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Cohomological vertex algebras

Colton Griffin

TL;DR

The paper develops cohomological vertex algebras (CVAs) as higher-dimensional analogs of vertex algebras, replacing the formal punctured disk with cohomology rings $H^ullet(D,\mathcal{O})$ to model $n$-dimensional coordinates and enable moduli-space constructions. It provides a rigorous definition of CVAs and cohomological vertex operator algebras (CVOAs), proves key structural properties such as locality, skew-symmetry, Jacobi identity, and various equivalent axiom systems, and reformulates the theory in terms of correlation functions via reconstruction. Through a reconstruction theorem, it constructs explicit CVA examples including a cohomological Virasoro vertex algebra, a $eta\gamma$-system analogue, a Heisenberg analogue, and affine CVAs, along with BRST reduction leading to cohomological W-algebras. The framework aims to connect higher-dimensional chiral/factorization-type structures to CVAs and their modules, with potential implications for higher-dimensional field theories and moduli problems.

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 $βγ$-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.

Cohomological vertex algebras

TL;DR

The paper develops cohomological vertex algebras (CVAs) as higher-dimensional analogs of vertex algebras, replacing the formal punctured disk with cohomology rings to model -dimensional coordinates and enable moduli-space constructions. It provides a rigorous definition of CVAs and cohomological vertex operator algebras (CVOAs), proves key structural properties such as locality, skew-symmetry, Jacobi identity, and various equivalent axiom systems, and reformulates the theory in terms of correlation functions via reconstruction. Through a reconstruction theorem, it constructs explicit CVA examples including a cohomological Virasoro vertex algebra, a -system analogue, a Heisenberg analogue, and affine CVAs, along with BRST reduction leading to cohomological W-algebras. The framework aims to connect higher-dimensional chiral/factorization-type structures to CVAs and their modules, with potential implications for higher-dimensional field theories and moduli problems.

Abstract

Vertex algebras (and their modules) can be described as vector spaces together with a linear operator-valued series in one parameter . With the interpretation of as a coordinate at a point on a curve, one can construct algebraic structures on the moduli space of curves from -modules. Here we propose a generalization of vertex algebras involving linear operators in parameters . One may interpret these as being the components of a set of coordinates on an -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 -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 -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.

Paper Structure

This paper contains 24 sections, 29 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Plan of the paper
  4. Preliminaries on formal calculus and local systems of fields
  5. Formal Calculus
  6. Taylor expansions and correlation functions
  7. Fields and locality
  8. Cohomological vertex algebras
  9. The main definitions
  10. Basic properties of cohomological vertex algebras
  11. Equivalence of various axioms for vertex algebras
  12. The case of characteristic 0
  13. Equivalent presentations of the Jacobi identity
  14. Replacing translation covariance and weak commutativity
  15. Vertex algebras in terms of correlation functions
  16. ...and 9 more sections

Key Result

Lemma 2.1.3

For all $\bf j,\bf k\geq \bf 0$ we have $\partial^{(\bf j)}_{\bf z}\partial^{(\bf k)}_{\bf z} = \binom{\bf j+\bf k}{\bf j}\partial^{(\bf j+\bf k)}_{\bf z}$. When $R$ is a $\mathbb Q$-algebra, then

Theorems & Definitions (68)

  • Remark 2.1.2
  • Lemma 2.1.3
  • Definition 2.1.5
  • Lemma 2.1.6
  • proof
  • Remark 2.1.7
  • Lemma 2.2.1
  • Definition 2.2.2
  • Lemma 2.2.3
  • Definition 2.3.1
  • ...and 58 more