Vertex operator algebras (VOAs) are mathematical structures that arise in the study of two-dimensional conformal field theory, algebraic structures, and number theory. They play a significant role in various areas of mathematics and theoretical physics, particularly in the study of string theory, modular forms, and representation theory.
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A vertex operator algebra is an integer-graded vector space equipped with a state-field correspondence , a vacuum, translation operator and conformal vector satisfying vacuum, translation, locality and compatibility axioms. It packages the local operator product expansions of a chiral conformal field theory. A string vertex operator is an individual state insertion, rather than this complete algebraic structure.