Double Loop Groups and Algebras, Central Extensions, and their Representations

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The proposed research is to study representation theory of double loop groups and algebras. The main idea of this project is that the "right" objects to study are not the groups nor algebras themselves, but their categorical central-extensions. The PI and his collaborator have already constructed these central-extensions and their representations. The PI will continue to study the finer structures on these extensions and their representations, and their interaction with algebraic geometry, K-theory, and topology. The PI also proposed to study various aspects of the Geometrical Langlands Duality. The proposal plans to understand the geometry of moduli space of local systems (the Langlands parameter), in particular the wild ramified local systems.Representation theory is a branch of mathematics that studies symmetries via linear algebra (or linear vector spaces), which is so far proved to be a very brilliant idea and a very powerful tool in the study of many other disciplines, such as physics and number theory. However, in recent years, there are many clues that there are symmetries that are complicated enough so that one should study them not via linear vector spaces, but via linear categories. The PI's research provides (probably the first) examples of such complicated symmetry which is better studied by categories rather than vector spaces. It is also hoped that studying this symmetry would shade lights on two-dimensional Langlands duality and four-dimensional quantum field theory.