Cohomological vertex algebras

TL;DR

This paper introduces cohomological vertex algebras (CVAs), generalizing traditional vertex algebras to multiple parameters, and explores their structural properties, examples, and applications in algebraic geometry and mathematical physics.

Contribution

It proposes the concept of CVAs, establishes their foundational theorems, and provides explicit examples and applications, extending the framework of vertex algebras to higher-dimensional cohomological settings.

Findings

Defined cohomological vertex algebras and vertex operator algebras.

Constructed examples including the $eta ext{-}eta$ system and affine Kac-Moody CVAs.

Connected CVAs to BRST reduction and W-algebras.

Abstract

Vertex algebras (and their modules) can be described as vector spaces together with a linear operator-valued series in one parameter $z$. With the interpretation of $z$ as a coordinate at a point on a curve, one can construct algebraic structures on the moduli space of curves from $V$-modules. Here we propose a generalization of vertex algebras involving linear operators in parameters $z_1,\ldots,z_n$. One may interpret these as being the components of a set of coordinates on an $n$-dimensional algebraic variety. These are referred to as cohomological vertex algebras (CVAs): the formal punctured 1-disk underlying a vertex algebra is replaced by a ring modeling the cohomology of certain modifications of the formal $n$-disk. We prove several structural theorems for CVAs and give a definition of cohomological vertex operator algebras (CVOAs). Using a reconstruction theorem for CVAs, we provide basic examples such as the $\beta\gamma$-system, the Heisenberg CVA, and the affine Kac-Moody CVAs. We use these constructions to describe BRST reduction, leading to an analog of W-algebras.