Prefactorization to AQFT: The Infinity-Categorical Converse Gap

There is no converse gap in the strict additive, Cauchy-constant, time-orderable one-categorical setting: a theorem reconstructs an AQFT and gives an inverse equivalence. The remaining gap is the genuinely homotopical version, where Cauchy maps are weak equivalences rather than invertible maps and their inverses must be chosen with all higher coherences. A generic factorization algebra also lacks the causality, involution, positivity, and completion data needed for an operator-algebraic QFT.

Required background. AQFT to prefactorization on time-orderable covers supplies the proved forward functor, and Weiss descent distinguishes local-to-global reconstruction from multiplication.

Helpful background. Counterexamples and nonconverses supplies the missing-hypothesis method, while equivalence and comparison notions distinguishes strict inverse functors from an $\infty$-categorical equivalence.

The proved strict reconstruction

Let $\mathfrak F$ be an additive, Cauchy-constant, time-orderable prefactorization algebra on globally hyperbolic spacetimes, valued in a suitable symmetric monoidal one-category. To define multiplication on $\mathfrak F(M)$, choose two small Cauchy neighborhoods $U_-,U_+\subset M$ with $U_-$ earlier than $U_+$. Cauchy constancy makes

$$\mathfrak F(U_\pm)\xrightarrow{\cong}\mathfrak F(M)$$

invertible. Move two inputs back along these isomorphisms, apply the time-orderable product $\mathfrak F(U_-,U_+\to M)$, and obtain

$$\mu_M:\mathfrak F(M)\otimes\mathfrak F(M) \longrightarrow\mathfrak F(M).$$

Independence of the chosen neighborhoods follows by common refinement and Cauchy constancy. Associativity follows by using three ordered Cauchy neighborhoods. Additivity is crucial for naturality under general spacetime embeddings and for Einstein causality. The resulting functor $\mathbb A$ is proved in Benini, Perin, and Schenkel 2020, Theorem 3.11; together with the forward functor it gives the strict equivalence ibid., Theorem 5.1.

Thus a page that simply says “the converse is unknown” would be outdated and false. The hypotheses must appear in the statement, because dropping any of time-orderability, additivity, or strict Cauchy constancy changes the problem.

Where higher coherence enters

For cochain-valued gauge theories, the time-slice axiom naturally says that a Cauchy map is a quasi-isomorphism. It has no canonical strict inverse. Choosing one inverse, homotopies witnessing the two inverse laws, homotopies between those homotopies, and choices natural in $M$ produces an unbounded coherence problem. Defining $\mu_M$ with an arbitrary chain-homotopy inverse can make associativity hold only up to an unspecified homotopy and can destroy covariance.

The current structural approach replaces additivity by a decomposition into compatible spacetime-wise problems and establishes the strict one-categorical equivalence in broad targets Benini et al. 2024, Theorems 3.3–3.4. For homotopical theories, it constructs localized semi-model categories, reduces the global comparison to spacetime-wise comparisons, and strictifies the AQFT homotopy time-slice condition ibid., Theorems 4.19–4.20 and 5.1. The time-orderable-prefactorization side still has a remaining spacetime-wise problem, stated explicitly as Benini et al. 2024, Open Problem 5.6. That is the infinity-categorical converse gap named here.

For natural collections of free BV theories, an explicit solution is known: BV and Moyal–Weyl quantizations yield isomorphic time-orderable prefactorization algebras Benini, Musante, and Schenkel 2024, Theorem 4.9. This model theorem does not solve the general interacting homotopical problem.

Testing an E₁ factorization algebra

Take a locally constant factorization algebra on oriented intervals determined by an associative $E_1$ algebra $A$. It has excellent Weiss descent and ordered interval multiplication. To claim a Haag–Kastler theory, however, ask axiom by axiom:

• What functor assigns data to all globally hyperbolic spacetimes, not just intervals in one line?
• Which maps implement Cauchy evolution and are they strict isomorphisms or quasi-isomorphisms?
• Where do causal commutation and covariance enter?
• Is there an order-reversing involution $a\mapsto a^*$ and a positive state?
• Which topology and completion produce the claimed operator algebra?

An arbitrary algebra $A$ answers none of the last questions. Even a $*$-algebra can lack positive representations. The physical comparison belongs at Wightman, Euclidean, Local-Algebraic, Constructive, and Perturbative Frameworks.

Counterexample to a descent-only converse

Choose an associative algebra with no selected involution and form its locally constant factorization algebra on the line. It satisfies Weiss descent, yet no $*$-AQFT can be recovered functorially because the missing antilinear order-reversing operation is not determined by multiplication. More sharply, a Euclidean factorization algebra can have products for all disjoint disks but no Lorentzian causal-order functor. Perfect descent therefore does not imply the hypotheses of the strict reconstruction theorem.

The strongest licensed conclusion is conditional: strict additive Cauchy-constant time-orderable data reconstruct an AQFT; selected free homotopical models admit explicit comparisons; the fully general homotopical equivalence remains open at the cited problem.

Exercises

Where is strict invertibility used in the multiplication formula?

Solution

Each input in $\mathfrak F(M)$ must be transported backward to $\mathfrak F(U_-)$ or $\mathfrak F(U_+)$. This uses actual inverses of the Cauchy maps. A quasi-isomorphism supplies an inverse only in the homotopy category and does not by itself provide coherent chain-level multiplication.

Why does adding an arbitrary involution not finish the reconstruction?

Solution

It must be natural, compatible with reversed multiplication and causal products, and admit the desired positive states or representations. An arbitrary antilinear map need satisfy none of these properties.

References

• Benini, Marco, Victor Carmona, Alastair Grant-Stuart, and Alexander Schenkel. “On the Equivalence of AQFTs and Prefactorization Algebras.” 2024. arXiv:2412.07318.
• Benini, Marco, Giorgio Musante, and Alexander Schenkel. “Quantization of Lorentzian Free BV Theories: Factorization Algebra vs Algebraic Quantum Field Theory.” Letters in Mathematical Physics 114 (2024): 30. doi:10.1007/s11005-024-01784-1.
• 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. arXiv:1903.03396.