MIZUTANI Tadayoshi Saitama University, Dept.of Math., Professor, 理学部, 教授 (20080492)
SAKAI Fumio Saitama Univ., Dept.of Math., Professor, 理学部, 教授 (40036596)
EGASHIRA Shinji Saitama Univ., Dept.of Math., Assistant Professor, 理学部, 助手 (00261876)
SAKAMOTO Kunio Saitama Univ., Dept.of Math., Professor, 理学部, 教授 (70089829)
OKUMURA Masafumi Saitama Univ., Dept.of Math., Professor, 理学部, 教授 (60016053)
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).