Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to the complex projective n-space minus n+2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a…

大体积处的同调镜像对称

Homological mirror symmetry at large volume

本文将大体积镜像构造为 Weinstein 辛流形，证明其 Fukaya 范畴与环面簇凝聚层范畴等价，并由此推出 Seidel 猜想的 Viterbo 限制局部到整体原理。

代数环面中完全交的同调镜像对称

Homological mirror symmetry for complete intersections in algebraic tori

本文证明所有维度代数环面中完全交的同调镜像对称一个方向，通过分解为高维裤衩对乘积的有限覆盖，建立 Fukaya 范畴与 LG 模型矩阵分解范畴的等价。

关于环带退化内蕴同调镜像对称性

On intrinsic homological mirror symmetry for toric degenerations

本文针对Calabi--Yau流形的极大单能退化，利用固定点Floer上同调构造候选镜像族，并在热带拉格朗日截面存在时建立导出范畴到Fukaya范畴的完全忠实嵌入，验证了若干环带退化情形。

Noncommutative homological mirror functor

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified under…

对曲面对称积的扇形分解

Sectorial Decompositions of Symmetric Products of Surfaces

本文研究了曲面对称积的扇形分解，并用几何方法证明了复二维耳裤的同调镜像对称性。

Mirror symmetry for honeycombs

We prove a homological mirror symmetry equivalence between an $A$-brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a $B$-brane category for a mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the $A$-mo…

Floer homology of cotangent bundles and the loop product

We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas-Sullivan loop product on the singular homology of the loop space of M. We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy-Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique.

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…

Orbifold Kodaira-Spencer 映射与穿孔 Riemann 曲面的闭弦镜像对称性

Orbifold Kodaira-Spencer maps and closed-string mirror symmetry for punctured Riemann surfaces

本文针对带有限阿贝尔群作用的 Weinstein 流形提出等变 TQFT 型镜像构造，并将其用于穿孔 Riemann 曲面，证明相应 orbifold Landau-Ginzburg 模型的闭弦镜像对称性。

Topological Fukaya category and mirror symmetry for punctured surfaces

In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface $Σ$ via the topological Fukaya category. We prove that the topological Fukaya category of $Σ$ is equivalent to the category of matrix factorizations of the mirror LG model $(X,W)$. Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which might be of independent interest.

镜像 P=W 猜想与扩展的 Fano/Landau-Ginzburg 对应

Mirror P=W conjecture and extended Fano/Landau-Ginzburg correspondence

本文面向 Fano 对提出混合 LG 模型以统一刻画边界除子分支的镜像，研究其反常滤过与 Hodge 理论，给出乘法性上界，并由相对同调镜像对称导出镜像 P=W 猜想。

来自曲面Fukaya范畴的矩阵分解的典范形式

Canonical form of matrix factorizations from Fukaya category of surface

该研究针对pair-of-pants曲面与奇点$xyz=0$的同调镜像对称，证明高重数带型极大Cohen-Macaulay模对应闭测地线高秩局部系统，给出对应矩阵分解显式典范形式并解释相关代数运算的几何意义。