Field Theory and infinite dimensional (toroidal) algebra

我们专注于（无限维度）对称性，它构成了理论物理的中心支柱之一。特别地，我们研究了出现在高维KWZW（K\ \ ahler-Wess-Zumino-Witten）模型和与Seiberg-Witten （S-W）理论相关的可积系统，即椭圆Calogero-Moser（C-M）模型中的环面代数。通过本研究阐明了理论物理，特别是场论和量子力学中各种形式的对称代数。这项为期三年的研究的主要目的是：1。无限维系统中环面代数的实现（场论）（退化）环面代数在有效场论（S-W理论等）和多粒子动力系统中的实现从环面代数到仿射代数在lax表示水平上的转换和由此得到的守恒量的澄清，3。在椭圆C-M系统和（仿射）Toda理论的经典（和量子）解水平上理解从环面代数到仿射代数转换的动力学意义，Sasaki：在第一年，他专注于C-M模型，这是在经典和量子水平上有限自由度的可积动力系统。第二年，研究了量子C-M系统。在第三年，增加了自旋自由度，哈伯德型模型，所谓的“相对论泛化”，准精确可积扩展，即所谓的Inozemtsev模型。Inami：第一年，研究了出现在各种可积模型中的对称代数与它们的动力学之间的关系。在第二年和第三年，对同样的研究对象进行了更详细、更深层次的调查。研究了具有扩展超对称的三维非线性sigma模型的紫外散度，以及非线性sigma模型的kikn解和块解。

We focused on (infinite dimensiona) symmetries, which constitute one of the center pillars of theoretical physics. In particular, we investigated the toroidal algebras appearing in the higher dimensional KWZW(K\"ahler-Wess-Zumino-Witten) models and integrable systems associated with Seiberg-Witten (S-W) theory, namely, elliptic Calogero-Moser(C-M) models. Various forms of symmetry algebras in theoretical physics, in particular, field theory and quantum mechanics are clarified through this research.Main aims of the three year research were :1. realisation of toroidal algebra in infinite dimensional systems (field theories)2. realisation of (degenerate) toroidal algebra in effective field theories (S-W theory, etc) and multi-particle dynamical systems3. clarification of the transition from the toroidal to affine algebras at the levels of Lax-representation and the conserved quantities obtained from it,4. understanding the dynamical meaning of the transition from the toroidal to affine algebras at the levels of classical (and quantum) solutions of elliptic C-M systems and (affine) Toda theories,Sasaki : In the first year, he focused on the C-M models, which are integrable dynamical systems of finite degrees of freedom at the classical and quantum levels. In the second year, quantum C-M systems were investigated. In the third year, the addition of spin degrees of freedom, Hubbard type models, so-called "relativistic generalisation", the quasi-exact integrable extension, i.e. so-called Inozemtsev models, were pursued.Inami : In the first year, the relationship between symmetry algebras appearing in various integrable models and their dynamics was pursued. In the second and third years the same subjects were investigated in more detail and at deeper levels. The ultra violet divergences in three dimensional non-linear sigma models with extended supersymmetry, and kikn solutions, lump solutions of non-linear sigma models were investigated.