Related papers: Jacobi Identity for Vertex Algebras in Higher Dime…
We prove a generalized rationality property and a new identity that we call the ``Jacobi identity'' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of…
The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that…
In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds' notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These…
In this paper we study cohomology and deformations of Jacobi-Jordan algebras. We develop their formal deformation theory. In particular, we introduce a method to construct a versal deformation for a given Jacobi-Jordan algebra, which can…
The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A criterion for simple currents…
The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX-relation). In…
These are the notes of an informal talk in Bonn describing how to define an analogue of vertex algebras in higher dimensions.
It was shown by the author [International Journal of Theoretical Physics 36 (1997), 1099-1131] in synthetic differential geometry that what is called the general Jacobi identity obtaining in microcubes underlies the Jacobi identity of…
Jacobi fields of classical solutions of a Hamiltonian mechanical system are quantized in the framework of vertical-extended Hamiltonian formalism. Quantum Jacobi fields characterize quantum transitions between classical solutions.
A higher dimensional analogue of the notion of vertex algebra is formulated in terms of formal variable language with Borcherds' notion of $G$-vertex algebra as a motivation. Some examples are given and certain analogous duality properties…
We encapsulate the basic notions of the theory of vertex algebras into the construction of a comonad on an appropriate category of formal distributions. Vertex algebras are recovered as coalgebras over this comonad.
We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The…
The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the…
Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie…
We give a short introduction to generalized vertex algebras, using the notion of polylocal fields. We construct a generalized vertex algebra associated to a vector space h with a symmetric bilinear form. It contains as subalgebras all…
Vertex algebras can be defined over any differential commutative ring. We develop the general descent theory for vertex algebras over such bases. We apply this to the classification of twisted forms of affine and Heisenberg vertex algebras,…
The quadratic rank two Jacobi algebra is identified from the relations obeyed by the bispectral operators of the two variable Jacobi polynomials orthogonal on the triangle. It is seen to admit as subalgebras Racah and Jacobi algebras of…
We study a certain linear automorphism of a vertex operator algebra induced by the formal change of variable $f(x)=e^x-1$ and describe examples showing how this relates the theory of vertex operator algebras to Bernoulli numbers, Bernoulli…
A framework is developed enabling the global analysis of the stability of cosmological models using the local geometric characteristics of the infinite-dimensional superspace, i.e. using the generalised Jacobi equation reformulated for…
We study spaces of conformal blocks associated with line bundles over elliptic curves, with coefficients in a vertex algebra. For vertex algebras satisfying suitable finiteness and semisimplicity conditions, which are met by all admissible…