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An operadic approach to vertex algebra and Poisson vertex algebra cohomology

Bojko Bakalov, Alberto De Sole, Reimundo Heluani, Victor G. Kac

Abstract

We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces a vertex algebra cohomology complex. Likewise, the associated graded of the chiral operad leads to a classical operad, which produces a Poisson vertex algebra cohomology complex. The latter is closely related to the variational Poisson cohomology studied by two of the authors.

An operadic approach to vertex algebra and Poisson vertex algebra cohomology

Abstract

We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces a vertex algebra cohomology complex. Likewise, the associated graded of the chiral operad leads to a classical operad, which produces a Poisson vertex algebra cohomology complex. The latter is closely related to the variational Poisson cohomology studied by two of the authors.

Paper Structure

This paper contains 54 sections, 50 theorems, 404 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on vector superspaces and the symmetric group
  3. Vector superspaces, tensor products and linear maps
  4. The action of the symmetric group on tensor powers
  5. Composition of permutations
  6. $\circ_i$-products of permutations
  7. Shuffles
  8. Superoperads and the associated $\mathbb{Z}$-graded Lie superalgebras
  9. Definition of a superoperad
  10. Universal Lie superalgebra associated to an operad
  11. The operad governing Lie superalgebras
  12. The operad governing Lie conformal superalgebras
  13. Lie conformal superalgebras
  14. The $\mathcal{C}hom$ operad
  15. Lie conformal superalgebras and the operad $\mathcal{C}hom$
  16. ...and 39 more sections

Key Result

Lemma 2.1

Let $g_i\colon U_i\to V_i$, $i=1,\dots,n$, be linear maps of vector superspaces, and let $u_i\in U_i$, $i=1,\dots,n$. For every $\sigma\in S_n$, we have

Theorems & Definitions (123)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 113 more