Field Theory and infinite dimensional (toroidal) algebra

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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.