A universal approach to vertex algebras

Abstract

In this paper, we characterize vertex algebras (in the appropriate sense) as algebras over a certain graded co-operad. We also discuss some applications and categorical ramifications of this characterization.

On discrete field theory properties of the dimer and Ising models and their conformal field theory limits

“…We will show only the construction of the map (17) in the case whenΣ is closed (i.e. where the definitions of the two Pfaffian lines are different).…”

Tree algebras: An algebraic axiomatization of intertwining vertex operators

“…Let F be any field of characteristic 0. We recall from [9] the ordered configuration space C(n) = C(z 1 , ..., z n ) of n distinct points z 1 , ..., z n in A 1 . Denote the coefficient ring of the affine variety C(n) by C(n).…”

“…It is advantageous to consider C(n) = C(z 1 , ..., z n ) a Z-graded ring with grading by total degree of a homogeneous rational function. In [9], it is shown that the system (C(n)) is a graded co-operad: the comultiplication (…”

“…and expanding in increasing powers of t ij and t i ′ j ′ . Moreover, it is shown in [9] that vertex algebras in a suitable sense can be characterized as graded algebras over the graded co-operad C. Define, for Z-graded F -vector spaces A, B, a Z-graded vector space…”

“…Although Huang [12] has obtained a more complicated generalization of the Jacobi identity for intertwining operators, devicing a workable system of axioms from this approach seems daunting. To get a handle on the structure in the present paper, we make substantial use a simple interpretation of the vertex algebra axioms obtained by Hortsch, Kriz and Pultr [9]: a graded vertex algebra is essentially the same thing as an algebra over a certain co-operad (the "correlation function co-operad"), which can be easily described explicitly.…”