Research on the global structure of manifold

“The head investigator介绍了纸上旋转D1 q D1结构的概念”旋转D1 q D1结构（J. Math. Soc. Japan， 47（1995）， 93-119）。在项目中，相关扭曲空间的某些属性，其中扭曲空间的概念是Penrose、Salamon等的通用化。were investigated， mainly， related to the operation of adiabatic limit.The total space of a Spin D1q d1-bundle divided by Spin d1c D1 is called the twistor space， which possesses a natural spin D1c D1 structure。在特别的情况下，扭曲的捆绑在一个古怪的维度旋转的D1 q D1 manifold是非常有趣的。In the case， the Dirac operators D We studied the relation between the adiabatic limit limイイD2EイイD2 ￣イD20イイD2（DイD3c（\）EイD3）和（DイD1qイD1）， where DイD3c（\）EイD3 is the adiabatic version of DイD1cイD1。“极限将对物理学中的全球异常现象产生正确的影响。”“The above are reported in the paper”Nagase ： SpinイD1 qイD1， twistor and SpinイD1 cイD1（Commun。Math. Phys，1997年， 107-126年。Moreover，研究开发人员进入Nagase ：扭曲空间和Dirac运算符的偶联限制（预印），Nagase ：旋转D1 q D1 Manifolds签名运算符的偶联限制（预印），Nagase ：扭曲空间和Seiberg-Witten equation （预印）。

The head investigator introduced the concept of SpinィイD1qィエD1 structure in the paper "SpinィイD1qィエD1 structures (J. Math. Soc. Japan, 47(1995), 93-119)". In the project, some properties of the associated twistor space, which is a generalization of the concept of twistor space due to Penrose, Salamon, etc., were investigated, mainly, related to the operation of adiabatic limit.The total space of a SpinィイD1qィエD1-bundle divided by SpinィイD1cィエD1 is called the twistor space, which possesses a natural SpinィイD1cィエD1 structure. In particular, the twistor bundle over an odd-dimensional SpinィイD1qィエD1 manifold is very interesting. In the case, the Dirac operators DィイD1cィエD1, DィイD1qィエD1 associated to the SpinィイD1cィエD1, SpinィイD1qィエD1 structures produce the non-zero eta-invariants η(DィイD1cィエD1), η(DィイD1qィエD1). We studied the relation between the adiabatic limit limィイD2EィエD2→ィイD20ィエD2η(DィイD3c(/)EィエD3) and η(DィイD1qィエD1), where DィイD3c(/)EィエD3 is the adiabatic version of DィイD1cィエD1. (The limit will correspond to the global anomaly in physics.) The above are reported in the paper "Nagase : SpinィイD1qィエD1, twistor and SpinィイD1cィエD1(Commun. Math. Phys., 189(1997), 107-126)". Moreover, the research develops into・Nagase : Twistor spaces and the adiabatic limits of Dirac operators (preprint),・Nagase : The adiabatic limits of signature operators for SpinィイD1qィエD1 manifolds (preprint),・Nagase : Twistor space and the Seiberg-Witten equation (preprint).