Vertex (Lie) algebras in higher dimensions

Vertex algebras provide an axiomatic algebraic description of the operator product expansion (OPE) of chiral fields in 2-dimensional conformal field theory. Vertex Lie algebras (= Lie conformal algebras) encode the singular part of the OPE, or, equivalently, the commutators of chiral fields. We discuss generalizations of vertex algebras and vertex Lie algebras, which are relevant for higher-dimensional quantum field theory. 1 Vertex algebras and Lie conformal algebras In the theory of vertex algebras [1, 2, 3], the (quantum) fields are linear maps from V to V [[z]][z−1], where z is a formal variable. They can be viewed as formal series a(z) = ∑n∈Z a(n) z−n−1 with a(n) ∈ EndV such that a(n)b = 0 for n large enough. Let Res a(z) = a(0); then the modes of a(z) are given by a(n) = Res z na(z). The locality condition for two fields (z−w)ab [a(z),b(w)] = 0, Nab ∈ N is equivalent to the commutator formula [a(z),b(w)] = Nab−1 ∑ j=0 c j(w)∂ j wδ (z−w)/ j! for some new fields c j(w), where δ (z−w) is the formal delta-function (see [2]). The operator product expansion (OPE) can be written symbolically a(z)b(w) = ∑ j∈Z c j(w)(z−w) − j−1 (see [2] for a rigorous treatment). The new field c j is called the j-th product of a,b and is denoted a( j)b. The Wick product (= normally ordered product)