Factorization Algebras and Local-to-Global Observables

Factorization algebras combine observables in disjoint regions and recover larger regions by descent. This chapter explains that local-to-global structure, its quantization, and the additional conditions needed for comparison with algebraic QFT.

Enter this chapter

Start with disjoint structure maps, then Weiss descent. Ordinary sheaf gluing is a useful analogy but uses a different cover condition. 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.

Topics

1. Prefactorization and Factorization Algebras — Defines disjoint-region products and their coherence.
2. Weiss Descent and Local-to-Global Observables — Uses covers that contain every finite configuration in some member.
3. Elliptic Complexes and Factorization Observables — Builds observable complexes from elliptic field data.
4. Classical Observables and Poisson Factorization — Adds the classical Poisson structure to local observables.
5. BV Quantization and Obstruction–Deformation Complexes — Treats quantization as an obstruction problem.
6. Quantum Observables, Renormalization, and Factorization — Constructs renormalized quantum structure maps.
7. Locally Constant Factorization Algebras and Eₙ Algebras — Identifies the extra local constancy behind Eₙ structures.
8. Factorization Homology and Manifold Invariants — Integrates local algebraic data over manifolds.
9. Holomorphic, Topological, and Mixed Factorization Theories — Distinguishes topological and holomorphic directions of dependence.
10. Factorization Comparison Theorems and Their Limits — States comparison results with their exact analytic and categorical scope.
11. AQFT to Prefactorization on Time-Orderable Covers — Constructs time-ordered prefactorization data from suitable Lorentzian nets.
12. Prefactorization to AQFT: The Infinity-Categorical Converse Gap — Explains why a higher-categorical converse needs additional proof.

How the ideas fit together

Disjoint multiplication gives prefactorization; descent is a separate condition Costello and Gwilliam 2017, Chapters 2–3. Local constancy then licenses the Eₙ comparison Lurie 2017, Theorem 5.4.5.9 (PDF). In Lorentzian geometry, the AQFT comparison instead uses additivity, Cauchy constancy, and time-orderability Benini, Perin, and Schenkel 2020, Theorems 3.11, 4.7, and 5.1 (PDF). A cochain-level equivalence does not automatically add positive states or an operator-algebra completion.

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).

Review the chapter

Can an ordinary open cover fail to be a Weiss cover?

Solution

Yes. Two disjoint nonempty components cover their union, but a finite subset containing one point in each component lies in neither member. Weiss descent tests finite configurations jointly, so ordinary pointwise coverage is insufficient.

Where to go next

Continue to derived and deformation methods for the complexes and obstruction classes underlying perturbative quantization.

References

• 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.
• Lurie, Jacob. Higher Algebra. September 18, 2017. Author PDF.