Homological mirror symmetry of $\mathbb{F}_1$ via Morse homotopy

通过Morse-Bott理论的等变Floer同伦

Equivariant Floer homotopy via Morse-Bott theory

本研究将Cohen-Jones-Segal构造推广至Morse-Bott情形，定义流范畴Morse-Bott类似物的标架并关联稳定同伦型，进而在Floer理论中构造球谱系数辛上同调的圆周等变模型，证明余切丛零截面情形下的等变Viterbo同构定理。

Planar Homological Mirror Symmetry

In this article, we formulate a planar limited version of the B-side in homological mirror symmetry that formularizes Chern-Simons-type topological open string field theory using homotopy associative algebra ($A_{\infty}$ algebra). This formulation is based on the works by Dijkgraaf and Vafa. We show that our formularization includes gravity/gauge theory correspondence which originates in the AdS/CFT duality of Dijkgraaf-Vafa theory.

复射影平面的SYZ镜像的多值Morse同伦

Multi-valued Morse homotopy for the SYZ mirror of the complex projective plane

提出多值Morse同伦范畴的定义，并证明其全子范畴与复射影平面上的全纯向量丛范畴的A_∞等价，应用于全纯切丛的整体截面的镜像描述。

On homotopy Poisson actions and reduction of symplectic Q-manifolds

We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.

Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space

We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make ex…

Canonical models of filtered $A_\infty$-algebras and Morse complexes

The purpose of this paper is two-fold. First we explain the construction of the canonical model of filtered $A_\infty$-algebras given in the authors' book [FOOO]. The canonical model plays a crucial role in the study of Lagrangian Floer theory on toric manifolds in our recent papers, arXiv:0802.1703 and arXiv:0810.5654. Then using a variation of the arguments used in that construction, we define a natural filtered $A_\infty$-structure on the Morse complex of a Morse function and its $A_\infty$ homotopy to the $A_\infty$-algebras on a Lagrangian submanifold constructed in [FOOO]. The correspond…

Homological mirror symmetry and torus fibrations

In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point of view on the origin of torus fibrations is based on the standard differential-geometric picture of collapsing Riemannian manifolds as well as analogous considerations for Conformal Field Theories. It seems to give a description of mirror manifolds much more transparent than th…

重探Morse-Bott理论中的Cohen-Jones-Segal构造

Revisiting the Cohen-Jones-Segal construction in Morse-Bott theory

该研究重探Morse-Bott理论中的Cohen-Jones-Segal构造，严格构造稳定法标架并验证其稳定同伦型，拓展了相容标架对应的Thom谱结果，还补充了模空间角结构、延拓映射构造及标架条件放宽等相关内容。

An $A_\infty$-structure for lines in a plane

As an explicit example of an $A_\infty$-structure associated to geometry, we construct an $A_\infty$-structure for a Fukaya category of finitely many lines (Lagrangians) in $\R^2$, ie., we define also {\em non-transversal} $A_\infty$-products. This construction is motivated by homological mirror symmetry of (two-)tori, where $\R^2$ is the covering space of a two-torus. The strategy is based on an algebraic reformulation of Morse homotopy theory through homological perturbation theory (HPT) as discussed by Kontsevich and Soibelman in math.SG/0011041, where we introduce a special DG category whi…

The Picard-Lefschetz theory of complexified Morse functions

Given a closed manifold N and a self-indexing Morse function f: N --> R with up to four distinct Morse indices, we construct a symplectic Lefschetz fibration pi: E --> C which models the complexification of f on the disk cotangent bundle, f_C : D(T*N) --> C, when f is real analytic. By construction, pi: E --> C comes with an explicit regular fiber M and vanishing spheres V_1,...,V_m in M, one for each critical point of f. Our main result is that (E,pi) is a good model for the complexification (D(T*N),f_C) in the sense that N embeds in E as an exact Lagrangian submanifold, and in addition, pi|N…

Fukaya's conjecture on $S^1$-equivariant de Rham complex

Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of $A_\infty$ structures. We extend and prove Fukaya's conjecture relating this Witten's deformed equivariant de Rham complexes, to a new Morse theoretical $A_\infty$ complexes defined by counting gradient trees with jumping which are closely related to the $\mathbb{S}^1$ equivariant symplectic cohomology proposed by S…

Morse 流范畴作为出口路径范畴

Morse flow categories as exit path categories

本文证明光滑闭流形上 Morse-Smale 对 $(f,ξ)$ 诱导的拓扑流范畴 $\mathcal{M}$ 作为 $\infty$-范畴等价于 Lurie 的关于稳定流形分层的出口路径 $\infty$-范畴 $\mathrm{Sing}_A(X)$。