2007 Fiscal Year Final Research Report Summary
Geometric structures on the moduli spaces in gauge theory and its applications to topology
| Project/Area Number |
18540094
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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 | Keio University |
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Principal Investigator |
KAMETANI Yukio Keio University, Faculty of Science and Technology, Associate Professor (70253581)
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| Co-Investigator(Kenkyū-buntansha) |
FURUTA Mikio University of Tokyo, Graduate Schooll of Mathematical Sciences, Professor (50181459)
MAEDA Yoshiaki Keio University, Faculty of Science and Technology, Professor (40101076)
MORIYOSHI Hitoshi Keio University, Faculty of Science and Technology, Associate Professor (00239708)
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| Project Period (FY) |
2006 – 2007
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| Keywords | gauge theory / geometric quantization |
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| Research Abstract |
Gauge theory has still played a central role in low dimensional topology. In particular Seiberg-Witten theory has been studied from the view point of its applications to topology. In this area M. Furuta has introduced a finite dimensional approximation to capture the equation in equivariant homotopy theory, by which he has obtained a refinement of the invariant and the 10/8-inequality for closed spin 4-manifolds.
This research consists of three parts. In the first part we studied this invariant by using homotopy theory. In the second part we gave a geometric interpretation for this invariant. In the third part we studied the index theorem as an application to the geometric quantization conjecture of Guillmen and Sternberg. The detail is given below.
The first one is a joint work with Norihiko Minami. We used a result by Nishida to get a vanishing theorem of the refined Seiberg-Witten invariants for the connected sum of 4-manifolds. In contrasts with this result, we have got a nonvanishing theorem. Then this result tells us that the Seiberg-Witten invariants cannot distinguish differential structure for the connected sum of too many 4-manifolds. In the second one we studied equivariant spin structures on the moduli spaces to Seiberg-Witten equations for a spin 4-manifold. Moreover we extended this result to a spin 4-manifold with positive first Betti number. These imply that we can reprove the 10/8-inequlatiy and its extension from the moduli space with the spin structure. As to the third one, the head investigator got a report that the Guilleman-Sternberg conjecture for Hamitonian torus actions can be reproved from the localization of the Riemann-Roch number.
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Research Products
(16 results)
