DEFORMATION QUANTIZATION AND ITS APPLICATION

这项研究有两个方面；(i)通过Weyl流形的变形量化的几何方面，（ii）关于变形参数h的收敛变形量化的研究。我们得到以下结果。(1) Weyl流形的模空间是基流形的第2上同调类中带系数的形式幂级数。利用与Weyl流形相对应的上同调，构造了一个包含Weyl流形作为子束的接触Weyl流形。在接触Weyl流形上，我们还构造了一个连接，其曲率形式决定了Weyl流形的上同调类。我们证明了该连接是Fedosov连接的扩展，并证明了由曲率给出的上同调类与Weyl流形的上同调类重合，从而证明了Weyl流形的poincarel - cartan类与Fedosov连接的曲率的上同调类是一回事。（ii）利用Moyal积公式，在Moyal积绝对收敛的多维复平面上，建立了某些全纯函数的Frechet空间。介绍了全纯函数的奇异指数，其星积打破了乘积的结合律。我们还研究了二次函数的星型指数函数。虽然Frechet空间不包含二次函数的指数，但二次函数与指数小于奇点的全纯函数之间的星积有很好的定义。研究了由二次指数函数生成的群的若干性质。特别地，这个群被认为是特殊线性群的扩展。

This researchment is two fold ; (i) geometric aspect of Deformation quantization via Weyl manifolds, (ii) investigation of convergent deformation quantization with repect to the deformation parameter h. We obtain the following. (i) The moduli space of Weyl manifolds are the formal power serires with coefficients in the 2nd cohomology classes of the base manifol. Using the cohomolgy corresponding to the Weyl manifold, we construct a contact Weyl manifold which contains Weyl manifold as a subbundle. On contact Weyl manifold, we also constuct a connection whose curvature form determines the cohomology class of the Weyl manifold. We show this connection is an extension of Fedosov connection and proved that the cohomolgy class given by the curvature coincides with the cohomology class of the Weyl manifold, hence we show the Poincare-Cartan class of Weyl manifold and the cohomology class of the curvature of Fedosov connection are the same thing. (ii) Using the Moyal product formula, we set certain Frechet space of certain holomophic functions on the multidimensional complex plane where the Moyal products are absolutely convergent. Singular exponent of holomorphic functions are introduced with respect which the star products breaks the associativity of product. We also investigate a star exponential functions of quadratic functions. Althoug the Frechet space does not contain the exponentials of the quadratic functions, the star product is well defined between the quadratic exponentials and holomorphic function having the exponent less than the singular one. Certain properties are investigated for the group generated by the quadratic exponential functions. Especially, the group is considered an extension of the special linear groups.