Renormarization of two dimensional random fields with rich symmetry

具有丰富对称性的二维随机场的重整化

• 批准号: 10640122
• 负责人: IWATA Koichiro
• 金额: $ 1.15万
• 依托单位: Hiroshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640122/
• 关键词: random fields; modular forms; Cauchy-Riemann equation; multiple Wiener integrals; 等角構造; wick積; コーシーリマン方程式

The aim of the present research project is to study conformally invariant random fields which arises as unique solution of inhomogeneous Cauchy-Riemann equation. The upper half space in the complex plain parameterizes the conformal structure of the two dimensional torus on which the modular group acts. The latter action yields the modular covariance of the random fields. As a consequence, the moment functions of the random field evaluated at rational points are automorphic. In several cases one can prove that the moment functions are actually modular functions. With the help of the expression of the solution in terms of elliptic functions, it is possible to construct functionals, called the renormalized product, of the field with higher weight relative to the modular group action. This extends the class of modular functions which admits integral representation by the random field. This is related to the fact that there exist a large class of local functionals, called the Wick products, in two dimensional quantum field theory. Their existence reflects the logarithmic singularity of the Green function. The origin of the rather mild singularity is the conformal structure of the two dimensional space. So it is natural to ask how the conformal structure determines the renonnalized products and how the conformal structure determines class of modular functions which admits integral representation. A natural way to attack these problem is as follows : Study the action of the modular group on the configuration space of the pair of the evaluation points and the wight of the renormalized products and the action of the modular group on the cusps. So far the space of modular forms spanned by Eisenstein series are studied and it is proved that Eisenstein series are represented by moments of the random fields provided the weight is not greater than 10.

本研究项目的目的是研究作为非齐次柯西-黎曼方程的唯一解而出现的共形不变随机场。复平面中的上半空间参数化模群作用的二维环面的共形结构。后一个动作产生随机场的模协方差。因此，在有理点处计算的随机场的矩函数是自守的。在某些情况下，我们可以证明矩函数实际上是模函数。借助椭圆函数的解表达，可以构造相对于模群作用具有更高权重的域的泛函（称为重整化积）。这扩展了模函数的类别，允许随机场进行积分表示。这与二维量子场论中存在一大类局域泛函（称为威克积）有关。它们的存在反映了格林函数的对数奇异性。相当温和的奇点的起源是二维空间的共形结构。因此，很自然地要问共形结构如何确定重整化产物以及共形结构如何确定允许积分表示的模函数的类。解决这些问题的自然方法如下：研究模群对评估点对的配置空间和重正化乘积的权重的作用以及模群对尖点的作用。至此，我们对爱森斯坦级数跨越的模形式空间进行了研究，并证明了在权重不大于10的情况下，爱森斯坦级数可以用随机场的矩来表示。

项目成果

K.Iwata: "Markov property and cokernels of local oterators"J.London Math.Soc.. 56. 657-672 (1997)

K.Iwata：“马尔可夫性质和本地操作者的核心”J.London Math.Soc.. 56. 657-672 (1997)

K. Iwata: "Markov property and cokernek of local operators"J. London Math. Soc.. 56. 657-672 (1997)

K. Iwata：“本地运营商的马尔可夫性质和 cokernek”J.

K.Iwata: "Markov property and cokernels of local operators" J.London Math Soc.56. 657-672 (1997)

K.Iwata：“马尔可夫性质和本地运营商的核心”J.London Math Soc.56。