Teichmueller空间中的拟对称缝合
Quasisymmetric sewing in rigged Teichmueller space
摘要：
共形场论（CFT）中的基本几何对象之一是黎曼曲面的模空间，其$n$边界被解析参数化“操纵”。基本的操作是使用参数化来识别点来缝合这样的表面。另一种模型是$n$ -穿孔黎曼曲面的模空间以及穿孔处的局部双全纯坐标。我们把这两个模空间称为“操纵黎曼模空间“。通过推广到拟对称边界参数化，并在边界图和穿刺图中定义了操纵Teichmueller空间，证明了如下结果：（1）以$n$闭曲线为边界的亏格- $g$曲面的Teichmueller空间覆盖了同类型曲面的操纵Riemann和操纵Teichmueller模空间，并在其上诱导出复流形结构. (2)有了这种复杂的结构，缝纫操作是全纯的。(3)Rig模空间和Rig Teichmueller空间的边界图和穿刺图是双全纯等价的。这些结果对于严格定义CFT（G. Segal），以及从顶点算子代数构造CFT。
英文摘要：
One of the basic geometric objects in conformal field theory (CFT) is the the moduli space of Riemann surfaces whose $n$ boundaries are ''rigged'' with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alternative model is the moduli space of $n$-punctured Riemann surfaces together with local biholomorphic coordinates at the punctures. We refer to both of these moduli spaces as the "rigged Riemann moduli space". By generalizing to quasisymmetric boundary parametrizations, and defining rigged Teichmueller spaces in both the border and puncture pictures, we prove the following results: (1) The Teichmueller space of a genus-$g$ surface bordered by $n$ closed curves covers the rigged Riemann and rigged Teichmueller moduli spaces of surfaces of the same type, and induces complex manifold structures on them. (2) With this complex structure the sewing operation is holomorphic. (3) The border and puncture pictures of the rigged moduli and rigged Teichmueller spaces are biholomorphically equivalent. These results are necessary in rigorously defining CFT (in the sense of G. Segal), as well as for the construction of CFT from vertex operator algebras.
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