Motivation and Related Algebraic Structures
The axioms of a vertex algebra are obtained from abstracting away the essentials of the operator product expansion of operators in a 2D Euclidean chiral conformal field theory. The two dimensional Euclidean space is treated as a Riemann sphere with the point at infinity removed. V is taken to be the space of all operators at z = 0. The operator product expansion is holomorphic in z and so, we can make a Laurent expansion of it. 1 is the identity operator. We treat an operator valued holomorphic map over C\{0} as a formal Laurent series. This is denoted by the notation V((z)). A holomorphic map over C is given by a Taylor series and as a formal power series, this is denoted by .
The operator b(0) is abstracted to b and the operator a(z) to Y(a,z). The derivative a'(z) is abstracted to -Ta.
If one considers only the singular part of the OPE in a Vertex algebra, one arrives at the definition of a Lie conformal algebra. Since one is often only concerned with the singular part of the OPE, this makes Lie conformal algebras a natural object to study.
Read more about this topic: Vertex Operator Algebra
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