Bigraded differential algebra for vertex algebra complexes

TL;DR

This paper develops a bigraded differential algebra structure for vertex algebra complexes, revealing new cohomology invariants and algebraic relations that deepen understanding of vertex algebra cohomology.

Contribution

It introduces a novel bigraded differential algebra framework for vertex algebra complexes, linking cohomology invariants with algebraic structures.

Findings

Bigraded differential algebra structure is established for vertex algebra bicomplexes.

New cohomology invariants are generated by algebra commutation relations.

The algebraic structure forms a continual Lie algebra with a grading-restricted vertex algebra.

Abstract

For an infinite chain bicomplex we show that the orthogonality and grading conditions provide it with the structure of a bigraded differential algebra with respect to a natural multiplication of several elements bicomplex spaces. Corresponding bigraded algebra commutation relations generate a sequence of non-vanishing cohomology invariants associated to vertex algebras. In particular, we apply this result to the bicomplex of grading-restricted vertex algebra cohomology endowed with a multiplication we introduce. We provide examples associated to various choices of vertex algebra bicomplex subspaces. The generators and commutation relations of the bigraded differential algebra form a continual Lie algebra with the root space provided by a grading-restricted vertex algebra.