Product-type classes for vertex algebra cohomology of foliations on complex curves

TL;DR

This paper develops a new product structure for vertex algebra cohomology associated with foliations on complex curves, extending classical cohomological concepts into the vertex algebra framework.

Contribution

It introduces a novel product for double complex spaces in vertex algebra cohomology and establishes a vertex algebra analogue of classical cohomological classes.

Findings

Defined a product of double complex spaces for vertex algebra cohomology.

Introduced a vertex algebra version of classical cohomological classes.

Established orthogonality conditions for elements in the double complex spaces.

Abstract

We define a product of pairs of double complex spaces $C^n_m(V, \mathcal{W}, \mathcal{F})$ for gradig-restricted vertex algebra cohomology of codimension one foliation on a complex curve. We introduce a vertex algebra counterpart of the classical %product-type cohomological class using the orthogonality conditions on elements of double complex spaces with respect to the product we introduced.