Four dimensional realization of two dimensional current groups (Q1924034)

Affine Lie groups are realized as quotients of central extensions of the current groups over Riemann surfaces. This is achieved by means of the Leray residue theory which replaces de Rham cohomology in the classical Wess-Zumino-Novikov-Witten (WZNW) construction for affine Lie groups. A review of the results on central extension of current algebras and current groups in two dimensions is first given. Then, these results are generalized to currents over Riemann surfaces with punctures. The four dimensional realization of two dimensional current groups is obtained by using Leray theory. The case of a punctured sphere is considered as an illustrative example.