Reserch on loop spaces related to mathematical physics

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(1) Research on loop group bundlesi. Characteristic map of a loop group bundle is realized as a matrix valued function.ii. Lifting and descent of a vector bundle or a loop group bundle over M to a loop group bundle or a vector bundle over OMEGAM or MXS^1 are defined.(2) Research on characteristic classes of loop group bundles (string classes)i. Differential geometric definition of string classes is given.ii. Relations between string classes and Chern classes of a bundle and its lifting or descent are computed.iii. String classes are expressed as WZNW terms of characteristic maps.(3) By using above results and non-abelian de Rham theory, relations between Chern-Simons gauge theory and topological field theory are studied.(4) To extend above results for loop groups over exceptiohal groups, concrete realizations of exceptional groups are done.(5) To get more advanced information in this direction, Eisenstein serieses of nurmber theoretical symmetric spaces are studied.(6) Research on current group bundles , I.Noncommutative connections. The notion of noncommutative connection is introduced and several results such as reduction to U_1-bundles and noncommutative Poincare lemma, are get.(7) Research on current group bundles, II.Connections with respect to the Dirac operator. Connections with respect to the Dirac operator is defined. They give families of Dirac operator and their n-functions give bundle invariants.