Constructing models of vertex algebras in higher dimensions

Vertex algebras in higher dimensions correspond to models of Quantum Field Theory (Wightman axioms) with Global Conformal Invariance. We review how such a vertex algebra can be generated from a collection of local fields, or from a vertex Lie algebra. The one-dimensional restriction of a vertex algebra in higher dimensions to a time-like line gives a chiral vertex algebra endowed with an action of the (higher-dimensional) conformal Lie algebra. We announce that from such data one can reconstruct the initial vertex algebra. 1 Definition of a vertex algebra on C One of the main problems of mathematical physics, which has remained open for more than fifty years, is to construct models of interacting quantum fields in higher dimensions. In the presence of the so-called global conformal invariance [14] this problem becomes purely algebraic. The corresponding algebraic structures are vertex algebras in higher dimensions [13] (see also [17] for an earlier work and [9] for a more general definition). These algebras appear as a straightforward generalization of the vertex algebras from two-dimensional conformal field theories, first introduced axiomatically in [8] (see [11] for an excellent introduction to the subject). A (boson) vertex algebra (on C) consists of the following data [13]: a vector space V (the space of states), a state-field correspondence (defined below), infinitesimal translations Tμ ∈ EndV (μ = 1, . . . , D), and a vector 1 ∈ V (vacuum). These data have to satisfy certain conditions, or axioms, which are called: locality, translation covariance, and the vacuum axiom. We will now explain the