On the equivalence of AQFTs and prefactorization algebras
Marco Benini, Victor Carmona, Alastair Grant-Stuart, Alexander Schenkel
TL;DR
The paper addresses the problem of linking algebraic quantum field theories (AQFT) and time-orderable prefactorization algebras (tPFAs) on globally hyperbolic Lorentzian manifolds, introducing a structural additivity framework using the Loc^{rc} subcategory to reduce global equivalence to spacetime-wise problems. It builds a 1-categorical equivalence via a tPFA/AQFT comparison Φ and Haag–Kastler/Costello–Gwilliam 2-functors, then extends to a homotopical setting with model structures, left Bousfield localizations, and operadic localizations to formulate a spacetime-wise reduction theorem. The core contributions include a generalized 1-categorical equivalence over Loc^{rc}, a homotopical reduction that decomposes global problems into local ones on each spacetime M, and a strictification result for the AQFT homotopy time-slice axiom that simplifies spacetime-wise AQFTs. An open problem remains: proving the full spacetime-wise ∞-categorical equivalence between strictified AQFTs and tPFAs, even after the spacetime-wise reduction and strictification steps. The work provides a robust framework for transferring techniques between AQFT and tPFA approaches, with potential implications for gauge theories and ∞-categorical QFT formalisms.
Abstract
This paper revisits the equivalence problem between algebraic quantum field theories and prefactorization algebras defined over globally hyperbolic Lorentzian manifolds. We develop a radically new approach whose main innovative features are 1.) a structural implementation of the additivity property used in earlier approaches and 2.) a reduction of the global equivalence problem to a family of simpler spacetime-wise problems. When applied to the case where the target category is a symmetric monoidal $1$-category, this yields a generalization of the equivalence theorem from [Commun. Math. Phys. 377, 971 (2019)]. In the case where the target is the symmetric monoidal $\infty$-category of cochain complexes, we obtain a reduction of the global $\infty$-categorical equivalence problem to simpler, but still challenging, spacetime-wise problems. The latter would be solved by showing that certain functors between $1$-categories exhibit $\infty$-localizations, however the available detection criteria are inconclusive in our case.