A contour integral representation for the dual five-point function and a symmetry of the genus-4 surface in ℝ6
A contour integral representation for the dual five-point function and a symmetry of the genus-4 surface in ℝ6
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对偶五点函数的轮廓积分表示和 ℝ6 中 genus-4 曲面的对称性
DOI：
10.1088/0305-4470/39/10/017
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Ji
中科院分区：
文献类型：
--
作者：
A. Hanson;Ji
“双共振模型”N 点函数 BN 的发明推动了当前弦理论的发展。这些模型中最简单的四点函数 B4 是经典的 Euler Beta 函数。许多单变量复分析的标准方法已被应用于阐明 Euler Beta 函数的性质，例如，导致了解析连续公式，例如 Pochhammer 在 1890 年获得的轮廓积分表示。然而，预期的多复变量推广到 BN 的精确特征尚未得到系统研究。在这里，我们探讨对偶五点函数 B5 的几何结构，这是欧拉 Beta 函数的最简单推广。定义 B5 的原始被积函数导致嵌入 中的五交叉帽表面的多面体结构，该结构具有 12 个五边形面和 120 阶对称群。我们找到了 B5 的类似 Pochhammer 的表示，它是 中沿属 5 的曲面的轮廓积分。中的五交叉帽表面的对称嵌入被 中的 genus 4 表面的相应对称嵌入双重覆盖，该表面具有 24 个五边形面和 240 阶对称群的多面体结构。这些对称性使得能够构建这些表面的优雅可视化。本文的核心思想是实现五点交比集的紧化形成一个光滑的实代数子簇，即 中的五交帽面。正是在该曲面的复化中，我们构建了 B5 的轮廓积分表示。我们的方法原则上可以推广到更高的维度，因此应该对进一步的研究感兴趣。
The invention of the 'dual resonance model' N-point functions BN motivated the development of current string theory. The simplest of these models, the four-point function B4, is the classical Euler Beta function. Many standard methods of complex analysis in a single variable have been applied to elucidate the properties of the Euler Beta function, leading, for example, to analytic continuation formulae such as the contour-integral representation obtained by Pochhammer in 1890. However, the precise features of the expected multiple-complex-variable generalizations to BN have not been systematically studied. Here we explore the geometry underlying the dual five-point function B5, the simplest generalization of the Euler Beta function. The original integrand defining B5 leads to a polyhedral structure for the five-crosscap surface, embedded in , that has 12 pentagonal faces and a symmetry group of order 120 in . We find a Pochhammer-like representation for B5 that is a contour integral along a surface of genus 5 in . The symmetric embedding of the five-crosscap surface in is doubly covered by a corresponding symmetric embedding of the surface of genus 4 in that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in . These symmetries enable the construction of elegant visualizations of these surfaces. The key idea of this paper is to realize that the compactification of the set of five-point cross-ratios forms a smooth real algebraic subvariety that is the five-crosscap surface in . It is in the complexification of this surface that we construct the contour integral representation for B5. Our methods are generalizable in principle to higher dimensions, and therefore should be of interest for further study.