Calabi-Yau流形上的量子BCOV理论和高亏格B模型
Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model
摘要：
Bershadsky-Cecotti-Ooguri-Vafa（BCOV）提出镜像对称的B模型应该用卡-丘簇上的量子场论来描述，他们称之为Kodaira-Spenser理论（我们称之为BCOV理论）。这是我们构建和分析量子BCOV理论的三篇论文中的第一篇。本文构造了任意维Calabi-Yau簇上的经典域论，定义了量子化的含义，分析了Gromov-Witten理论的Givental辛形式，证明了椭圆曲线上量子化的唯一性，并证明了椭圆曲线上的Virasoro约束.第二篇论文（arXiv：1112.4063）证明了量子BCOV理论在椭圆曲线上的分拆函数与镜像椭圆曲线上的Gromov-Witten理论是等价的。第三篇论文，正在进行中，构造量子BCOV理论上的一般卡-丘。
英文摘要：
Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed that the B-model of mirror symmetry should be described by a quantum field theory on a Calabi-Yau variety, which they called the Kodaira-Spenser theory (we call it the BCOV theory). This is the first of three papers in which we construct and analyze the quantum BCOV theory. In this paper, we construct the classical field theory on a Calabi-Yau variety of arbitrary dimension; define what it means to give a quantization; analyze the relation Givental's symplectic formalism for Gromov-Witten theory; prove uniqueness of the quantization on an elliptic curve; and prove the Virasoro constraints on an elliptic curve. The second paper (arXiv:1112.4063) proves that the partition function of the quantum BCOV theory on the elliptic curve is equivalent to the Gromov-Witten theory of the mirror elliptic curve. The third paper, in progress, constructs the quantum BCOV theory on a general Calabi-Yau.
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