Prefactorization and Factorization Algebras

A prefactorization algebra assigns observables to open regions and supplies a coherent product whenever several input regions are pairwise disjoint inside a larger one. It becomes a factorization algebra only after a local-to-global condition is proved for a specified class of covers. The distinction is essential: disjoint multiplication is algebraic input, whereas descent is a theorem about reconstructing observables with arbitrarily complicated finite support.

Required background. QFT Frameworks, Object Classes, and Typed Maps supplies the source category, target category, and direction of every structure map.

Helpful background. Natural transformations and subtheory embeddings clarifies covariance; causal factorization gives the Lorentzian comparison; typed comparisons of nets, fields, and factorization prevents an unlicensed equivalence claim; and microcausal functionals provide a common class of examples.

Disjoint-region multiplication

Let $M$ be a manifold and $(\mathcal C,\otimes,\mathbb 1)$ a symmetric monoidal category, usually cochain complexes with quasi-isomorphisms as weak equivalences. A prefactorization algebra $\mathcal F$ consists of an object $\mathcal F(U)$ for every open $U\subset M$ and, for every finite family of pairwise disjoint opens $U_1,\ldots,U_n\subset V$, a structure map

$$\mu^V_{U_1,\ldots,U_n}: \mathcal F(U_1)\otimes\cdots\otimes\mathcal F(U_n) \longrightarrow \mathcal F(V).$$

The empty family gives a unit $\mathbb 1\to\mathcal F(V)$. A one-element family gives covariance under inclusions. Permuting the inputs agrees with the symmetry of $\mathcal C$, and composing disjoint configurations in stages gives the same map as composing them at once. These are operadic coherence laws, not commutativity of products inside one region. Indeed, the locations of little input regions can retain ordering, braiding, or higher-homotopy information. This definition and its operator-product interpretation are given precisely in Costello and Gwilliam 2023, Definition 1 and §1.

For classical fields, an observable supported in $U$ is a function on fields whose functional derivatives are supported in $U$. If $F_i$ have disjoint supports, their pointwise product is supported in their union, furnishing $\mu$. Quantum observables replace that pointwise product by a renormalized or cochain-level product appropriate to the theory. The same outer structure can therefore carry commutative classical observables, formal quantum observables, or chiral operations, but those target categories and differentials must not be mixed.

Weiss descent

A Weiss cover $\{U_i\}_{i\in I}$ of $U$ is stronger than an ordinary open cover: every finite subset $S\subset U$ lies in some $U_i$. The condition is tuned to observables with finite collections of insertion points. A factorization algebra is a prefactorization algebra for which the canonical map from the homotopy-coherent Čech object of every Weiss cover to $\mathcal F(U)$ is an equivalence,

$$\underset{\varnothing\ne J\subset I}{\operatorname{hocolim}} \mathcal F\!\left(\bigcap_{j\in J}U_j\right) \xrightarrow{\simeq}\mathcal F(U),$$

with the appropriate derived version when $\mathcal C$ is homotopical. Equivalent formulations often use a factorizing basis of disks and their finite disjoint unions. The exact descent axiom, including the differentiable-vector-space qualifications used in field theory, is developed in Costello and Gwilliam 2017, Chs. 2–3; an accessible formulation appears in Costello and Gwilliam 2023, Definitions 2–3.

Descent is not a statement about states or Hilbert spaces. It says that the observable complex on $U$ is recovered, up to the selected weak equivalence, from compatible observables supported in sufficiently small configurations. A global vacuum, a positive representation, or a measure on the full field space requires separate input.

Free scalar observables as a first construction

For a Euclidean free scalar on $\mathbb R^d$, take compactly supported test sections in $U$, form linear observables $\ell_f(\phi)=\int_U f\phi\,\mathrm d^dx$, and then the graded symmetric algebra generated by them, with a differential implementing the linear equation of motion. Extension by zero is covariant, and multiplication of polynomial functionals supplies the disjoint structure maps. The coherence check reduces to associativity of multiplication and functoriality of extension by zero.

This produces a prefactorization algebra before any descent theorem is invoked. For the standard free complex, compactly supported sections satisfy cosheaf descent and the symmetric construction preserves the relevant homotopy colimits under the usual nuclearity/completion hypotheses, so one obtains factorization. The physical Gaussian construction is developed at Gaussian Fields and Sources; this page isolates the local-to-global structure it supplies.

A prefactorization failure test

Disjoint products alone do not force descent. For example, enlarge the value on one large open $U$ by a new generator $x$ while letting every proper member of a Weiss cover see only the unit sector; define all incoming structure maps to miss $x$. Units, symmetry, and iterated disjoint multiplication can still be coherent, but the Čech homotopy colimit has no class mapping to $x$. The descent map is therefore not essentially surjective on cohomology. This simple construction exposes the missing hypothesis without pretending that every physically motivated assignment fails in this way.

The reliable naming rule is thus: specify opens, target category, disjoint maps, and coherence to claim prefactorization; identify the cover topology and prove the derived local-to-global map to claim factorization.

Exercises

Let $U_1,U_2,U_3$ be pairwise disjoint in $V$. Show that multiplying $U_1$ and $U_2$ first and then multiplying with $U_3$ gives the same map as the three-input operation.

Solution

Choose an intermediate open $W$ containing $U_1\sqcup U_2$ and disjoint from $U_3$. The composition axiom identifies $\mu^V_{W,U_3}(\mu^W_{U_1,U_2}\otimes\mathrm{id})$ with $\mu^V_{U_1,U_2,U_3}$. Applying the same axiom to the other bracketing gives that same three-input map. No binary product on $\mathcal F(V)$ was assumed.

Why is an ordinary cover by small balls generally not Weiss?

Solution

Two sufficiently separated points can lie in no single small ball, although each point lies in some member of the ordinary cover. A Weiss cover must contain one member embracing every prescribed finite configuration; finite disjoint unions of sufficiently small balls provide the standard repair.

References

• Costello, Kevin, and Owen Gwilliam. Factorization Algebras in Quantum Field Theory, Volume 1. Cambridge University Press, 2017. doi:10.1017/9781316678626.
• Costello, Kevin, and Owen Gwilliam. “Factorization Algebra.” Encyclopedia of Mathematical Physics, 2nd ed., 2023. arXiv:2310.06137.