The product on $\W$-spaces of rational forms

TL;DR

This paper introduces a geometric product on spaces of rational differential forms associated with vertex algebras, using Riemann sphere sewing, and studies its algebraic properties within chain complexes.

Contribution

It defines a new geometric product on rational form spaces related to vertex algebras and analyzes its properties within chain complexes.

Findings

The product is well-defined via Riemann sphere sewing.

The product preserves invariance under complex parameter transformations.

Properties of the product are characterized within the chain complex structure.

Abstract

Let $V$ be a quasi-conformal grading-restricted vertex algebra, $W$ be its module, and $\W_{z_1, \ldots, z_n}$ be the space of rational differential forms with complex parameters $(z_1, \ldots, z_n)$ for $n \ge 0$. Using geometric interpretation in terms of two Riemann spheres sewing we define a product of elements of two spaces $\W_{x_1, \ldots, x_k}$ and $\W_{y_1, \ldots, y_n}$, and study its properties. A product is introduced also for elements of two spaces $C^k_m(V, \W)$ $\times$ $C^n_{m'}(V, \W)$ $\to$ $C^{k+n}_{m+m'}(V, \W)$ of the corresponding chain complex of rational differential forms invariant with respect to transformations of complex parameters.