On free conformal and vertex algebras

These are the notes of an informal talk in Bonn describing how to define an analogue of vertex algebras in higher dimensions.

We develop methods for computation of Poisson vertex algebra cohomology. This cohomology is computed for the free bosonic and fermionic Poisson vertex (super)algebras, as well as for the universal affine and Virasoro Poisson vertex algebras. We establish finite dimensionality of this cohomology for conformal Poisson vertex (super)algebras that are finitely and freely generated by elements of positive conformal weight.

In this paper we develop a formalism for working with twisted realizations of vertex and conformal algebras. As an example, we study realizations of conformal algebras by twisted formal power series. The main application of our technique is the construction of a very large family of representations for the vertex superalgebra $\goth V_\Lambda$ corresponding to an integer lattice $\Lambda$. For an automorphism $\^\sigma:\goth V_\Lambda\to\goth V_\Lambda$ coming from a finite o...

We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields.

This paper investigates the algebraic structure of indecomposable $\mathbb{N}$-graded vertex algebras $V = \bigoplus_{n=0}^{\infty} V_n$, emphasizing the intricate interactions between the commutative associative algebra $V_0$, the Leibniz algebra $V_1$ and how non-degenerate bilinear forms on $V_0$ influence their overall structure. We establish foundational properties for indecomposability and locality in $\mathbb{N}$-graded vertex algebras, with our main result demonstrati...

Using the Zhu algebra for a certain category of $\mathbb{C}$-graded vertex algebras $V$, we prove that if $V$ is finitely $\Omega$-generated and satisfies suitable grading conditions, then $V$ is rational, i.e. has semi-simple representation theory, with one dimensional level zero Zhu algebra. Here $\Omega$ denotes the vectors in $V$ that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal...

We develop the notion of a (pro-) conformal pseudo operad and apply it to the construction of the basic cohomology complex of a vertex algebra. The paper heavily uses the ideas and constructions of the work of Tamarkin [Tam02]

We characterize vertex algebras (in a suitable sense) as algebras over a certain graded co-operad. We also discuss some examples and categorical implications of this characterization.

Bakalov, Kac and Voronov introduced Leibniz conformal algebras (and their cohomology) as a non-commutative analogue of Lie conformal algebras. Leibniz conformal algebras are closely related to field algebras which are non-skew-symmetric generalizations of vertex algebras. In this paper, we first introduce $Leib_\infty$-conformal algebras (also called strongly homotopy Leibniz conformal algebras) where the Leibniz conformal identity holds up to homotopy. We give some equivalen...

In this note we show how to apply the Gr\"obner--Shirshov bases (GSB) method for modules over an associative algebra to the study of vertex algebras defined by generators and relations. We compute GSBs for a series of vertex algebras and study the problem of embedding of a left-symmetric algebra into a vertex one preserving the normally ordered product.