用户: Cybcat/数学物理学家用的数学/Anomaly

The present article is a gentle introduction to some mathematical aspects of the BCOV holomorphic anomaly equations, which represent a beautiful generalization of the classical $g=0$ mirror symmetry. The classical $g=0$ mirror symmetry states that counting the rational curves in a Calabi–Yau threefold $X^{\vee }$ (A-model) is equivalent to studying the variation of Hodge structures of its mirror Calabi–Yau threefold $X$ (B-model). Higher genus mirror symmetry is concerned with counting the higher genus curves in a Calabi–Yau threefold. While Gromov– Witten theory rigorously defines a mathematical theory of counting curves of any genus and thus higher genus A-model makes sense at all genera, the higher genus B-model, a generalization of the theory of variation of Hodge structures, has been much more mysterious.

1特殊 Kahler 几何

我们的设定如下: $\mathcal{M}$ 是某光滑 CY 3-流形复结构的模空间 $X$, 记 $n:=\dim \mathcal{M}=h^{2,1}(X)$. 向量场$$\mathcal{H}:=R^3{\pi }_{\ast }\underline{\mathbb{C}}\otimes {\mathcal{O}}_{\mathcal{M}}.$$带有 Gauss–Manin 联络 $\nabla$ 和自然的权 $3$ 的 Hodge 分次 $F^{\ast }$, 具体地即$$\mathcal{H}:=\underset{p+q=3}{⨁}{\mathcal{H}}^{p,q},{\mathcal{H}}^{p,q}:=F^p\mathcal{H}\cap \overline{F^q\mathcal{H}}.$$然后全纯线丛 $\mathcal{L}:=F^3\mathcal{H}={\mathcal{H}}^{3,0}$ 被称为真空丛 (Vacuum bundle). 另外局部上我们选取一个 $[X]\in \mathcal{M}$ 并将 $\mathcal{H}$ 的纤维等同 $H^3(X,\mathbb{C})$. 其上带有 (辛) 配对 $(\alpha ,\beta ):=i{\int }_X\alpha ⌣\beta$. 在此配对下可选择一组 (辛) 基 $\{{\alpha }_i,{\beta }^j{\}}_{i,j=0}^n\subset H^3(X,\mathbb{Z})$ 以及它们的对偶基 $\{{\alpha }^i,{\beta }_j{\}}_{i,j=0}^n$ 满足 $({\alpha }_i,{\beta }^j)=(-{\beta }^j,{\alpha }_i)={\delta }_{ij}$.