Projective Flat Connections on Moduli Spaces of Riemann Surfaces and the Knizhnik–Zamolodchikov Equations

A global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus with marked points is developed. Here, the globality means that one uses the Krichever–Novikov algebras of gauge and conformal symmetries (i.e., of global symmetries) instead of the loop and Virasoro algebras, which are local in this context. A thorough account of the global approach with all necessary details from the theory of Krichever–Novikov algebras and their representations was given by the author earlier (Usp. Mat. Nauk, 1999, vol. 54, no. 1; 2004, vol. 59, no. 4). This paper focuses on the geometric ideas that underlie our construction of conformal blocks. We prove the invariance of these blocks with respect to the (generalized) Knizhnik–Zamolodchikov connection and the projective flatness of this connection.