2012 Fiscal Year Final Research Report
A generalization of the Atiyah-Singer Index Theorem on Noncommutative manifolds
| Project/Area Number |
22540077
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Nagoya University |
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Principal Investigator |
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| Co-Investigator(Renkei-kenkyūsha) |
MAEDA Yoshiaki 慶應義塾大学, 理工学部, 教授 (40101076)
MIYAZAKI Naoya 慶應義塾大学, 経済学部, 教授 (50315826)
KATO Tsuyoshi 京都大学, 大学院・理学研究科, 教授 (20273427)
NATSUME Toshikazu 名古屋工業大学, 工学系研究科, 教授 (00125890)
MITSUMATSU Yoshihiko 中央大学, 理工学部, 教授 (70190725)
ONO Kaoru 北海道大学, 大学院・理学研究院, 教授 (20204232)
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| Project Period (FY) |
2010 – 2012
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| Keywords | 指数定理 / 非可換幾何学 / K理論 / エータ不変量 / 葉層多様体 / Godbillon-Vey類 / 巡回コホモロジー |
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| Research Abstract |
The purpose of the present research are;
1) to find an extension of the Atiyah-Singerindex theorem in the framework of Noncommutative Geometry;
2) to apply such anoncommutative index theorem to Geometry and String theory. Achieving the presentresearch we obtained finally the following results.
First we extended the classicalAtiyah-Patodi-Singer index theorem to foliated manifolds with boundary of higherdimensional leaves and obtained an index theorem involved with the Godbillon-Veyclass (a joint work with P. Piazza).
Second, by exploiting the framework of Noncommutative Geometry, we extended the domain of the Godbillon-Vey class (differentiability of foliations), and related our cocycle to the area cocycle defined by T.Tsuboi.
Third, we clarified relationship among the family index theorem of odd dimension, the Dixmier-Douady class (a characteristic class of gerbe) and the Godbillon-Vey class.
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