1993 Fiscal Year Final Research Report Summary
Reserch on loop spaces related to mathematical physics
| Project/Area Number |
03640031
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Shinshu University |
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Principal Investigator |
ASADA Akira Shinshu University, Faculty of Science, Professor, 理学部, 教授 (00020652)
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| Co-Investigator(Kenkyū-buntansha) |
ABE Kojun Shinshu Univ.General Education Professor, 教養部, 教授 (30021231)
SAITO Shiroshi , 教授(元) (10020645)
YOKOTA Ichiro , 教授(元) (20020638)
HIRONAKA Yumiko Shinshu University, Faculty of Science, Associate Professor, 理学部, 助教授 (10153652)
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| Project Period (FY) |
1991 – 1993
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| Keywords | Loop spaces / Loop group bundles / Strlng classes / WZNW terms / Current group bundles / Noncommutative connections / Non-abelian de Rham theory / Dirac operators |
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| Research Abstract |
(1) Research on loop group bundles
i. Characteristic map of a loop group bundle is realized as a matrix valued function.
ii. Lifting and descent of a vector bundle or a loop group bundle over M to a loop group bundle or a vector bundle over OMEGAM or MXS^1 are defined.
(2) Research on characteristic classes of loop group bundles (string classes)
i. Differential geometric definition of string classes is given.
ii. Relations between string classes and Chern classes of a bundle and its lifting or descent are computed.
iii. String classes are expressed as WZNW terms of characteristic maps.
(3) By using above results and non-abelian de Rham theory, relations between Chern-Simons gauge theory and topological field theory are studied.
(4) To extend above results for loop groups over exceptiohal groups, concrete realizations of exceptional groups are done.
(5) To get more advanced information in this direction, Eisenstein serieses of nurmber theoretical symmetric spaces are studied.
(6) Research on current group bundles , I.Noncommutative connections. The notion of noncommutative connection is introduced and several results such as reduction to U_1-bundles and noncommutative Poincare lemma, are get.
(7) Research on current group bundles, II.Connections with respect to the Dirac operator. Connections with respect to the Dirac operator is defined. They give families of Dirac operator and their n-functions give bundle invariants.
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Research Products
(8 results)
