Factorization Algebras and Local-to-Global Observables

Factorization algebras organize observables by where they are supported. Their basic operation combines observables from pairwise disjoint regions; their decisive local-to-global condition reconstructs observables on a large region from a cover adapted to finite configurations. This chapter develops that distinction from free and perturbative field theory through $E_n$ structures and factorization homology, then states exactly when Lorentzian factorization data and algebraic QFT can be compared. It does not identify every factorization algebra with a physical net.

Helpful background. Presheaves, sheaves, cosheaves, and Čech descent supplies the ordinary local-to-global language. Derived, higher, and factorization frameworks supplies homotopy colimits and higher coherence. From the local OPE to conformal data gives a physical comparison point without equating an OPE expansion with Weiss descent.

Enter this chapter

Begin every construction by naming four pieces of data: the category of regions, the symmetric monoidal target, the weak equivalences in that target, and the allowed disjoint configurations. A prefactorization algebra $\mathcal F$ then supplies maps

$$\mathcal F(U_1)\otimes\cdots\otimes\mathcal F(U_k) \longrightarrow \mathcal F(V), \qquad U_1\sqcup\cdots\sqcup U_k\subset V,$$

compatible with permutations and nested configurations. Those maps alone do not imply descent. To call $\mathcal F$ a factorization algebra, one must also specify the cover topology and prove that the corresponding derived Čech object reconstructs $\mathcal F(V)$. For observables supported on finite configurations, the relevant topology is normally the Weiss topology: every finite subset of $V$ must lie in one member of the cover Costello and Gwilliam 2017, Chs. 2–3.

Three distinctions guide the chapter. First, a Euclidean elliptic BV complex and a Lorentzian Green-hyperbolic theory have different analytic mechanisms even when their observable complexes can be compared. Second, local constancy is a strong topological condition, not a generic property of relativistic QFT. Third, a comparison theorem must retain its direction, target category, additivity, Cauchy constancy, and time-orderability hypotheses. Positivity, states, Hilbert-space representations, and operator-algebraic completion do not appear merely because a cochain-level comparison succeeds.

The dependency diagram separates the common construction from its specializations. Read the horizontal row from disjoint products through descent and field-theoretic observables to quantization. The lower branch records the topological and Lorentzian conclusions that become available only after their own additional assumptions.

Disjoint structure maps provide prefactorization data. Weiss descent is the independent local-to-global step. Elliptic or Green-hyperbolic field complexes then control support, equations of motion, and brackets; BV quantization adds a local obstruction class and renormalized quantum master equation. Locally constant $E_n$ structures, factorization homology, and the strict Lorentzian AQFT comparison are conditional branches, not automatic consequences for an arbitrary QFT. Every arrow is one-way; the diagram is schematic and not to scale. Structured description and source data (JSON)

The chapter sequence

Read the pages in this order.

1. Prefactorization and factorization algebras defines disjoint-region products and the additional descent condition.
2. Weiss descent and local-to-global observables explains why finite configurations require a cover stronger than an ordinary open cover.
3. Elliptic complexes and factorization observables constructs Euclidean observables from compactly supported elliptic BV data and tests zero-mode obstructions.
4. Classical observables and Poisson factorization compares the shifted BV bracket with the Lorentzian Peierls bracket and its wavefront restrictions.
5. BV quantization and obstruction–deformation complexes locates the quantum master obstruction in local cohomology.
6. Quantum observables, renormalization, and factorization constructs scale-dependent quantum complexes and checks the one-loop change in four-dimensional $\phi^4$ theory.
7. Locally constant factorization algebras and Eₙ algebras derives little-disks operations while retaining framing and higher coherence.
8. Factorization homology and manifold invariants integrates disk data by tensor excision and computes the circle value as Hochschild homology.
9. Holomorphic, topological, and mixed factorization theories separates translation invariance, holomorphic dependence, twists, and anomalies.
10. Factorization comparison theorems and their limits aligns geometry, target, weak equivalence, additivity, time-slice, and time ordering before any comparison.
11. AQFT to prefactorization on time-orderable covers constructs ordered products from Einstein causality and proves independence of the chosen causal order.
12. Prefactorization to AQFT: the infinity-categorical converse gap proves the strict reconstruction and isolates the still-unresolved homotopical extension.

The sequence moves from definitions to constructions and only then to comparisons. That order prevents a common circular argument: one cannot use a hoped-for equivalence with AQFT to supply the descent, analytic completion, or higher coherence needed to define the factorization object being compared.

Hypotheses and licensed conclusions

Domains, required hypotheses, directional results, and failure tests for factorization observables

Object and domain	Required hypotheses	Licensed result	Excluded converse or upgrade	Adversarial check
Assignment on opens	Symmetric monoidal target; coherent maps for finite disjoint families; unit and equivariance	Prefactorization algebra	Disjoint multiplication alone does not give local-to-global reconstruction	Add a class visible only on one large open and missed by every cover member
Prefactorization algebra on a manifold	Declared Weiss or factorizing covers; homotopically correct Čech object; suitable completion	Factorization algebra with derived local-to-global descent	Ordinary sheaf descent or underived colimits need not recover multilocal observables	Use a cover in which separated insertion points never occur together
Classical field complex	Elliptic Euclidean complex or Green-hyperbolic Lorentzian operator; support conditions; equation-of-motion quotient; wavefront domain where needed	Classical observable complex with the appropriate shifted BV or Peierls bracket	Ellipticity does not imply Lorentzian causality, and local functionals alone are not generally bracket-closed	Expose a zero mode, noncompact-support class, or forbidden wavefront product
Perturbative quantum observables	Gauge-fixed BV resolution; local renormalization; effective quantum master equation; vanishing or cancelled obstruction class	Quantum factorization algebra as a formal, scale-compatible deformation	No nonperturbative convergence, positive state, or regulator-independent completion follows	Compute a nonzero local anomaly class or introduce a nonlocal counterterm
Locally constant factorization algebra	Disk inclusions are weak equivalences; specified framing, orientation, or other tangential structure; presentable monoidal target	$E_n$ algebra and factorization homology by disk extension and excision	A generic metric-dependent QFT is not locally constant; an abstract $E_n$ algebra is not automatically a relativistic theory	Vary a mass-dependent correlator with disk size or transport framed data around an unlicensed rotation
Additive Cauchy-constant AQFT	Globally hyperbolic Lorentzian regions; Einstein causality; time-slice; time-orderable tuples; compatible monoidal target	Associated additive Cauchy-constant time-orderable prefactorization algebra	No Euclidean continuation, Weiss descent, positivity transfer, or BV model is supplied for free	Choose a causally cyclic tuple with no linear time order
Time-orderable prefactorization algebra	Additivity and Cauchy constancy in the proved strict setting; coherent multiplication reconstructed from Cauchy morphisms	Strict AQFT comparison and inverse functors in the stated one-category	The strict theorem does not establish the general infinity-categorical converse	Retain descent but remove time-slice or required higher coherence and test associativity

Structured table data (JSON) preserves the caption, scoped headers, rows, and reading order.

The locally constant recognition theorem is especially powerful and especially easy to overextend. On framed $n$-space, locally constant factorization algebras are equivalent, in a suitable homotopical target, to $E_n$ algebras Lurie 2017, Theorem 5.4.5.9. Factorization homology then extends disk data to manifolds by a colimit satisfying tensor excision; for an associative algebra $A$, integration over the circle recovers Hochschild homology Ayala and Francis 2015, Theorem 3.19. Neither theorem supplies reflection positivity or relativistic states.

The failure map asks where a proposed theorem first becomes ill-typed or loses a necessary implication. Its lower row records the strongest conclusion still available after each missing input.

Each dashed arrow removes a concrete hypothesis. Coherent disjoint products can miss global classes; ordinary covers can miss bilocal support; zero modes and forbidden distribution products can invalidate an observable complex; a nontrivial BV anomaly blocks the quantum master equation; omitted tangential data block manifold transport; and descent alone cannot replace Lorentzian additivity, Cauchy constancy, time ordering, or higher coherence. The diagram is schematic and not to scale. Structured description and source data (JSON)

Comparison boundary

The strongest model-independent result used here is precise and restricted. In a suitable bicomplete closed symmetric monoidal one-category, additive Cauchy-constant AQFTs and additive Cauchy-constant time-orderable prefactorization algebras are related by inverse functors Benini, Perin, and Schenkel 2020, Theorems 3.11, 4.7, and 5.1. For free Green-hyperbolic theories, explicit cochain-level natural transformations compare pAQFT nets and factorization observables Gwilliam and Rejzner 2020, Theorems 3.5–3.6 and 3.17.

These theorems do not yield a universal equivalence between Euclidean factorization algebras and Haag–Kastler nets. They do not automatically transport $C^*$-completion, positivity, states, or representations. The extension of the strict comparison to the intended infinity-categorical setting remains a genuine coherence problem. The chapter therefore states strict equivalence where proved, quasi-isomorphism where constructed, and an open problem where higher compatibility has not been established.

Review the chapter

Before applying a factorization result, answer the following.

1. What are the regions, admissible embeddings, target category, tensor product, and weak equivalences?
2. Are only disjoint products defined, or has descent been proved for a named topology?
3. Is the Čech construction underived, derived, or completed in a specified functional-analytic category?
4. Does the field complex use elliptic Euclidean data or Green-hyperbolic Lorentzian data?
5. Which support and wavefront conditions make every product and bracket defined?
6. What cohomology group contains the quantization obstruction, and has it actually vanished?
7. Is local constancy proved, and which tangential structure permits transport between disks?
8. Does a factorization-homology calculation use tensor excision in a target preserving the required colimits?
9. For an AQFT comparison, are additivity, Cauchy constancy, time-orderability, and direction explicit?
10. Is the result strict, derived, completed, representation-level, or still conjectural?

Synthesis exercise

Suppose a cochain-valued assignment has coherent products on disjoint opens, is locally constant on disks, and agrees in cohomology with the local observables of a free Lorentzian scalar theory. May one conclude that it is a factorization algebra, that it defines an oriented $E_n$ algebra, and that it is equivalent to the scalar Haag–Kastler net?

Solution

Not from the stated data. The assignment is prefactorization, but factorization additionally requires derived Weiss descent. Local constancy on framed disks gives framed $E_n$ data only after the target and coherence hypotheses are met; an oriented theory needs compatible rotation data. Agreement in cohomology with local scalar observables does not identify off-shell complexes, products, $*$-structures, states, or operator-algebraic completions. The strict Lorentzian comparison further requires additivity, Cauchy constancy, time-orderability, and the specified target category.

References

• Ayala, David, and John Francis. “Factorization Homology of Topological Manifolds.” Journal of Topology 8 (2015): 1045–1084. DOI; Open PDF.
• Benini, Marco, Marco Perin, and Alexander Schenkel. “Model-Independent Comparison between Factorization Algebras and Algebraic Quantum Field Theory on Lorentzian Manifolds.” Communications in Mathematical Physics 377 (2020): 971–997. DOI; Open PDF.
• Costello, Kevin, and Owen Gwilliam. Factorization Algebras in Quantum Field Theory, Volume 1. Cambridge: Cambridge University Press, 2017. DOI.
• Gwilliam, Owen, and Katarzyna Rejzner. “Relating Nets and Factorization Algebras of Observables: Free Field Theories.” Communications in Mathematical Physics 373 (2020): 107–174. DOI; Open PDF.
• Lurie, Jacob. Higher Algebra. September 18, 2017. Author PDF.