Covariantly functorial wrapped Floer theory on Liouville sectors

Ganatra, Sheel; Pardon, John; Shende, Vivek

doi:10.1007/s10240-019-00112-x

Cited by 102 publications

(256 citation statements)

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“…The upshot is that a split-generating subcategory I can be found as the image under ı σ of a split-generating subcategory I ⊂ W(σ). This reduces the problem of split-generation of partially wrapped Fukaya categories with strongly nondegenerate stops to that of fully wrapped Fukaya categories, for which [16] gives a good answer.Concretely, consider the Landau-Ginzburg model (M, W ) = (C 3 , xyz), which is mirror to the pair of pants. Recall from Section 1.2.2 that this gives rise to a stop σ W which is symplectomorphic to the generic fiber of W .…”

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confidence: 97%

“…This allows us to avoid a large-energy homotopy which would fall outside the scope of Lemma 4.32. In the presence of a better confinement lemma, such as [16, Lemma 4.11], one could probably weaken the condition on σ from strong nondegeneracy to ordinary nondegeneracy.On the other hand, it is unlikely that the condition that σ is nondegenerate could be completely eliminated. That said, very little is currently known about degenerate Liouville domains, so producing a counterexample would be difficult at the moment.…”

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On partially wrapped Fukaya categories

Sylvan

2019

Journal of Topology

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We define a new class of symplectic objects called 'stops', which, roughly speaking, are Liouville hypersurfaces in the boundary of a Liouville domain. Locally, these can be viewed as pages of a compatible open book. To a Liouville domain with a collection of disjoint stops, we assign an A∞-category called its partially wrapped Fukaya category. An exact Landau-Ginzburg model gives rise to a stop, and the corresponding partially wrapped Fukaya category is meant to agree with the Fukaya category one is supposed to assign to the Landau-Ginzburg model. As evidence, we prove a formula that relates these partially wrapped Fukaya categories to the wrapped Fukaya category of the underlying Liouville domain. This operation is mirror to removing a divisor. 373is the open-closed map. Here, the action filtration on Hochschild homology is induced from the action filtration on Floer cochain groups; for a precise definition, see Section 4.3.In particular, any punctured Riemann surface other than C and any cotangent bundle is strongly nondegenerate.Remark 1.1. A nondegenerate Liouville domain F is one for which some element e ∈ HH * (W(F )) satisfies OC(e) = 1. This condition is Abouzaid's generation criterion [2], which says that such an F admits a finite collection of Lagrangians which split-generate W(F ). Ganatra further proved that in this case, OC is an isomorphism [15], which in particular implies that e is unique.An approximate version of Theorem 4.21 can be stated as follows: Theorem 1.2. Let M be a Liouville domain, and let σ be a strongly nondegenerate stop in M . Let B ⊂ W σ (M ) be the full subcategory of objects supported near σ. Then the inclusion W σ (M ) → W(M ) induces a fully faithful functorwhere the quotient is a quotient of an A ∞ category by a full subcategory in the sense of .At the most basic level, Theorem 1.2 shows that the partially wrapped Fukaya category, together with B, really knows at least as much as the wrapped Fukaya category. That is, restricting to the zero-filtered part but remembering the stop does not lose information. At an intuitive level, this happens because the 'nice' presentation of W(M ) giving rise to the stop filtration uses a contact form with a large number of canceling Reeb chords. These chords live at different levels in the filtration, so passing to the zero-filtered part results in them no longer canceling. In a more minimal presentation, these chords could be eliminated geometrically.The key ingredient of the proof of Theorem 1.2 is an auxiliary filtration on W(M ), presented as the trivial quotient A = W(M )/B. The benefit of this quotient presentation is that it naturally contains the category A 0 = W σ (M )/B(σ) as the minimally filtered part, which makes it possible to build a homotopy which retracts A onto A 0 . The homotopy itself requires a filtered version of the annulus trick, which was introduced in [2] and extended in a forthcoming paper by Abouzaid and Auroux. Specifically, one factors the identity operation as a composition of a product and a coproduct, where ...

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On partially wrapped Fukaya categories

Sylvan

2019

Journal of Topology

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“…However, this assumption can actually be removed since the Fukaya category

A (π)

can also be defined directly on

E

using a particular class of Hamiltonian perturbations specified in , without passing to the double branched cover

\overset{E}{true}

. Alternatively,

A (π)

can be defined as a version of partially wrapped Fukaya category .…”

Section: Formality Of A∞‐structuresmentioning

confidence: 99%

“…This is because the Fukaya category A(π) is defined to be the Z/2-invariant subcategory of the Fukaya category F( E), where E is a double cover of E branched along the smooth fiber M * . However, this assumption can actually be removed since the Fukaya category A(π) can also be defined directly on E using a particular class of Hamiltonian perturbations specified in [58], without passing to the double branched cover E. Alternatively, A(π) can be defined as a version of partially wrapped Fukaya category [27].…”

Section: Suspension Of a Lefschetz Fibrationmentioning

confidence: 99%

Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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Let M be a Liouville 6‐manifold which is the smooth fiber of a Lefschetz fibration on C4 constructed by suspending a Lefschetz fibration on C3. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category F(M) and the wrapped Fukaya category W(M) are related through A∞‐Koszul duality, by identifying them with cyclic and Calabi–Yau completions of the same quiver algebra. This implies the split‐generation of the compact Fukaya category F(M) by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi‐dilations in the sense of Seidel–Solomon are obtained.

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