The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, I

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 $(\partial_k+T[e_k])Φ=0$ are given, where the elements $\{e_k\}$ are a basis of the subspace, and $T$ is the affine 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.