Reduction cohomology of Riemann surfaces

Abstract

We study the algebraic conditions leading to the chain property of complexes for vertex operator algebra n-point functions (with their convergence assumed) with differential being defined through reduction formulas. The notion of the reduction cohomology of Riemann surfaces is introduced. Algebraic, geometrical, and cohomological meanings of reduction formulas are clarified. A counterpart of the Bott–Segal theorem for Riemann surfaces in terms of the reductions cohomology is proven. It is shown that the reduction cohomology is given by the cohomology of n-point connections over the vertex operator algebra bundle defined on a genus g Riemann surface [Formula: see text]. The reduction cohomology for a vertex operator algebra with formal parameters identified with local coordinates around marked points on [Formula: see text] is found in terms of the space of analytical continuations of solutions to Knizhnik–Zamolodchikov equations. For the reduction cohomology, the Euler–Poincaré formula is derived. Examples for various genera and vertex operator cluster algebras are provided.