# On partially wrapped Fukaya categories

@article{Sylvan2019OnPW, title={On partially wrapped Fukaya categories}, author={Zachary Sylvan}, journal={Journal of Topology}, year={2019} }

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 $A_\infty$-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…

## 67 Citations

Orlov and Viterbo functors in partially wrapped Fukaya categories.

- Mathematics
- 2019

We study two functors between (partially) wrapped Fukaya categories. The first is the Orlov functor from the Fukaya category of a stop to the Fukaya category of the ambient sector. We give a…

The Fukaya category pairs with Lagrangian cobordisms

- Mathematics
- 2016

Fix a suitably convex, exact symplectic manifold M. We consider the stable oo-category Lag(M) of non-compact Lagrangians whose (higher) morphisms are (higher) Lagrangian cobordisms between them. We…

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

- Mathematics
- 2017

We prove that the wrapped Fukaya category of any $2n$-dimensional Weinstein manifold (or, more generally, Weinstein sector) $W$ is generated by the unstable manifolds of the index $n$ critical points…

On the Rouquier dimension of wrapped Fukaya categories

- Mathematics
- 2021

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs. We describe upper and lower bounds for this invariant in terms of various quantities of interest in…

Localization and flexibilization in symplectic geometry

- Mathematics
- 2021

We introduce the critical Weinstein category the result of stabilizing the category of Weinstein sectors and inverting subcritical morphisms and construct localizing ‘P-flexibilization’ endofunctors…

Microlocal Morse theory of wrapped Fukaya categories

- Mathematics
- 2018

Consider on the one hand the partially wrapped Fukaya category of a cotangent bundle stopped at an appropriately stratifiable singular isotropic. Consider on the other hand the derived category of…

Geometric and algebraic presentations of Weinstein domains

- Mathematics
- 2019

We prove that geometric intersections between Weinstein handles induce algebraic relations in the wrapped Fukaya category, which we use to study the Grothendieck group. We produce a surjective map…

Partially wrapped Fukaya categories of simplicial skeleta

- Mathematics
- 2017

A class of partially wrapped Fukaya categories in $T^* N$ are proven to be well defined and then studied. In the case of $N$ diffeomorphic to $\mathbb{R}^m \times \mathbb{T}^n$, it is shown that…

Structural results in wrapped Floer theory

- Mathematics
- 2018

We prove results relating different partially wrapped Fukaya categories, including a Kunneth formula, a 'stop removal' result relating partially wrapped Fukaya categories relative to different stops,…

Covariantly functorial wrapped Floer theory on Liouville sectors

- MathematicsPublications mathématiques de l'IHÉS
- 2019

We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors,…

## References

SHOWING 1-10 OF 36 REFERENCES

Symplectic cohomology and duality for the wrapped Fukaya category

- Mathematics
- 2012

Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that…

Quivers, Floer cohomology, and braid group actions

- Mathematics
- 2000

We consider the derived categories of modules over a certain family A_m of graded
rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which
are the Milnor fibres of…

Covariantly functorial wrapped Floer theory on Liouville sectors

- MathematicsPublications mathématiques de l'IHÉS
- 2019

We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors,…

A geometric criterion for generating the Fukaya category

- Mathematics
- 2010

Given a collection of exact Lagrangians in a Liouville manifold, we construct a map from the Hochschild homology of the Fukaya category that they generate to symplectic cohomology. Whenever the…

On the wrapped Fukaya category and based loops

- Mathematics
- 2009

Given an exact relatively Pin Lagrangian embedding Q in a symplectic manifold M, we construct an A-infinity restriction functor from the wrapped Fukaya category of M to the category of modules on the…

Constructible sheaves and the Fukaya category

- Mathematics
- 2006

Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper,…

An open string analogue of Viterbo functoriality

- Mathematics
- 2007

In this preprint, we look at exact Lagrangian submanifolds with Legendrian boundary inside a Liouville domain. The analogue of symplectic cohomology for such submanifolds is called “wrapped Floer…

A cotangent fibre generates the Fukaya category

- Mathematics
- 2010

We prove that the algebra of chains on the based loop space recovers the derived (wrapped) Fukaya category of the cotangent bundle of a closed smooth orientable manifold. The main new idea is the…

Symplectic cohomology and Viterbo's theorem

- Mathematics
- 2013

This is a research monograph on symplectic cohomology (disguised as an advanced graduate textbook), which provides a construction of this version of Hamiltonian Floer cohomology for cotangent bundles…

Wrapped microlocal sheaves on pairs of pants

- Mathematics, Physics
- 2016

Inspired by the geometry of wrapped Fukaya categories, we introduce the notion of wrapped microlocal sheaves. We show that traditional microlocal sheaves are equivalent to functionals on wrapped…