Boundary conditions for gavge coupled Dirac operators and their invariants.

(1)T.Kori研究了具有Grassman边界条件的规范耦合Dirac算符的指数理论，特别是给出了一种不用Atiyah-Patodi-Singer理论直接计算四维半球上这类问题的方法。(2)T.Kori.证明了规范耦合Dirac算符的手征反常公式。本文证明了S^4上规范耦合狄拉克算符的指数等于半球上几何狄拉克算符的指数，其中格拉斯曼边界条件是从矢势出发的，即规范的影响被吸收在边界条件中。这一结果将作为AMS当代数学系列丛书的一卷发表在偏微分方程几何方面的会议上。(3)解决了旋量作为零模旋量从边界延伸到内部或外部的问题。由此得到了零模旋量的洛朗展开定理的一个类比。由此引入了亚纯旋量及其余量的概念。他在S^4中证明了区域上的留数定理。许多与复变函数论中已知的定理相对应的定理有望在我们的旋量分析框架中成立。这将是我们的下一个项目。

(1) T.Kori investigated the theory of the index of a gauge coupled Dirac operator with Grassmannian boundary condition, especially he gave a direct method of calculations not using the Atiyah-Patodi-Singer theory for those problems on the four dimensional hemisphere.(2) T.Kori.proved a formula about the chiral anomaly of gauge coupled Dirac operators. Here he proved that the index of a gauge coupled Diracoperator on S^4 is equal to the index of the geometric Dirac operator on the hemisphere with a Grassmannian boundary condition comming from the vector potential, that is, the effect by the gauge is absorbed in the boundary condition. This result will be published in the Proceeding of the conference on Geometric Aspects of Partial Differential Equations as one volume of AMS Contemporary Mathematics series.(3) The problem of extension of spinors from the boundary to the interior or the exterior as zero mode spinors is solved. By this an analogy of the Laurent expansion theorem for zero mode spinors is obtained. Thus the concepts of meromorphic spinors and their residues are introduced. He proved the residue theorem on a domain in S^4. Many theorems that are counterparts of what are known in complex function theory are expected to hold in our framework of spinor analysis. This will be our next project.