Renormarization of two dimensional random fields with rich symmetry

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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.