FIELD MODELS FROM 2-COCYCLES ON INFINITE-DIMENSIONAL LIE-GROUPS AND SYMPLECTIC STRUCTURES - 2D-GRAVITY AND CHERN-SIMONS THEORY

In this paper the connection between field models and infinite-dimensi onal Lie groups is widely analysed on the bases of a new group quantiz ation approach. We also relate the Poincare-Cartan form of variational calculus to the symplectic current/structure of the covariant phase-s pace formulation of (higher-derivative) field theory. The Virasoro and Kac-Moody groups are considered. In the first case the action functio nal of the 2D-induced gravity in the light-cone formulation is derived . The hidden SL(2, R) simply appears as generated by the kernel of the Lie algebra two-cocycle and plays the role of a gauge-type symmetry. Nevertheless, it is shown that a proper space-like formulation is out of reach of the Virasoro group. The corresponding symplectic structure of the (non-local) action functional is determined showing that it is related to the symplectic structure associated with the SL(2,R)-Kac-M oody group. This unravels the proper geometrical meaning of the hidden symmetry and differs from the analysis in related works based on the coadjoint-orbit approach. The relation between the Kac-Moody groups an d the Chem-Simons gauge theory on a disc in the presence of a source i s considered using the new approach.