ADHM-construction of vector bundles and harmonic maps

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400074
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Meiji University
• Principal Investigator: NAGATOMO yasuyuki 明治大学, 理工学部, 教授 (10266075)
• Co-Investigator(Renkei-kenkyūsha): TAKAHASHI masaro 久留米工業高等専門学校, 一般科目理科系, 准教授 (70311107)
• Research Collaborator: KOGA Isami 九州大学, 大学院・数理学府, 博士研究員 (60782232) ; Oscar Macia University of Valenncia, Faculty of Mathematical Science, Profesor ayudante doctor
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: ベクトル束 / ゲージ理論 / 調和写像 / 正則写像 / モジュライ空間 / 表現論

Outline of Final Research Achievements

A generalisation of do Carmo-Wallach theory on harmonic maps into Grassmannians is more extended in the case that the domain of maps are compact Riemannian manifolds. This theory enables us to construct moduli spaces of harmonic maps in a similar way to the ADHM-construction of instantons on the 4-sphere.
This theory has a lot of applications. As one of them, we can construct moduli spaces of holomorphic isometric embeddings of the complex projective line into a complex quadric hypersurface of the projective space. Due to this, it turned out that the moduli space has a structure of foliation whose leaves are Kaehler quotients of flat spaces.
As another example, we can classify equivariant holomorphic maps of complex projective line into complex Grassmannian of 2-planes. In this case, our　problem reduces to classify invariant connections on vector bundles of rank 2.　In each case, our theory provides the compactification of the moduli space with a natural geomertic interpretation.

Free Research Field

微分幾何学