Kodaira-Spencer映射、Lagrangian Floer理论和轨道折叠Jacobian代数
Kodaira-Spencer map, Lagrangian Floer theory and orbifold Jacobian algebras
摘要：
镜像对称的一个版本预言了辛流形的量子上同调与朗道-金斯堡镜像的雅可比代数之间的环同构，而对于环面流形，谷-哦-太田-小野利用拉格朗日弗洛尔理论构造了一个称为Kodaira-Spencer映射的映射。当LG镜势$W$由$J$ -拉格朗日$L$上具有边界的全纯圆盘给出时，我们讨论了Kodaira-Spencer环同态的一般构造：在对$L$的温和假设下，我们找到了一个$A_{\infty}$ -代数$\mathcal{B}$，它的$m_1$ -复形是$W$的Koszul复形.闭-开映射给出了从量子上同调到$\mathcal{B}$的上同调代数的环同态，该上同调代数是$W$的雅可比代数.我们还构造了一个等变的轨道LG镜$(W,H)$。我们构造了一个从量子上同调到另一个$A_{\infty}$ -代数$(\mathcal{B}\rtimes H)^H$的Kodaira-Spencer映射，该代数的上同调代数与$(W,H)$的orbifold Jacobian代数同构.对于具有$\mathbb{Z}/3$ -作用量的Fermat三次函数所给出的轨道LG模型为镜像的$2$ -环面，我们计算了一个显式的Kodaira-Spencer同构.
英文摘要：
A version of mirror symmetry predicts a ring isomorphism between quantum cohomology of a symplectic manifold and Jacobian algebra of the Landau-Ginzburg mirror, and for toric manifolds Fukaya-Oh-Ohta-Ono constructed such a map called Kodaira-Spencer map using Lagrangian Floer theory. We discuss a general construction of Kodaira-Spencer ring homomorphism when LG mirror potential $W$ is given by $J$-holomorphic discs with boundary on a Lagrangian $L$: we find an $A_{\infty}$-algebra $\mathcal{B}$ whose $m_1$-complex is a Koszul complex for $W$ under mild assumptions on $L$. Closed-open map gives a ring homomorphism from quantum cohomology to cohomology algebra of $\mathcal{B}$ which is Jacobian algebra of $W$. We also construct an equivariant version for orbifold LG mirror $(W,H)$. We construct a Kodaira-Spencer map from quantum cohomology to another $A_{\infty}$-algebra $(\mathcal{B}\rtimes H)^H$ whose cohomology algebra is isomorphic to the orbifold Jacobian algebra of $(W,H)$ under an assumption. For the $2$-torus whose mirror is an orbifold LG model given by Fermat cubic with a $\mathbb{Z}/3$-action, we compute an explicit Kodaira-Spencer isomorphism.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
