Research on the global structure of manifold

The head investigator introduced the concept of Spin I D 1 q I D 1 structure in the paper“Spin I D 1 q I D 1 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 I D1q Ie D1-bundle divided by Spin y D1c i D1is called the twistor space，which possesses a natural Spin y D1 c e D1structure。In particular，the twistor bundle over an odd-dimensional Spin Ney D1qi D1manifold is very interesting.In the case，the Dirac operators D I D 1c i D 1，D I D 1 q D 1 associated to the Spin I D 1 c D 1，Spin I D 1 q D 1 structures produce the non-zero eta-invariantsη(D I D 1 C I D 1)，η(D I D 1 Q I D 1)。We studied the relation between the adiabatic limit lim y D2E I D2→I D 20 I D 2η(D I D 3 c(/)E I D 3)andη(D I D 1 Q I D 1)，where D D 3 c(/)E I D 3 is the adiabatic version of D I D 1.(The limit will correspond to the global anomaly in physics.)The above are reported in the paper“Nagase：Spin I D 1 q I D 1，twistor and Spin I D 1 c I D 1(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 I D1qi D1manifolds(Preprint)，·Nagase：Twistor space and the Seiberg-Witten@equation(Preprint)。

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).