论文标题
光滑的射曲比YAU的frobenius歧管和更高的残留配对的算法和算法
Smooth projective Calabi-Yau complete intersections and algorithms for their Frobenius manifolds and higher residue pairings
论文作者
论文摘要
本文的目的是提供明确的算法构建$ f $ - manifold结构，正式的Frobenius歧管结构以及在原始的中等维度的$ \ mathbb {h} $的原始中等维度配对上的较高的残留配对。 \ dots，g_k（\下划线x）$。我们的主要方法是分析某个DGBV（差异Gerstenhaber-batalin-vilkovisky）代数$ \ MATHCAL {a} $从计算$ \ MATHBB {H h} $的Twisted de Rham综合体获得的。更明确地，我们介绍了与$ \ Mathcal {a} $的毛勒 - 卡丹方程和Gauss-Manin Connection相关的\ textit {弱原始形式}的概念，该方程是saito的原始形式的弱版（\ cite {saito}）。此外，我们还为基于Gröbner基础的弱原始形式提供了一种明确的算法，以实现我们的目标。我们通过弱原始形式的方法可以看作是统一的链接（基于Witten的仪表线性的Sigma模型，\ cite \ cite {w93}）在Barannikov-Kontsevich通过DGBV代数（通过非线性Sigma模型的非线性Sigma模型）和SA的方法{BK}）在Barannikov-Kontsevich对Frobenius歧管的方法之间以及更高的残留配对（Landau-Ginzburg模型，\ cite {st}）。
The goal of this article is to provide an explicit algorithmic construction of formal $F$-manifold structures, formal Frobenius manifold structures, and higher residue pairings on the primitive middle-dimensional cohomology $\mathbb{H}$ of a smooth projective Calabi-Yau complete intersection variety $X$ defined by homogeneous polynomials $G_1(\underline x), \dots, G_k(\underline x)$. Our main method is to analyze a certain dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra $\mathcal{A}$ obtained from the twisted de Rham complex which computes $\mathbb{H}$. More explicitly, we introduce a notion of \textit{a weak primitive form} associated to a solution of the Maurer-Cartan equation of $\mathcal{A}$ and the Gauss-Manin connection, which is a weakened version of Saito's primitive form (\cite{Saito}). In addition, we provide an explicit algorithm for a weak primitive form based on the Gröbner basis in order to achieve our goal. Our approach through the weak primitive form can be viewed as a unifying link (based on Witten's gauged linear sigma model, \cite{W93}) between the Barannikov-Kontsevich's approach to Frobenius manifolds via dGBV algebras (non-linear topological sigma model, \cite{BK}) and Saito's approach to Frobenius manifolds via primitive forms and higher residue pairings (Landau-Ginzburg model, \cite{ST}).
