论文标题
Kodaira-Spencer地图，Lagrangian浮子理论和Orbifold Jacobian代数
Kodaira-Spencer map, Lagrangian Floer theory and orbifold Jacobian algebras
论文作者
论文摘要
镜像对称性的一种版本预测了与landau-ginzburg镜的量子歧管的量子同构和雅各布式代数之间的环形同构，以及用于福卡亚 - oh-ohta-ono的折叠式歧管，构建了一种称为Kodaira-spencer图的地图，使用lagrangian lagrangian lagrangian图。我们讨论了Kodaira-Spencer环的一般结构同构同构时，当LG镜面潜在$ W $由$ j $ - 圆形盘给带边界的Lagrangian $ l $：我们找到一个$ a _ {\ infty} $ - 代数 - 代数 - 代数$ \ nathcal $ \ nathcal {b} $ nmim $ _1 $ $ - $ $ - $ - $ $ n $ - 封闭图提供了一个从量子共同体学到$ \ Mathcal {B} $的共同体学代数的戒指同态，这是$ W $的Jacobian代数。 我们还为Orbifold LG Mirror $（W，h）$构建了一个均等版本。我们构建了一个Kodaira-spencer地图，从量子共同学到另一个$ a _ {\ infty} $ - 代数$（\ Mathcal {b} \ rtimes h）^H $，其同胞代数对Orbifold Jacobian Algebra的同构为同构成$（W，W，W，H）$。对于$ 2 $ -torus，其镜子是Fermat Cutic提供的Orbifold LG模型，并使用$ \ Mathbb {Z}/3 $ -Action，我们计算一个显式的Kodaira-Spencer同构。
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.
