Vertex Algebras and Factorization Algebras

Abstract

This thesis is about the relationship between vertex algebras and Costello-Gwilliam factorization algebras, two mathematical approaches to chiral conformal field theory. Many vertex algebras have already been constructed. Some of these are known to arise from holomorphic factorization algebras on the plane of complex numbers. We prove that every <strong>Z</strong>-graded vertex algebra arises from such a factorization algebra. <br /> First, we show that a <strong>Z</strong>-graded vertex algebra is the same thing as a geometric vertex algebra. Geometric vertex algebras serve as an intermediary between <strong>Z</strong>-graded vertex algebras and factorization algebras. Our factorization algebras take values in the symmetric monoidal category of complete bornological vector spaces. We describe how to obtain geometric vertex algebras from certain prefactorization algebras with values in the symmetric monoidal category of complete bornological vector spaces. Second, we attach a prefactorization algebra <strong>F</strong><em>V</em> to every geometric vertex algebra and show that the geometric vertex algebra associated with <strong>F</strong><em>V</em> is isomorphic to <em>V</em>. Third, we prove that <strong>F</strong><em>V</em> is in fact a factorization algebra.