Vertex Algebras

Vertex algebras are algebraic structures that provide a rigorous mathematical axiomatization of the chiral part of a two-dimensional conformal field theory. Central to this structure is the state-field correspondence, which associates each vector (or "state") in a vector space with a formal power series of operators known as a "field" or "vertex operator." The axioms of a vertex algebra, particularly the locality axiom, precisely capture the algebraic properties of operator product expansions from physics. These objects have profound connections to various areas of mathematics, including the representation theory of infinite-dimensional Lie algebras (such as the Virasoro and affine Kac-Moody algebras), modular forms, and the monstrous moonshine conjecture.

1. Foundations and Motivation
   1. Historical Context
      1. Origins in Conformal Field Theory
      2. Development from String Theory
      3. Connection to Quantum Field Theory
      4. Mathematical Formalization Efforts
   2. Physical Motivation
      1. Classical Field Theory Background
      2. Quantum Field Theory Concepts
      3. Operator-Valued Distributions
      4. Need for Rigorous Mathematical Framework
   3. Operator Product Expansions in Physics
      1. Physical OPE Concept
      2. Singular Behavior of Products
      3. Locality Requirements
      4. Transition to Mathematical Formulation