Geometry of Field Theory and Spinor Analysis

在“场论几何”项目中，我们给出了四维Wess-Zumino-维滕模型的构造。本文将Wess-Zumino-维滕作用定义为从共形平坦的四维流形范畴到有联络的线丛范畴的函子，并给出了它的构造，从而将四维球面上讨论的许多现象推广到有边界的共形平坦的四维流形上，特别是成功地得到了这类流形上的Polyakov-Wiegner公式.我们还得到了从3-球面到李群的光滑映射群的两种对偶阿贝尔扩张。这些结果发表在Journal of Geometry and Physics，47（2003）.在“Spinor Analysis”项目中，我们研究了共形平坦四维流形上调和旋量的各种性质，包括（1）积分表示（2）解的局部存在性，（3）Runnge逼近定理，Mittag-Lefler定理，（4）Dirac方程在区域上解的整体存在性.这些结果发表在日本数学学报，第28 -1卷（2002年），1-30。接着，我们继续研究了共形平坦流形上的亚纯旋量及其导子。（6）亚纯旋量上同调群的Riemann-Roch型定理。研究结果将发表在《数学趋势，分析和几何学进展》2上，作者是Birkhauser。

As for the project "Geometry of Field Theory" we gave a construction of the four-dimensional Wess-Zumino-Witten model. We proposed a definition of a Wess-Zumino-Witten action as a functor from the category conformally flat 4-manifolds to the category of line bundles with connection and gave a construction of it. This extends many phenomena discussed on 4-dimensional sphere to the class of conformally flat 4-manifolds with boundary, especially we succeeded to have Polyakov-Wiegner formula on such a manifold. We also obtained two dual types of abelian extension of the group of the smooth maps from 3-sphere to a Lie group. These results are published in Journal of Geometry and Physics, 47(2003).As for the project "Spinor Analysis" we investigated various properties of harmonic spinors on conformally flat 4-manifolds, comprising (1)Integral representation (2)local existence of the solution, (3)Runnge's approximation theorem, Mittag-Lefler theorem, (4)global existence of solutions on domains of the Dirac equation. These results are published in the Japanese Journao of Mathematics, vol.28-1(2002), 1-30. Then investigations of this subject continue to ; (5)the introduction of meromorphic spinors on conformally flat manifolds and their devisors. (6)Riemann-Roch type theorem for the cohomology groups of meromorphic spinors. The results will be published in "Trends in Mathematics, Advances in Analysis and Geometry2, by Birkhauser publ..