Reserch on loop spaces related to mathematical physics

(1)环群丛集的研究。循环群丛的特征映射被实现为矩阵值函数。定义了环M上的向量丛或环群丛到OMEGAM或MXS^1上的环群丛或向量丛的提升和下降.(2)环群丛(弦类)的特征类的研究.给出了弦类的微分几何定义.计算了丛的弦类和陈类之间的关系以及它的提升或下降。将弦类表示为特征映射的WZNW项。(3)利用上述结果和非阿贝尔德罗姆理论，研究了Chern-Simons规范理论和拓扑场论之间的关系。(4)将上述结果推广到例外群上的循环群，给出了例外群的具体实现。(5)为了在这方面得到更深入的信息，研究了Number理论对称空间的Eisenstein级数。(6)关于当前群丛的研究，I.非交换联络。引入了非对易联络的概念，得到了归结为U_1丛和非对易Poincare引理等几个结果。(7)关于当前群丛的研究，II.关于Dirac算子的联络。定义了关于狄拉克算子的联系。它们给出了狄拉克算子族，它们的n-函数给出丛不变量。

(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.