Vertex Algebras and Mirror Symmetry
Vertex Algebras and Mirror Symmetry
复制标题
顶点代数和镜像对称
DOI：
10.1007/s002200000312
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发表时间：
1998
影响因子：
2.4
通讯作者：
L. Borisov
中科院分区：
文献类型：
--
作者：
L. Borisov
摘要：环面簇中的Calabi-Yau超曲面的镜面对称性已被公认。然而，以前的方法并没有揭示镜像品种是镜像的潜在原因。我们能够显式地计算对应于Calabi-Yau超曲面的A和B模型的全纯部分的顶点代数以及环丛中的完全交。我们建立了镜像Calabi-Yau流形的这些顶点代数之间的关系。这最终应该允许我们用顶点代数的语言重写环面镜对称的整个故事。我们的方法是纯代数的，涉及环面几何和同调代数的简单技巧，以及顶点代数理论的一些基本结果。这篇文章也可能对物理学家感兴趣，因为它包含了Calabi-Yau超曲面的A和B模型的全纯部分，以及关于自由玻色子和费米子的完全交的显式描述。
Abstract: Mirror Symmetry for Calabi–Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi–Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi–Yau manifolds. This should eventually allow us to rewrite the whole story of toric Mirror Symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties.This paper could also be of interest to physicists, because it contains explicit description of holomorphic parts of A and B models of Calabi–Yau hypersurfaces and complete intersections in terms of free bosons and fermions.