Gluing data for factorization monoids and vertex ind-schemes

TL;DR

This paper develops a new framework for describing factorization algebras and monoids over the affine line using gluing data from OPE algebras, leading to the definition of vertex ind-schemes as conformal analogues of Lie groups.

Contribution

It introduces OPE monoids as a non-linear extension of OPE algebras and constructs vertex ind-schemes, connecting them to vertex algebras and Lie conformal algebras.

Findings

Explicit description of factorization algebras via gluing data

Introduction of OPE monoids as non-linear OPE algebra generalization

Vertex ind-schemes as conformal analogues of Lie groups

Abstract

We give an explicit description of factorization algebras over the affine line, constructing them from the gluing data determined by its corresponding OPE algebra. We then generalize this construction to factorization monoids, obtaining a description of them in terms of a non-linear version of OPE algebras which we call OPE monoids. In the translation equivariant setting this approach allows us to define vertex ind-schemes, which we interpret as a conformal analogue of the notion of Lie group, since we show that their linearizations yield vertex algebras and that their Zariski tangent spaces are Lie conformal algebras.