Renormarization of two dimensional random fields with rich symmetry

本研究的目的是研究非齐次柯西-黎曼方程的保形不变随机场的唯一解。复平面的上半空间参数化了模群所作用的二维环面的共形结构。后一种作用产生随机场的模协方差。因此，随机场在有理点处的矩函数是自同构的。在一些情况下，我们可以证明力矩函数实际上是模函数。借助椭圆函数的解表达式，可以构造相对于模群作用具有更高权重的域的泛函，称为重正化积。这扩展了允许随机场的积分表示的模函数类。这与二维量子场论中存在一大类局部泛函（称为Wick积）的事实有关。它们的存在反映了格林函数的对数奇异性。这个相当温和的奇点的起源是二维空间的共形结构。因此，人们自然要问，共形结构如何决定再规整积，以及共形结构如何决定允许积分表示的模函数的种类。解决这些问题的一种自然方法是：研究模群对评价点和重归一化积对的组态空间的作用以及模群对顶点的作用。研究了爱森斯坦级数张成的模形式空间，证明了在权值不大于10的情况下，爱森斯坦级数是由随机场的矩表示的。

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.