Higher Courant-Dorfman algebras and associated higher Poisson vertex algebras

TL;DR

This paper introduces higher Courant-Dorfman algebras and explores their relationship with graded symplectic geometry, extending classical structures to higher degrees and providing algebraic descriptions of BFV current algebras.

Contribution

It defines higher Courant-Dorfman and Poisson vertex algebras, establishing their properties and connections with graded symplectic geometry and differential-graded manifolds.

Findings

Higher Courant-Dorfman algebras generalize classical structures to higher degrees.

The paper relates these algebras to functions on dg symplectic manifolds.

An algebraic description of BFV current algebras is provided.

Abstract

In this paper, we consider a notion of a higher version of the relation between Courant-Dorfman algebras and Poisson vertex algebras. We define a higher Courant-Dorfman algebra, and study the relationship with graded symplectic geometry. In particular, we give graded Poisson algebras of degree $-n$ in the non-degenerate case. For higher Courant-Dorfman algebras coming from finite-dimensional vector bundles, they coincide with the algebras of functions of the associated differential-graded(dg) symplectic manifolds of degree $n$. We define a higher Lie conformal algebra and Poisson vertex algebra, and give a higher (weak) Courant-Dorfman algebraic structure arising from them. Moreover, we prove that the higher Lie conformal algebras and higher Poisson vertex algebras have properties like Lie conformal algebras and Poisson vertex algebras. As an example, we obtain an algebraic description of Batalin-Fradkin-Vilkovisky(BFV) current algebras.