Related papers: Curved String Topology and Tangential Fukaya Categ…
We give an introduction to partially wrapped Fukaya categories of surfaces with orbifold singularities. Dissecting an orbifold surface $\mathbf S$ into polygons, certain dissections give rise to formal generators, inducing a triangulated…
This paper establishes an isomorphism between endomorphism algebras from the wrapped Fukaya category of a type of punctured surface, and the class of A-infinity algebras related to bordered knot Floer homology, called star algebras, which…
The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side of mirror symmetry) with the category of holonomic modules over the quantized algebra of functions on the same symplectic manifold. We…
We embed triangulated categories defined by quivers with potential arising from ideal triangulations of marked bordered surfaces into Fukaya categories of quasi-projective 3-folds associated to meromorphic quadratic differentials. Together…
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, we develop a Fukaya $A_\infty$-category…
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 these categories provide homological mirrors…
The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya…
A sequel to arXiv:1111.1460, this paper elaborates on some of the themes in the above paper. Connections to Symplectic Field Theory (SFT) and mirror symmetry are explored.
Let $\mathfrak{Fuk}(T^*M)$ be the Fukaya category in the Fukaya's immersed Lagrangian Floer theory \cite{fukaya:immersed} which is generated by immersed Lagrangian submanifolds with clean self-intersections. This category is monoidal in…
In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from…
Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a…
We study the partially wrapped Fukaya category of a surface with boundary with an action of a group of order two. Inspired by skew-group algebras and categories, we define the notion of a skew-group $A_\infty$-category and let it play the…
This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify…
We give a complete description of partially wrapped Fukaya categories of graded orbifold surfaces with stops. We show that a construction via global sections of a natural cosheaf of A$_\infty$ categories on a Lagrangian core of the surface…
To a simple polarized hyperplane arrangement (not necessarily cyclic) $\mathbb{V}$, one can associate a stopped Liouville manifold (equivalently, a Liouville sector) $\left(M(\mathbb{V}),\xi\right)$, where $M(\mathbb{V})$ is the complement…
To a conical symplectic resolution with Hamiltonian torus action, Braden--Proudfoot--Licata--Webster associate a category O, defined using deformation quantization (DQ) modules. It has long been expected, though not stated precisely in the…
Operads often arise from geometry. The standard $A_\infty$ operad can be derived from the cellular chains on the Stasheff associahedra, and an $A_\infty$ algebra is an algebra over this operad. The notion of an $\mathbf{fc}$-multicategory,…
We define a class of non-compact Fano toric manifolds, called admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed…
Plumbing spaces have drawn significant attention among symplectic topologists due to their natural occurrence as examples of Weinstein manifolds. In our paper, we provide a general formula for the wrapped Fukaya category of plumbings (with…
We give a complete description of the A$_\infty$ deformation theory of partially wrapped Fukaya categories of graded surfaces. We show that any abstract A$_\infty$ deformation is "geometric", namely it is equivalent to the partially wrapped…