Representation Theory of Infinite-Dimensional Lie Superalgebras and W-algebras and Their Mathematical Applications

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Under this project, the author has made joint research with Professor Victor G. Kac of Massachusetts Institute of Technology and obtained some remarkable results on the structure and representation theory of affine Lie superalgebras and supercoformal algebras (SCA), which are described as follows :1.We gave explicit construction of basic representations of classical affine superalgebras sl(m|n)^and osp(m|n)^in terms of bosons and fermions. Using this construction, we obtained character formulas of Weyl-Kac type and of theta-function type and of quasi-particle-type for basic sl(m|n)^modules and deduced modular properties of characters of basic osp(m|n)^-modules.2.We found that the characters of basic representations of affine superalgebras sl(m|l)^are written in terms of Appell's elliptic functions and, by using them, deduced the asymptotics of their characters.3.Most important classes among infinite-dimensional superalgebras are "affine superalgebrad" and "superconformal algebras". But … More there are quite differences between them. For superconformal algebras there exist no invariant bilinear forms nor Weyl groups, which are most important tools for the study of representations of affine superalgebras. Because of this reason, the study of representation of superconformal algebras has been extremely difficult and there were only a few rigorous mathematical results about it. We succeeded to construct W-algebras in terms of the quantization of Drinfeld-Sokolov reduction, which enabled us to study the structure and representations of superconformal algebras in detail.4.Vertex algebras play important roles in the theory of W-algebras. But "Ramond-type particles" cannot be treated in the framework of vertex algebras. In order to treat them, we need to consider "twisted" vertex algebras. So we went further to extend our theory of W-algebras to twisted vertex algebras and constructed "twisted superconformal algebras" and studied their structures and representations.5.The theory of W-algebras is the "function" from affine superalgebras to superconformal algebras. Namely W-algebras do not only give construction of superconformal algebras, but also transform representations of affine superalgebras to representations of superconformal algebras. This gave remarkable progress in the representation theory of superconformal algebras. As its application, we gave determinant formulas, character formulas and free fields representations of superconformal algebras. Less