Boundary conditions for gavge coupled Dirac operators and their invariants.
Project/Area Number  09640134 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  Waseda University 
Principal Investigator 
KORI Toshiaki Waseda Univ.School of Sci.& Eng.Professor, 理工学部, 教授 (50063730)

CoInvestigator(Kenkyūbuntansha) 
福島 延久 早稲田大学, 理工学部, 助手 (00298168)
SUZUKI Takeru Waseda Univ.Sch.Sci.& Eng.Professor, 理工学部, 教授 (60047347)
KOSAMA Tokitake Waseda Univ.Sch.Sci.& Eng.Professor, 理工学部, 教授 (10063538)
TANAKA Kazunaga Waseda Univ.School of Sci.& Eng.Associate Prof., 理工学部, 助教授 (20188288)
KAJI Hajime Waseda Univ.School of Sci.& Eng.Associate Prof., 理工学部, 助教授 (70194727)
YONIDA Gen Waseda Univ.Sch.Sci.& Eng.Assistant, 理工学部, 助手 (90277848)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1997 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  Dirac operator / Index theorem / Spinor analysis / Chiral anomaly / ディラック作用素 / 指数定理 / スピノール解析 / カイラルアノマリ / Dirac作用素 / ゲージ項 / グラスマン境界条件 / インスタントン / ヤン・ミルズ・ヒグス理論 
Research Abstract 
(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 AtiyahPatodiSinger 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.

Report
(4results)
Research Output
(5results)