Reduction cohomology of Riemann surfaces

TL;DR

This paper introduces the concept of reduction cohomology for Riemann surfaces, linking algebraic, geometric, and cohomological aspects, and provides explicit calculations and examples for various genera and vertex operator algebras.

Contribution

It defines reduction cohomology for Riemann surfaces, proves a Bott-Segal type theorem, and relates it to solutions of Knizhnik-Zamolodchikov equations in vertex operator algebras.

Findings

Reduction cohomology is expressed via n-point connections on Riemann surfaces.

A Bott-Segal type theorem for Riemann surfaces is established.

Explicit examples for different genera and vertex operator algebras are provided.

Abstract

We study the algebraic conditions leading to the chain property of complexes for vertex operator algebra $n$-point functions 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 is 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 $\Sigma^{(g)}$. The reduction cohomology for a vertex operator algebra with formal parameters identified with local coordinates around marked points on $\Sigma^{(g)}$ is found in terms of the space of analytical continuations of solutions to Knizhnik-Zamolodchikov equations. For the reduction cohomology, the Euler-Poincare formula is derived. Examples for various genera and vertex operator cluster algebras are provided.