Geometry of Field Theory and Spinor Analysis

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As for the project "Geometry of Field Theory" we gave a construction of the four-dimensional Wess-Zumino-Witten model. We proposed a definition of a Wess-Zumino-Witten action as a functor from the category conformally flat 4-manifolds to the category of line bundles with connection and gave a construction of it. This extends many phenomena discussed on 4-dimensional sphere to the class of conformally flat 4-manifolds with boundary, especially we succeeded to have Polyakov-Wiegner formula on such a manifold. We also obtained two dual types of abelian extension of the group of the smooth maps from 3-sphere to a Lie group. These results are published in Journal of Geometry and Physics, 47(2003).As for the project "Spinor Analysis" we investigated various properties of harmonic spinors on conformally flat 4-manifolds, comprising (1)Integral representation (2)local existence of the solution, (3)Runnge's approximation theorem, Mittag-Lefler theorem, (4)global existence of solutions on domains of the Dirac equation. These results are published in the Japanese Journao of Mathematics, vol.28-1(2002), 1-30. Then investigations of this subject continue to ; (5)the introduction of meromorphic spinors on conformally flat manifolds and their devisors. (6)Riemann-Roch type theorem for the cohomology groups of meromorphic spinors. The results will be published in "Trends in Mathematics, Advances in Analysis and Geometry2, by Birkhauser publ..