Infinite hidden symmetries and integrability of N=4 super-Yang-Mills theory

代数几何方法在世界范围内弦、规范理论和可积模型的研究中发挥着越来越重要的作用。我和我的合作者通过研究N=2弦及其隐藏的Kac-Moody型对称性，超扭转空间上的拓扑弦及其相应的弦场理论，全纯chen - simons和超yang - mills理论，以及低维的可积模型，参与了这一努力。我们的研究使我们在扭转方法、微分几何、扩展超对称和可积性方面获得了深远的经验。这一能力将进一步用于改进对N=4超对称杨-米尔斯理论的无限隐对称性的处理。本课题拟从泊松几何、泊松-李群和修整变换的角度，阐述N=4超杨-米尔斯理论及其自对偶子扇区的扭曲描述。用扭扭方法结合泊松几何和泊松-李群的方法来考虑N=4超杨-米尔斯理论还没有得到解决。我们打算填补这一空白，研究N=4超对称杨-米尔斯理论的隐藏泊松-李对称性。

We are witnessing an increasing role of algebro-geometric methods in the worldwide study of strings, gauge theories and integrable models. Me and my collaborators have participated in this endeavors through the investigation of N=2 strings and their hidden Kac-Moody type symmetries, of topological strings on supertwistor spaces and the corresponding string field theories, of holomorphic Chern-Simons and super-Yang-Mills theories, as well as of integrable models in lower dimensions. Our studies have given us far-reaching experience with twistor methods, differential geometry, extended supersymmetry and integrability. This competence will be further employed to an improved handling of the infinite hidden symmetries of N=4 supersymmetric Yang-Mills theory.The present project intends to elaborate the twistor description of N=4 super-Yang-Mills theory and its self-dual subsector from the viewpoint of Poisson geometry, Poisson-Lie groups and dressing transformations. The consideration of N=4 super-Yang-Mills theory by twistor methods combined with the methods of Poisson geometry and Poisson-Lie groups remains yet unaddressed. We intend to fill this gap and to study hidden Poisson-Lie symmetries of N=4 supersymmetric Yang-Mills theory.