论文标题
更高的复杂结构和平坦的连接
Higher Complex Structures and Flat Connections
论文作者
论文摘要
在2018年，弗拉基米尔·福克（Vladimir Fock）和作者在表面上引入了一个几何结构，称为较高的复杂结构，其模量空间与希钦（Hitchin）的成分共享了多个属性。在本文中，我们建立了平坦连接与更高复杂结构之间的各种联系。 特别是，我们在配备了线路捆绑$ l $的捆绑包上研究了某种类别的连接，我们称之为$ l $ parabolic。这些连接的曲率为等级1。$ l $ - 撒布代谢连接的某个家族是由较高的复杂结构和cotangent变化的数据进行了参数。连接家族是平坦的，这意味着cotangent变化的兼容性条件。通过更改$ L $引起的量规转换实现了较高的差异性。建造这种连接的平坦家族与TODA集成系统有关。
In 2018, Vladimir Fock and the author introduced a geometric structure on surfaces, called higher complex structure, whose moduli space shares several properties with Hitchin's component. In this paper, we establish various links between flat connections and higher complex structures. In particular, we study a certain class of connections on a bundle equipped with a line subbundle $L$, which we call $L$-parabolic. The curvature of these connections is of rank 1. A certain family of $L$-parabolic connections is parametrized by the data of a higher complex structure and a cotangent variation. The family of connections being flat implies the compatibility condition of the cotangent variation. Higher diffeomorphisms are realized by the gauge transformation induced by changing $L$. Constructing flat families of connections of this kind is linked to Toda integrable systems.
