Factorization envelopes and enveloping vertex algebras
Yusuke Nishinaka
TL;DR
This work builds a general, analytic bridge between factorization algebras and vertex algebras by using factorization envelopes and the Dolbeault-Dolbeault framework. It shows that, starting from a Lie conformal algebra L, the factorization envelope yields an amenably holomorphic prefactorization algebra whose associated vertex algebra is isomorphic to the enveloping vertex algebra V(L), and it extends this to central extensions via 2-cocycles. The construction unifies and generalizes the Costello–Gwilliam and Williams approaches for Kac–Moody and Virasoro cases, and it naturally extends to vertex superalgebras, producing Neveu–Schwarz, N=2, and N=4 algebras in a higher-dimensional, geometric setting. The results provide a robust framework for generating factorization algebras corresponding to a wide class of vertex (super)algebras, including new examples, through a unified, analytic methodology. Overall, the paper advances the categorical and analytic correspondence between observables in chiral conformal field theories and their algebraic realizations.
Abstract
We construct a factorization algebra, via the factorization envelope, starting from a Lie conformal algebra, and prove that the associated vertex algebra is isomorphic to its enveloping vertex algebra. Our construction generalizes the Kac--Moody factorization algebra of Costello--Gwilliam and the Virasoro factorization algebra of Williams. Moreover, by considering the super analogue of this construction, we obtain new factorization algebras corresponding to vertex superalgebras, such as the Neveu--Schwarz vertex superalgebra and the $N=2$ vertex superalgebra.