Factorization algebras in quite a lot of generality
Clark Barwick
TL;DR
This work develops a minimalist, geometry-agnostic framework for factorization algebras by introducing isolability structures that encode when observables on a geometric object can combine. It builds a rich combinatorial foundation based on cographs, twofold symmetric monoidal structures, and para-isolability/envelopes to realize factorization algebras as sections of parallax diagrams, unifying holomorphic, topological, and arithmetic contexts. The paper constructs constructible and locally constant factorization algebras, factorization stacks, and a general Beilinson–Drinfeld Grassmannian in broad generality, with observer stacks enabling extended observables. While outlining limitations (e.g., not fully incorporating the manifold context or a full $eta$-version), it sets up a versatile framework with potential applications to arithmetic QFT and a deeper link to Ran spaces and stratified homotopy theory.
Abstract
This is a first stab at a mathematical framework in which one can study quantum field theories on spacetimes with quite general geometries. We will study these theories via their factorization algebras. The aim is to identify a minimalist formalism that makes sense of factorization algebras in any geometric context. This formalism extends the technology of factorization algebras to many new contexts, including those arising in arithmetic quantum field theories. In order to make sense of factorization algebras on a geometric object X, one needs two ingredients. First, one needs an additional piece of structure on X that we call an "isolability structure." This is the data required to say whether two (generalized) points of X are "distant." This is encoded as a functor from a certain combinatorial category of cographs. Second, one needs some sort of sheaf theory. The isolability structure then induces on the category of sheaves a kind of twofold symmetric monoidal structure. Factorization algebras are then defined in terms of this structure. This paper develops this formalism. We describe how some existing theories of factorization algebras fit into this framework, and we give a construction of the Beilinson-Drinfeld Grassmannian as a factorization stack that works in quite a lot of generality.