Higgs 场、非交换 Cauchy 核与 Goldman 辛结构
Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure
AI总结：
本文利用 Tyurin 数据构造非交换 Cauchy 核与平坦全纯联络，建立 Higgs 场余切丛到 GL_n 特征标簇的单值化映射，并证明其拉回保持 Goldman 辛结构。
摘要：
我们考察在固定的亏格 g≥2 的 Riemann 曲面上，秩为 n、次数为 ng 的向量丛的模空间。我们采用基于 Tyurin 数据的显式参数化。在该模空间中存在一个“非交换的”Theta 除子，由满足 h^1≥1 的向量丛组成。在此除子的补集上，我们利用 Tyurin 数据显式构造了一个非交换 Cauchy 核。借助一个额外的非特殊除子数据，我们能够构造一个参考的平坦全纯联络，该联络也全纯地依赖于向量丛的模。这使我们能够将 Higgs 场丛（即模空间的余切丛）等同于全纯联络的仿射丛，并给出一个映入 GL_n 特征标簇的单值化映射。我们证明，特征标簇上的 Goldman 辛结构沿该映射的拉回，就是余切丛上的复标准辛结构，因而也是仿射联络空间上的复标准辛结构。随后证明，Liouville 一形式到联络仿射丛的拉回是一个对数形式，其极点位于非交换 theta 除子上，留数由 h^1 给出。
英文摘要：
We consider the moduli space of vector bundles of rank $n$ and degree $ng$ over a fixed Riemann surface of genus $g\geq 2$. We use the explicit parametrization in terms of the Tyurin data. In the moduli space there is a "non-abelian" Theta divisor, consisting of bundles with $h^1\geq 1$. On the complement of this divisor we construct a non-abelian Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which is also dependent holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the ${\rm GL}_n$ character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of affine connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by $h^1$.
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