Spin^q structures and the adiabatic limit

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As for the infinitesimal chiral anomaly used in physics, the head investigator felt some ambiguity about how to define that, what should be investigated, etc., as a mathematical object. In the project, he proposed its mathematical definition and tried to withdraw its essential partHis previous study says that a Spin^q manifold possesses a canonical CP^1-fibration and its total space called a twistor space has a canonical Spin structure. The structure induces the Dirac operator θ. First, its infinitesimal variation δ_χθ in the X-direction, where X is a cross-section of a certain adjoint bundle, and its anomaly denoted log det δ_χθ were defined from the mathematical viewpoint. Since the corresponding spinor bundle also changes it is nonsense to take naively the variation of θ. Hence it was essential how to interpret δ_χθ. Second, he tried to withdraw an essential part of the anomaly. After the analogy of the physical twistor theory and the creating theory of the universe, he considered the operation of collapsing each fiber into one point (returning to the pre-universe), i.e., the operation of taking the adiabatic limit, to produce its essential part denoted lim_<ε→0> log det δ_xθ_ε. In the latter half of the project, to investigate the limit was the main purpose. He conjectured the essential part lim_<ε→0> log det δ_χθ_ε depends essentially on the behavior when ε → 0 of Tr(δ_χθ_ε・θ_εe^<-tθ^<^2_ε>>) under the condition 0 < t < ε^a (a>0) and has nearly finished its study.