ALGEBRAIC QUANTUM FIELD THEORY: A HOMOTOPICAL VIEW Victor Carmona* August 2, 2021 Abstract In this paper we define and compare several new Quillen model struc- tures which present the homotopy theory of algebraic quantum field theories. In this way, we expand foundational work of Benini et al. [7] by providing a richer framework to detect and treat homotopical phenomena in quantum field theory. Our main technical tool is a new extension model structure on operadic algebras which is constructed via (right) Bousfield localization. We expect that this tool is useful in other contexts. Contents 1 Introduction 2 2 Algebraic quantum field theories 4 2.1 Definitionandexamples . 4 2.2 Operadicpresentationinanutshell . 6 arXiv:2107.14176v2 [math-ph] 30 Jul 2021 3 Time-slice localization 9 3.1 Stricttime-sliceaxiom . 9 3.2 Weaktime-sliceaxiom . 11 *The author was partially supported by the Spanish Ministry of Economy under the grant MTM2016-76453-C2-1-P (AEI/FEDER, UE) and by the Andalusian Ministry of Economy and Knowledge and the Operational Program FEDER 2014-2020 under the grant US-1263032. He was also partly supported by Spanish Ministry of Science, Innovation and Universities, grant FPU17/01871. 1 4 Extension model structure 16 5 Local-to-global cellularization 23 5.1 TheextensionmodelonAQFTs . 23 5.2 Mixinglocalizationandcellularization . ... 24 6 Comparison of homotopy theories 29 1 Introduction The problem of applying quantum mechanical principles to classical systems of fields was the foundational idea which motivates the emersion of Quantum Field Theories. However, a general formulation of what is precisely a Quantum Field Theory seems to be rather elusive. This fact gave rise to the appearance of a broad variety of competing definitions to deal with Quantum Field Theories formally. A very rough classification of such candidates could be: perturbative models, which use the knowledge about simpler free models ([26]); functorial approaches, which are centered on the study of spaces of fields ([23, 30]); and algebraic formulations, which deal with the possible algebraic structures that the collection of observables may carry ([20]). In this work, we focus on the algebraic picture, i.e. on the formal study of algebras of observables. A recent development in the field of Algebraic Quantum Field Theories (usu- ally denoted by AQFTs) has been undertaken in the program of Benini and col- laborators [4, 5, 8, 9, 10]. Their fundamental contribution was to identify operadic constructions, and their importance, on previous axiomatics and foundations, e.g. interprete Haag-Kastler axioms and recognize generalizations [33] or justify the Fredenhagen’s local-to-global principle [14], as well as detecting its weakness. Reasonably, a good deal of new questions arises by the introduction of operad theory into the branch of AQFTs, as addresed in the survey [7]. Operad theory [24] has been proven to be a fruitful mathematical machinery to deal with algebraic structures. Its importance in mathematics is doubtless due to its multidisciplinarity, as it appears in homotopy theory, algebraic topology, geometry, category theory, mathematical physics or mathematical programming. For instance, this theory has also been a valuable tool in Kontsevich’s work on quantisation of Poisson manifolds [22] or Costello-Gwilliam’s factorization alge- bras in perturbative field theories [17]. One of the main advantages of working with operads is the possibility of ma- nipulate the homotopy theory of AQFTs quite explicitly, as observed in [9]. On the one hand, embracing homotopical mathematics permits a better understanding of non discrete phenomena such as gauge symmetries or the presence of anti-fields 2 and anti-ghosts in the BV-BRST formalism (e.g. [6, 8]). These facts motivate a deeper immersion into the homotopy theory of AQFTs. On the other hand, the presence of a Quillen model structure brings to our disposal an assorted toolkit to understand a particular homotopy theory. In [10, 13], the algebraic structure of an AQFT is encoded via a naturally de- fined operad, and so it is reasonable to define the homotopy theory of AQFTs as a canonical homotopy theory of operadic algebras [32]. However, such approx- imation is not well suited to introduce further axioms on AQFTs which are not structural. For example, it does not take into account dynamic axioms properly (time-slice axiom, Definitions 3.1 and 3.6), as noted in [7, Open Problem 4.14], or local-to-global principles (Definition 5.1). The present work exploits the operadic description of AQFTs one step further to find more advanced homotopy theories of Algebraic Quantum Field Theories which fix these problems. Our main results are the construction and comparisons of several model cate- gory structures summarized in Section 6. First, we propose two different model structures to deal with AQFTs satisfying a homotopy meaningful time-slice axiom (Definition 3.6) which turn out to be equivalent. This can be summarized in the following theorem (see Subsection 3.2). Theorem A (Propositions 3.11, 3.14 and Theorem 3.15). Let C⊥ be an orthogo- nal category (Definition 2.1) and S a set of maps in C. Then, the homotopy theory of algebraic quantum field theories over C⊥ satisfying the weak S-time-slice ax- ⊥ iom is presented by the Quillen equivalent model structures QFT w S(C ) and ⊥ LS QFT (C ). Both models provide different aspects of the homotopy theory of AQFTs sat- isfying such dynamical axiom. In fact, below Theorem 3.15 we expand on this to answer [7, Open Problem 4.14]. The local-to-global principle (Definition 5.1) is incorporated in a new model structure that is an instance of a general construction explained in Section 4 which culminates in Theorem 4.9. ⊥ ⊥ Theorem B (Theorem 5.3). Let C⋄ ֒→ C be an inclusion of a full orthogonal subcategory. Then, the homotopy theory of algebraic quantum field theories over ⊥ C⋄ ⊥ C which satisfy the C⋄-local-to-global principle is modelled by QFT (C ). Finally, for the combination of both axioms, time-slice plus local-to-global, we find again two model structures presenting the associated homotopy theory which are equivalent as expected. ⊥ ⊥ Theorem C (Theorems 5.9, 5.11, 5.13). Let C⋄ ֒→ C be an inclusion of a full orthogonal subcategory and S a suitable set of maps in C. Then, the homotopythe- ory of algebraic quantum field theories over C⊥ satisfying the weak S-time-slice 3 axiom and the C⋄-local-to-global principle is presented by the Quillen equivalent C⋄ ⊥ C⋄ ⊥ model structures QFT w S(C ) and LS QFT (C ). Coming back to the general construction of Section 4, it is mandatory to com- ment that it has its own importance in abstract homotopy theory and so several applications are expected. Indeed, in [16], the homotopy theory of factorization algebras is investigated by these means. It is worth noting that most of the results in this work are valid for a closed symmetric monoidal model category V satisfying some general requirements con- sisting on a combination of hypothesis appearing in Section 4 and in [15] (or [16, Section 6]). However, we have chosen to state them for chain-complex valued algebraic field theories over a (commutative) ring R which contains the rationals for simplicity and due to connections with the existing literature. This restriction is not made explicit in order to enforce the cited generality. For instance, Sections 2 and 4 are written for a general V, whereas the rest of the sections should be interpreted under the assumption V = Ch(R). Outline: We include a brief summary of the contents of this work. Section 2 presents basic definitions in the theory of AQFTs and a straightforward construc- tion of the canonical operad developed in [10] and [12]. The rest of the paper is organized depending on which further axiom of AQFTs is under study. Section 3 is devoted to the analysis and discussion of the time-slice axiom and its associated homotopy theories, going from the strict version to the homotopical one, which is seen as part of structure or as a property of AQFTs. The local-to-global principle introduced in the work of Benini and collaborators, and its interaction with the time-slice axiom, is the focus on Section 5. To fully understand the constructions in this last section, the content of Section 4 is necessary. It yields a machine to construct cellularizations of model categories of operadic algebras. Despite this fact, this homotopical discussion can be used as a necessary black box, since it re- produces the local-to-global principle model categorically. At the end, we include a table and a diagram (Section 6) condensing the field theory related material ob- tained in this paper. 2 Algebraic quantum field theories 2.1 Definition and examples Algebraic quantum field theory is an axiomatic approach to quantum physics which focuses on the assignation of algebras of observables to spacetime regions. 4 The fundamental idea of such objects is that the algebraic structure on the ob- servables is supposed to capture quantum information, so it is not commutative in general, although observables coming from “disjoint spacetime regions" do com- mute. The notion of orthogonal category formalizes the concept of spacetime regions and the relation of being disjoint. Definition 2.1. [33, Definition 8.2.1] An orthogonal category C⊥ is a pair (C, ⊥) given by a category C with choices of sets of morphisms ⊥ = ⊥ ({U, W}; V) ⊆ C(U, V) × C(W, V) U,W,V∈ob C which are stable by composition in C and are symmetric in (U, W). It is said that two C-morphisms f and g are orthogonal, denoted f ⊥ g, if they have the same codomain and (f, g) belongs to ⊥.