赋范矩阵分解的数值不变量
Numerical invariants of normed matrix factorizations
AI总结：
本文构造赋范矩阵分解范畴与 Dirac factorization，通过边界上链分类定义数值不变量，并在单形情形验证其与实射影空间的开 Gromov-Witten 型不变量一致。
摘要：
我们定义了一个赋范矩阵分解范畴，以及该范畴中对象的边界上链概念。我们对球形对象的边界上链按规范等价进行分类，并利用这一分类定义数值不变量。这些不变量预期在镜像对称下对应于仅带边界约束的开 Gromov-Witten 不变量，后者由第二作者与 Tukachinsky 针对 Lagrangian 有理上同调球面定义。对于每个 Delzant 多面体，我们构造一个赋范矩阵分解范畴。对于满足组合相对自旋条件的 Delzant 多面体，我们构造该范畴中的一个对象，称为 Dirac factorization。Dirac factorization 预期在镜像对称下对应于一个 Lagrangian 子流形，即与该 Delzant 多面体相关联的环面辛流形的实轨迹。在 n 为奇数的 n-单形情形中，我们证明 Dirac factorization 是球形的，这反映了 RP^n 是有理上同调球面这一事实。对于 n=1，我们证明 Dirac factorization 的数值不变量与包含于 CP^1 中的 RP^1 的开 Gromov-Witten 不变量一致。对于 n=3，在低次数下，计算机计算验证 Dirac factorization 的数值不变量与包含于 CP^3 中的 RP^3 的开 Gromov-Witten-Welschinger 不变量一致。尽管在任意特征零域上，RP^n 在 CP^n 的 Fukaya 范畴中是平凡的，但上述结果可被视为 Novikov 环上镜像对称的一种体现。
英文摘要：
We define a normed matrix factorization category and a notion of bounding cochains for objects of this category. We classify bounding cochains up to gauge equivalence for spherical objects and use this classification to define numerical invariants. These invariants are expected to correspond under mirror symmetry to the open Gromov-Witten invariants with only boundary constraints of Lagrangian rational cohomology spheres defined by the second author and Tukachinsky. For each Delzant polytope, we construct a normed matrix factorization category. For Delzant polytopes satisfying a combinatorial relative spin condition, we construct an object of this category called the Dirac factorization. The Dirac factorization is expected to correspond under mirror symmetry to the Lagrangian submanifold given by the real locus of the toric symplectic manifold associated to the Delzant polytope. In the case of the $n$-simplex for $n$ odd, we show that the Dirac factorization is spherical, mirroring the fact that $\mathbb{R} P^n$ is a rational cohomology sphere. For $n = 1,$ we show the numerical invariants of the Dirac factorization coincide with the open Gromov-Witten invariants of $\mathbb{R} P^1 \subset \mathbb{C} P^1.$ For $n = 3$ in low degrees, computer calculations verify that the numerical invariants of the Dirac factorization coincide with the open Gromov-Witten-Welschinger invariants of $\mathbb{R}P^3 \subset \mathbb{C} P^3.$ Although $\mathbb{R} P^n$ is trivial in the Fukaya category of $\mathbb{C} P^n$ over any field of characteristic zero, the above results can be seen as a manifestation of mirror symmetry over a Novikov ring.
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