R6中对偶五点函数的围道积分表示与亏格四曲面的对称性
A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6
摘要：
“对偶共振模型”N点函数BN的发明推动了当前弦论的发展。这些模型中最简单的四点函数B4是经典的欧拉Beta函数。许多单变量复分析的标准方法已经被应用于阐明欧拉Beta函数的性质，例如，导致解析延拓公式，如Pochhammer在1890年获得的轮廓积分表示。在这里，我们探索的几何基础的对偶五点函数B5，最简单的推广欧拉贝塔函数。通过分析B5被积函数，可以得到嵌入RP5中的五交叉曲面的多面体结构，它有12个五边形面和PGL（6）中的120阶对称群。我们发现一个Pochhammer表示B5是沿曲面的轮廓积分属五。R5中的五交叉曲面的对称嵌入被R6中亏格为4的曲面的对称嵌入双重覆盖，R6具有24个五边形面的多面体结构和O（6）中的240阶对称群。这些方法似乎可推广到所有N，并且所得结构似乎与任意维度的缔合多面体有关。
英文摘要：
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 formulas such as the contour-integral representation obtained by Pochhammer in 1890. Here we explore the geometry underlying the dual five-point function B5, the simplest generalization of the Euler Beta function. Analyzing the B5 integrand leads to a polyhedral structure for the five-crosscap surface, embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120 in PGL(6). We find a Pochhammer-like representation for B5 that is a contour integral along a surface of genus five. The symmetric embedding of the five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the surface of genus four in R6 that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in O(6). The methods appear generalizable to all N, and the resulting structures seem to be related to associahedra in arbitrary dimensions.
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