Torsor structure of level-raising operators

TL;DR

This paper explores the torsor structure of level-raising operators within families of reductive complexes, providing explicit descriptions of their multiple cohomology and establishing connections to automorphism groups in complex geometry.

Contribution

It introduces a novel explicit computation of the torsor structure of multiple cohomology in complexes related to level-raising operators, linking algebraic and geometric frameworks.

Findings

Explicit torsor structure of multiple cohomology computed

Cohomology characterized as factor spaces of reduction functions

Establishes equivalence between cohomology and automorphism group actions

Abstract

We consider families of reductive complexes related by level-raising operators and originating from an associative algebra. In the main theorem it is shown that the multiple cohomology of that complexes is given by the factor space of products of reduction operators. In particular, we compute explicit torsor structure of the genus $g$ multiple cohomology of the families of horizontal complexes with spaces of of canonical converging reductive differential forms for a $C_2$-cofinite quasiconformal strong-conformal field theory-type vertex operator algebra associated to a complex curve. That provides an equivalence of multiple cohomology to factor spaces of products of sums of reduction functions with actions of the group of local coordinates automorphisms.