De Rham Theory of Loop Spaces and Quantum Topological Invariants

我们研究了配置空间环空间同调的代数结构，并阐明了它与辫子的有限型拓扑不变量的关系。特别地，我们研究了配置空间的环空间与有限类型拓扑不变量的同调。重点讨论了复上半平面上Fuchsian群作用下轨道构形空间的环空间的同调问题。我们证明了这种环空间的全同调与商曲面上的水平弦图的代数同构。基于Browder运算，给出了轨道位形空间迭代环空间同调的Poisson代数结构，并用超几何积分的方法完整地描述了黎曼球面上共形场理论的共形块空间.特别地，我们将积分圈定义为局部有限链同调中的可正则化圈，并研究了各种模空间及其相关模群的结构，如黎曼曲面映射类群的模空间和自由群的图-外自同构群的模空间。村上对一个关于纽结的Jones不变量的渐近性、纽结补的双曲体积和三维流形的几何结构的猜想提出了新的观点。

We investigated the algebraic structure of the homology of the loop spaces of configuration spaces and clarified its relation to finite type topological invariants for braids. Especially, we studied the homology of the loop spaces of configuration spaces and finite type topological invariants. We focused on the homology of the loop spaces of orbit configuration spaces associated with the action of Fuchsian groups on the complex upper half plane. We showed that the total homology of such loop space is isomorphic to the algebra of horizontal chord diagrams on the quotient surface. We introduced a structure of a Poisson algebra for the homology of the iterated loop space of the orbit configuration space based on the Browder operation.We gave a complete description of the space of conformal blocks for the conformal field theory on the Riemann sphere in terms of hypergeometric integrals. In particular, we clarified the integration cycles as the regularizable cycles in the homology of locally finite chains with coefficients in a certain local system defined over the complement a disrciminantal arrangement.Morita investigated the structure of various moduli spaces as well as their associated modular groups, such as the moduli space of Riemann surface -mapping class groups and the moduli space of graphs -outer automorphism group of free groups. Murakami gave a new point of view on a conjecture concerning the asymptotics of the Jones invariants for knots, the hyperbolic volume of the knot complement, and the geometric structure of 3-manifolds.