split-generates the target.

A third point concerns the role of Reeb chords. In (Broćić et al., 15 Jun 2026), the absence of Reeb chords does not itself produce the final contradiction. Rather, it enables the exact embedding $SH_b$2, and that embedding makes the filtered restriction map available. The contradiction is then algebraic: the restriction annihilates the image of top-degree classes and, under $SH_b$3-invertibility, forces the unit to vanish, while the filtered Künneth theorem and string topology show that the corresponding unit in the cotangent-bundle model is non-zero.

Taken together, these works present the Viterbo restriction functor as a structure-preserving passage from ambient to subdomain geometry, visible at several levels: filtered symplectic cohomology, persistence modules, linearized Legendrian homology, wrapped Fukaya categories, and graph-correspondence formalisms. The common theme is that the restriction map is strongest when it retains enough filtration, homotopical, or $SH_b$4-structure to convert geometric exclusion statements into rigid algebraic consequences (Broćić et al., 15 Jun 2026, Sylvan, 2019, Gao, 2020).