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Haag-Kastler stacks

Marco Benini, Alastair Grant-Stuart, Alexander Schenkel

TL;DR

This work reframes general local covariance for AQFTs through a Haag-Kastler 2-functor HK: Loc^op -> CAT, whose objects are Haag-Kastler AQFTs on each Lorentzian spacetime M and whose morphisms are pullback functors along spacetime embeddings. It establishes a tight link between locally covariant AQFTs and the points of HK, while exposing descent issues for various Haag-Kastler and time-slice variants; to remedy this, it introduces HK^{rc} and HK^{rc,W} built from relatively compact opens and Cauchy localization, and develops an 'improvement' procedure using locally presentable categories to produce stacks. The Klein-Gordon field is shown to produce a point of the resulting stacks, validating the construction and suggesting applicability to explicit QFTs. The framework blends 2-categorical descent with operadic left Kan extensions and Lorentzian geometry, offering a flexible path to local-to-global reconstruction and potential extensions to higher-categorical AQFTs and factorization-algebra formalisms.

Abstract

This paper provides an alternative implementation of the principle of general local covariance for algebraic quantum field theories (AQFTs) which is more flexible than the original one by Brunetti, Fredenhagen and Verch. This is realized by considering the $2$-functor $\mathsf{HK} : \mathbf{Loc}^\mathrm{op} \to \mathbf{CAT}$ which assigns to each Lorentzian manifold $M$ the category $\mathsf{HK}(M)$ of Haag-Kastler-style AQFTs over $M$ and to each embedding $f:M\to N$ a pullback functor $f^\ast = \mathsf{HK}(f) : \mathsf{HK}(N) \to \mathsf{HK}(M)$ restricting theories from $N$ to $M$. Locally covariant AQFTs are recovered as the points of the $2$-functor $\mathsf{HK}$. The main advantages of this new perspective are: 1.) It leads to technical simplifications, in particular with regard to the time-slice axiom, since global problems on $\mathbf{Loc}$ become families of simpler local problems on individual Lorentzian manifolds. 2.) Some aspects of the Haag-Kastler framework which previously got lost in locally covariant AQFT, such as a relative compactness condition on the open subsets in a Lorentzian manifold $M$, are reintroduced. 3.) It provides a radically new perspective on descent conditions in AQFT, i.e. local-to-global conditions which allow one to recover a global AQFT on a Lorentzian manifold $M$ from its local data in an open cover $\{U_i \subseteq M\}$.

Haag-Kastler stacks

TL;DR

This work reframes general local covariance for AQFTs through a Haag-Kastler 2-functor HK: Loc^op -> CAT, whose objects are Haag-Kastler AQFTs on each Lorentzian spacetime M and whose morphisms are pullback functors along spacetime embeddings. It establishes a tight link between locally covariant AQFTs and the points of HK, while exposing descent issues for various Haag-Kastler and time-slice variants; to remedy this, it introduces HK^{rc} and HK^{rc,W} built from relatively compact opens and Cauchy localization, and develops an 'improvement' procedure using locally presentable categories to produce stacks. The Klein-Gordon field is shown to produce a point of the resulting stacks, validating the construction and suggesting applicability to explicit QFTs. The framework blends 2-categorical descent with operadic left Kan extensions and Lorentzian geometry, offering a flexible path to local-to-global reconstruction and potential extensions to higher-categorical AQFTs and factorization-algebra formalisms.

Abstract

This paper provides an alternative implementation of the principle of general local covariance for algebraic quantum field theories (AQFTs) which is more flexible than the original one by Brunetti, Fredenhagen and Verch. This is realized by considering the -functor which assigns to each Lorentzian manifold the category of Haag-Kastler-style AQFTs over and to each embedding a pullback functor restricting theories from to . Locally covariant AQFTs are recovered as the points of the -functor . The main advantages of this new perspective are: 1.) It leads to technical simplifications, in particular with regard to the time-slice axiom, since global problems on become families of simpler local problems on individual Lorentzian manifolds. 2.) Some aspects of the Haag-Kastler framework which previously got lost in locally covariant AQFT, such as a relative compactness condition on the open subsets in a Lorentzian manifold , are reintroduced. 3.) It provides a radically new perspective on descent conditions in AQFT, i.e. local-to-global conditions which allow one to recover a global AQFT on a Lorentzian manifold from its local data in an open cover .
Paper Structure (30 sections, 50 theorems, 186 equations)

This paper contains 30 sections, 50 theorems, 186 equations.

Table of Contents

  1. Introduction and summary
  2. Remarks about AQFTs taking values in $C^\ast$-algebras:
  3. Preliminaries
  4. Orthogonal categories and AQFTs
  5. Pseudo-functors and stacks
  6. Locally presentable categories
  7. Haag-Kastler 2-functors
  8. The case of causally convex opens
  9. Definition and properties
  10. Comparison to locally covariant AQFT
  11. Time-slice axiom
  12. The case of relatively compact causally convex opens
  13. Definition and properties
  14. Comparison to additivity properties
  15. Additivity in Haag-Kastler-style AQFTs:
  16. ...and 15 more sections

Key Result

Proposition 2.3

For every orthogonal functor $F :\overline{\mathbf{C}}\to \overline{\mathbf{D}}$, the pullback functor in eqn:Fpullbackfunctor admits a left adjoint, i.e. one has an adjunction between the corresponding AQFT categories. The left adjoint $F_!$ is called operadic left Kan extension along $F$, see also Appendix app:operadicLKE for an explicit model.

Theorems & Definitions (110)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 100 more