In this paper, we develop the general approach, introduced in [1], to Lax operators on algebraic curves. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct orthogonal and symplectic analogs of Lax operators, prove that they form almost graded Lie algebras, and construct local central extensions of these Lie algebras.
We show" how to obtain from highest-weight representations of Krichever-Novikov algebras of affine type (also called higher-genus affine Kac-Moody algebras) representations of centrally extended Krichever-Novikov vector field algebras via the Sugawara construction. This generalizes classical results, where one obtains representations of the Virasoro algebra. Relations between the weights of the corresponding representations axe given, and Casimir operators are constructed. In the Appendix, the Sugawara construction for the multi-point situation is outlined.
Abstract. Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus g are given. Sheaves of representations of affine Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann surfaces (respectively of smooth, projective complex curves) with N marked points are introduced. It is shown that the tangent space of the moduli space at an arbitrary moduli point is isomorphic to a certain subspace of the Krichever-Novikov vector field algebra given by the data of the moduli point. This subspace is complementary to the direct sum of the two subspaces containing the vector fields which vanish at the marked points, respectively which are regular at a fixed reference point. For each representation of the affine algebra 3g − 3 + N equations (∂ k + T [e k ])Φ = 0 are given, where the elements {e k } are a basis of the subspace, and T is the Sugawara representation of the centrally extended vector field algebra. For genus zero one obtains the Knizhnik-Zamolodchikov equations in this way. The coefficients of the equations for genus one are found in terms of Weierstraß-σ function.
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