Sugawara Construction for Higher Genus Riemann Surfaces

By the classical genus zero Sugawara construction one obtains from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type) representations of the Virasoro algebra. In this lecture first the classical construction is recalled. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in joint work with O.K. Sheinman. We were able to show that also in the higher genus, multi-point situation one obtains from representations of the global algebras of affine type representations of a centrally extended algebra of meromorphic vector fields on Riemann surfaces. The latter algebra is the generalization of the Virasoro algebra to higher genus. Invited lecture at the XVI${}^{th}$ workshop on geometric methods in physics, Bialowieza, Poland, June 30 -- July 6, 1997.