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

本文主要研究双环群和代数的表示理论。这个项目的主要思想是，研究的“正确”对象不是群也不是代数本身，而是它们的范畴中心扩展。PI和他的合作者已经构造了这些中心扩展和它们的表示。PI将继续研究这些扩展及其表示的更精细结构，以及它们与代数几何、k理论和拓扑学的相互作用。PI还建议研究几何朗兰兹对偶的各个方面。该提案计划了解局部系统（朗兰兹参数）的模空间几何，特别是野生分支局部系统。表征理论是通过线性代数（或线性向量空间）研究对称性的数学分支，迄今为止，它被证明是一个非常聪明的想法，也是研究许多其他学科（如物理学和数论）的一个非常强大的工具。然而，近年来，有许多线索表明，存在足够复杂的对称性，以至于人们不应该通过线性向量空间，而是通过线性范畴来研究它们。PI的研究提供了（可能是第一个）这种复杂对称性的例子，这种对称性最好通过分类而不是向量空间来研究。人们还希望对这种对称性的研究能给二维朗兰兹对偶和四维量子场论带来启示。

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.