Research on the global structure of manifold

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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).