论文标题
全态浮子理论和燃料方程
Holomorphic Floer Theory and the Fueter Equation
论文作者
论文摘要
我们概述了与HyperKähler歧管$ M $相关的$ 2 $ -CAGETORY $ \ MATHRM {FUET} _M $的建议，该$ M $分类为复杂Lagrangians产生的福卡亚类别的子类别。在此$ 2 $ - 类别中，形态形式是福卡亚 - 单态象征性动作功能的seidel类别。因此，$ \ mathrm {fuet} _m $基于将图表计算到满足fueter方程的$ m $，并在Holomorphic Lagrangians上使用边界值。我们通过建立一些有关Fueter图的基本分析结果（例如能量结合和最大原理）来迈出构建此类别的第一步。当$ m = t^*x $是kähler歧管$ x $和$ x $和$（l_0，l_1）$的cotangengent捆绑包时，是零部分，是全态函数$ f：x \ t to \ mathbb {c} $的差异图，我们证明了所有fueter maps $ fuueter for $ fuueter of yout x $ for $ for $ fip of yout x $ for $ fip $ fi $ for $ fi $ fipories unt $ fi $ fipories unt $ for $ fi $ f。福卡亚 - $ f $的seidel类别。这是对cotangent束中伪旋晶条的Floer定理的复杂化。在整个论文中，我们建议可能对该主题感兴趣的分析师和几何图形的问题和研究方向。
We outline a proposal for a $2$-category $\mathrm{Fuet}_M$ associated to a hyperkähler manifold $M$, which categorifies the subcategory of the Fukaya category of $M$ generated by complex Lagrangians. Morphisms in this $2$-category are formally the Fukaya--Seidel categories of holomorphic symplectic action functionals. As such, $\mathrm{Fuet}_M$ is based on counting maps to $M$ satisfying the Fueter equation with boundary values on holomorphic Lagrangians. We make the first step towards constructing this category by establishing some basic analytic results about Fueter maps, such as the energy bound and maximum principle. When $M=T^*X$ is the cotangent bundle of a Kähler manifold $X$ and $(L_0, L_1)$ are the zero section and the graph of the differential of a holomorphic function $F: X \to \mathbb{C}$, we prove that all Fueter maps correspond to the complex gradient trajectories of $F$ in $X$, which relates our proposal to the Fukaya--Seidel category of $F$. This is a complexification of Floer's theorem on pseudo-holomorphic strips in cotangent bundles. Throughout the paper, we suggest problems and research directions for analysts and geometers that may be interested in the subject.
