Related papers: Spectral Floer theory and tangential structures

We provide an explicit formula for localizing $A^1$-homotopy invariants of topological Fukaya categories of marked surfaces. Following a proposal of Kontsevich, this differential $\mathbb Z$-graded category is defined as global sections of…

Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;\Lambda) for one Lagrangian or…

This is an expository article on the A-side of Kontsevich's Homological Mirror Symmetry conjecture. We give first a self-contained study of $A_\infty$-categories and their homological algebra, and later restrict to Fukaya categories, with…

Let $(M,\omega)$ be a symplectic manifold compact or convex at infinity. Consider a closed Lagrangian submanifold $L$ such that $\omega |_{\pi_2(M,L)}=0$ and $\mu|_{\pi_2(M,L)}=0$, where $\mu$ is the Maslov index. Given any Lagrangian…

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…

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…

We introduce an $A_\infty$ map from the cubical chain complex of the based loop space of Lagrangian submanifolds with Legendrian boundary in a Liouville Manifold $C_{*}(\Omega_{L} \mathcal{L}\mathit{ag})$ to wrapped Floer cohomology of…

We show: the Floer homology over the Novikov ring of (nonexact!) rational Lagrangians in an (nonexact!) integral symplectic manifold can be computed in terms of exact Lagrangians in an exact filling of the prequantization bundle. As a…

For a symplectic manifold satisfying some topological condition,we define a special class of modules over the deformation quantization algebra. For any two such modules we construct an infinity local system of morphisms. We construct such…

We prove several combinatorial results on path algebras over discrete structures related to directed graphs. These results are motivated by Morse theory on a manifold with boundary and, more generally, by Floer theory on a configuration…

We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable $\infty$-categories. Building on this, we study the relative Waldhausen S-construction $S_\bullet(F)$ of a spherical functor $F$ and…

We define partial quasi-morphisms on the group of Hamiltonian diffeomorphisms of the cotangent bundle using the spectral invariants in Lagrangian Floer homology with conormal boundary conditions, where the product compatible with the PSS…

A symplectic manifold gives rise to a triangulated A-infinity category, the derived Fukaya category, which encodes information on Lagrangian submanifolds and dynamics as probed by Floer cohomology. This survey aims to give some insight into…

We establish a novel relation between the cluster categories associated with marked surfaces and the topological Fukaya categories of the surfaces. We consider a generalization of the triangulated cluster category of the surface by a…

This paper is an introduction to Homological Mirror Symmetry, derived categories, and topological D-branes aimed mainly at a mathematical audience. In the paper we explain the physicists' viewpoint of the Mirror Phenomenon, its relation to…

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper "Symplectic topology as the geometry of generating functions," they have been defined in various contexts, mainly via…

This is a survey on two closely related subjects. First, we review the study of topological structure of `finite type' components of spaces of Bridgeland's stability conditions on triangulated categories. The key is to understand…

We recapture Douglas' framework for twisted parametrized stable homotopy theory in the language of $\infty$- categories. A twisted spectrum is essentially a section of a bundle of presentable stable $\infty$-categories whose fiber is the…

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.

In this paper, we study a family of symplectic manifolds introduced by Woodward. These manifolds belong to the broader class of \emph{multiplicity-free} Hamiltonian $G$-manifolds, a generalization of toric manifolds for non-abelian…