A REALIZATION OF THE KAC-MOODY ALGEBRA ON HOLOMORPHIC FIELDS

We discuss the loop group G(S1, G) of mappings of the circle S1 into a compact Lie group G. We show that there exists a realization of projective unitary representation of G in the Hilbert space of functionals of holomorphic fields with values in G(C) (the complexification of G). We show that this representation belongs to the discrete series of positive energy representations of G. We discuss the quantum field theory of holomorphic fields.