Renormarization of two dimensional random fields with rich symmetry

本研究的目的是研究作为非齐次Cauchy-Riemann方程唯一解的共形不变随机场。复杂平面中的上半空间将模群作用于其上的二维环面的共形结构参数化。后一种作用产生随机场的模协方差。因此，在有理点上求值的随机场的矩函数是自同构的。在某些情况下，我们可以证明矩函数实际上是模函数。借助于椭圆函数形式的解的表达式，可以构造相对于模群作用具有较高权重的场的泛函，称为重整化积。这扩展了允许用随机场积分表示的模函数类。这与二维量子场论中存在一大类称为Wick积的局域泛函有关。它们的存在反映了格林函数的对数奇异性。这种相当温和的奇点的起源是二维空间的共形结构。因此，人们很自然地会问，共形结构如何决定重归化积，以及共形结构如何决定允许积分表示的模函数类。解决这些问题的一个自然方法是：研究模群在重整化乘积的赋值点和鬼魂对的位形空间上的作用以及模群在尖点上的作用。到目前为止，我们研究了Eisenstein级数所跨越的模形式空间，证明了Eisenstein级数可以用随机场的矩来表示，只要它的权不大于10。

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.