Factorization envelopes and enveloping vertex algebras

Abstract

We develop a bornological version of Costello and Gwilliam's procedure for extracting vertex algebras from suitable prefactorization algebras on the complex plane. Using bornological complex analysis, we remove the discreteness condition imposed in their extraction theorem. We then construct, from a suitable Lie conformal algebra, a prefactorization algebra to which this extraction procedure applies, and prove that the resulting vertex algebra is isomorphic to the enveloping vertex algebra of the original Lie conformal algebra. Our construction uses a factorization envelope and extends the construction of Costello--Gwilliam in the affine vertex algebra case and that of Williams in the Virasoro vertex algebra case. Moreover, a super analogue yields new prefactorization algebras corresponding to vertex superalgebras, such as the Neveu--Schwarz vertex superalgebra, the $N=2$ vertex superalgebra, and the $N=4$ vertex superalgebra.