Colliding holes in Riemann surfaces and quantum cluster algebras
Colliding holes in Riemann surfaces and quantum cluster algebras
复制标题
黎曼曲面和量子簇代数中的碰撞空穴
DOI：
10.1088/1361-6544/aa9729
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发表时间：
2018
期刊：
影响因子：
1.7
通讯作者：
Chekhov L
中科院分区：
文献类型：
--
作者：
Chekhov L
在本文中，我们描述了一种新的手术的非紧黎曼曲面，自然出现碰撞时，两个洞或同一个洞的两侧在一个定向黎曼曲面的边界（可能orbifold点）。由于这种手术，边界尖点出现在黎曼曲面的边界分量上。在庞加莱单值化中，这些有边尖点对应于基本域中的理想三角形。我们引入了加边尖点Teichmüller空间的概念，并赋予它一个泊松结构，量子化的实现与规范的量子序。我们给出了一个完整的组合描述的有边尖Teichmüller空间通过引入最大尖迭的概念，一个迭由测地弧之间的有边尖点和封闭的测地线同伦的边界，使它三角化的黎曼曲面。我们证明了每个有界尖点都带有一个自然的装饰，即选择一个horocycle，因此在最大尖点叠层中的弧的长度被定义为Thurston-Penner术语中的λ-长度。我们根据这些λ-长度显式地计算了Goldman括号，并证明了翻转态射的广群充当了广义簇代数突变。从物理的角度来看，我们的建设提供了一个明确的协调开/闭弦世界表及其量子化的模空间。
In this paper, we describe a new type of surgery for non-compact Riemann surfaces that naturally appears when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincaré uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichmüller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichmüller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesic arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, ie a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as λ-lengths in Thurston–Penner terminology. We compute the Goldman bracket explicitly in terms of these λ-lengths and show that the groupoid of flip morphisms acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.