Factorization algebras in quite a lot of generality

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.

💡 Research Summary

**
The paper proposes a highly general framework for factorization algebras that works on a wide variety of geometric objects, far beyond the traditional settings of manifolds and complex curves. The central new ingredient is an “isolability structure,” a piece of data that records, for any pair of points (or more generally any configuration of points) on a space X, the ways in which those points can be made mutually “distant” without affecting observables. Formally, for each T‑point x, y : T→X one assigns a higher groupoid ⟦x ≁ y⟧; these groupoids assemble into a configuration space X^{⊕1}→X×X. This construction generalizes the familiar complement of the diagonal in a manifold and the Ran space in holomorphic contexts, but it also accommodates more sophisticated clustering patterns (e.g., septuples split into three clusters) encoded by a combinatorial category D of cographs.

The category D carries two independent symmetric monoidal structures: a left ⊕ given by disjoint union of cographs, and a right ⊕ given by a “connected sum” operation. The two monoidal products do not coincide (otherwise Eckmann–Hilton would force them to be the same); instead there is a non‑invertible interchange map (A⊕B)⊕(C⊕D)→(A⊕C)⊕(B⊕D). This makes D a two‑fold symmetric monoidal category (also known as a duoidal category). The authors use this structure to define a “parallax” symmetric monoidal functor: a lax symmetric monoidal functor D→Cat (or equivalently D→SymMonCat). For a given geometric context (X, A) – where X is a category of geometric objects and A is a sheaf of symmetric monoidal categories (e.g., QCoh, constructible sheaves) – the isolability data X· yields a diagram A(X·) indexed by D. This diagram carries two tensor products: the usual one obtained by pulling back the external tensor product along the diagonal Δ:X→X², and a second one obtained by pulling back along the complement X^{⊕1}→X². The interaction of these two tensors encodes precisely the locality principle that underlies factorization algebras.

A factorization algebra on X with coefficients in A is then defined as a right‑symmetric‑monoidal functor 1·→A(X·). This abstract definition recovers many known examples. For instance, when X=ℝⁿ and the isolability structure is chosen so that ⟦x ≁ x⟧≅S^{n‑1}, the resulting factorization algebras are exactly En‑algebras, reproducing the classical identification of locally constant factorization algebras on ℝⁿ with En‑algebras. The construction also matches Cepek’s combinatorial description of the Ran space, showing that the Ran space is a special case of the isolability framework.

The paper further introduces the notion of an “observer stack” O_X, which records the possible supports of observables (e.g., Hilbert schemes of subschemes, embedding stacks). This allows the theory to treat observables not only at points but along higher‑dimensional subspaces, a feature essential for many quantum field theories. The authors discuss how this idea connects with recent work of Hennion, Melani, and Vezzozi on higher Ran spaces.

As a concrete application, the authors construct a generalized Beilinson–Drinfeld Grassmannian as a factorization stack. Given any object X and a pointed stack B on X (for example, the classifying stack of a group scheme G), together with an observer stack O_X, they define a Grassmannian factorization stack Gr_B on O_X. When X is a complex curve and B=BG, this recovers the classical Beilinson–Drinfeld Grassmannian. Moreover, by taking X to be the Fargues–Fontaine curve and O_X the “mirror curve” Div₁, one obtains a factorization enhancement of the Scholze–Weinstein construction, suggesting a route toward arithmetic quantum field theories.

The authors acknowledge several limitations: (1) the traditional Costello–Gwilliam manifold factorization algebras are not yet incorporated; (2) a “β‑version” that would allow more intricate combinations of extended observables remains speculative; (3) the Koszul dual perspective on Lie‑algebraic factorization structures is not explored; and (4) concrete computational examples are limited. They outline future directions, including integrating the manifold case, developing β‑structures, studying Koszul duality, and linking the two‑fold monoidal formalism with a full six‑functor formalism.

Overall, the paper offers a minimalist yet powerful categorical machinery that unifies existing factorization algebra theories, extends them to new geometric settings (including arithmetic geometry), and opens a pathway toward a systematic study of quantum field theories on highly generalized spacetimes.