Poisson Geometry, Contact Geometry and Quantization Problems

本研究计画针对泊松几何、接触几何中的几个问题，以及量子化问题。特别地，我们的项目着重于从几何的角度处理量子化问题。非对易微分几何是本课题的研究目标之一。在这个项目中，我们有几个收敛变形量子化问题的结果。Grebes出现自然从建设的星星指数函数的二次型。也就是说，我们考虑复Weil代数中的二次型的集合，它构成一个同构于sp（n，C）的李代数。当我们考虑这些对象的esponential函数时，我们可能会期待元李群的复版本。由于这是单连通的，我们无法处理它。我们通过显式计算来投资这个对象，并将其置于具有乘法的grebes范畴中。然而，它可以描述更多的几何连接方面。第二个问题是研究不变形变量子化问题，构造ax+B群情形的收敛星星积。我们得到了普适的星星乘积公式。第三个结果是研究了闭星星积。给出了利用Stokes公式得到Hochschild上循环的一般方法.这一结果仍在研究中，我们希望它能与无限维情况下的形变量子化问题联系起来。为了取得这些成果，我们邀请国内外研究人员与研究伙伴一起举办了许多研讨会。研究成果丰硕，在国际上享有较高的评价，并在此基础上建立了该领域的国际研究网络。该项目仍在进行中，并将继续进行下一个项目。

This research project aims several problems in Poisson geometry and contact geometry, and quantization problems. Specially, our project focus on the treatments for the quantization problems geometrical point of view. The noncommutative differential geometry is one of our target of our research project. In this project, we had several results on convergent deformation quantization problem. Grebes appears naturally from the construction of the star exponential functions of quadratic forms. Namely, we consider the set of quadratic forms in the complex Weil algebra which forms a Lie algebra isomorphic to sp(n, C). When we consider the esponential functions for these objects, we might expect the complex version of the metaplectic Lie group. Since this is simply connected, we could not handle it. We invested this object by using the explicit computations and have it is in the category of grebes with multiplications. However, it can be described more geometry in terms of the connections. The second problem is to study the invariant deformation quantization problems and construct a convergent star product for ax+b group case. We obtain the universal star product formula. The third result is to study the closed star product. We describe a general settings for obtain how to get the Hochschild cocycle via Stokes formula. This results is still working on and we expect it should be related to the deformation quantization problems for infinite dimensional case. To obtain these results, we have lot of workshops by inviting overseas and domestic researchers together with the research partners. As conclusions, we have fruitful research results which have high evaluation internationally, and also establish the international research network for this area by this grant. This project is still working and will continue for the next project.