Algebraic field theory operads and linear quantization
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Abstract
We generalize the operadic approach to algebraic quantum field theory (arXiv:1709.08657) to a broader class of field theories whose observables on a spacetime are algebras over any singlecolored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose left adjoint describes the quantization of linear field theories. We also analyze homotopical properties of the linear quantization adjunction for chain complex valued field theories, which leads to a homotopically meaningful quantization prescription for linear gauge theories.
Keywords
Algebraic quantum field theory Locally covariant quantum field theory Colored operads Universal constructions Gauge theory Model categoriesMathematics Subject Classification
81Txx 18D50 18G551 Introduction and summary
Algebraic quantum field theory is a wellestablished and successful framework to axiomatize and investigate quantum field theories on the Minkowski spacetime and also on more general Lorentzian manifolds, see e.g. [11, 25] for overviews. In this setting a theory is described by a functor \(\mathfrak {A}: \mathbf {C}\rightarrow \mathbf {Alg}_{\mathsf {As}}\) from a suitable category \(\mathbf {C}\) of spacetimes to the category of associative and unital algebras, which is required to satisfy some physically motivated axioms. For instance, in locally covariant algebraic quantum field theory [13, 18], \(\mathbf {C}=\mathbf {Loc}\) is the category of globally hyperbolic Lorentzian manifolds with morphisms given by causal isometric embeddings and the physical axioms are Einstein causality and the timeslice axiom. Einstein causality demands that any two observables, i.e. elements of the algebras assigned by \(\mathfrak {A}\), that are causally disjoint commute with each other, which encodes the idea that two measurements in causally disjoint spacetime regions do not influence each other. The timeslice axiom demands that the algebra maps \(\mathfrak {A}(f): \mathfrak {A}(M)\rightarrow \mathfrak {A}(M^\prime )\) associated with Cauchy morphisms, i.e. spacetime embeddings \(f:M\rightarrow M^\prime \) such that \(f(M)\subseteq M^\prime \) contains a Cauchy surface of \(M^\prime \), are isomorphisms, which encodes a concept of time evolution. The framework of algebraic quantum field theory can also be adapted to obtain a novel point of view on classical field theories, see e.g. [5, 12, 14, 22], where in contrast to associative and unital algebras one assigns Poisson algebras of classical observables to spacetimes. The classical analog of Einstein causality then demands that the Poisson bracket between any two causally disjoint observables is zero.
The aim of this paper is to develop an operadic framework that generalizes [8] to a very broad and flexible class of field theories, see Definition 3.3. This includes as special instances the various flavors of algebraic quantum field theory [11, 13, 18, 25] and their classical analogs [5, 12, 14, 22]. Our two main motivations for this work are as follows: (1) Describing field theories in terms of algebras over colored operads provides an excellent framework to discover and study universal constructions. This has already lead to a refinement of Fredenhagen’s universal algebra construction for quantum field theories [19, 20, 21, 31] in terms of a socalled operadic left Kan extension [8], which technically behaves better than the original construction as it respects the quantum field theory axioms. In this paper, we will show that the quantization of linear field theories may be expressed in terms of an operadic left Kan extension too, which allows us to understand and describe the interplay between quantization and other universal constructions. (2) Operadic techniques are particularly useful and powerful when working with chain complex valued field theories, e.g. gauge theories described in the BRST/BV formalism [22, 23, 29]. The reason for this is that chain complexes are naturally compared by quasiisomorphisms and hence one is only allowed to perform constructions that preserve quasiisomorphisms. Operad theory provides a huge toolbox to develop such constructions, which in technical language are called derived functors, see, e.g. [27, 28] and also [9] for applications to quantum field theory. In this paper, we apply these techniques to investigate homotopical properties of the linear quantization functor. A similar construction in the context of factorization algebras [15] has been recently investigated in [24]. As a simple example, we present a quantization of linear Chern–Simons theory on oriented surfaces that is compatible with quasiisomorphisms.
Let us now present a more detailed outline of the content of this paper. In Sect. 2, we shall fix our notations by recalling the necessary background material on colored operads and their algebras. In Sect. 3, we introduce our broad and flexible framework for field theories in Definition 3.3. A field theory is described by a functor \(\mathfrak {A}: \mathbf {C}\rightarrow \mathbf {Alg}_\mathcal {P}\) from a small category \(\mathbf {C}\) to the category of algebras over a singlecolored operad \(\mathcal {P}\), which is required to satisfy a suitable generalization of the Einstein causality axiom. (The timeslice axiom will be formalized via localization techniques in Sect. 4.2.) One should interpret \(\mathbf {C}\) as a category of spacetimes and \(\mathcal {P}\) as the operad controlling the algebraic structure of the observables on a fixed spacetime. For example, quantum field theories are obtained by choosing the associative operad \(\mathcal {P}=\mathsf {As}\) and classical field theories by choosing the Poisson operad \(\mathcal {P}= \mathsf {Pois}\). Linear field theories, which we describe in terms of Heisenberg Lie algebras of presymplectic vector spaces, are obtained by choosing the unital Lie operad \(\mathcal {P}= \mathsf {uLie}\). One of the key results of this section (see Theorem 3.12) is that such field theories are precisely the algebras over a colored operad that we denote by \(\mathcal {P}^{(r_1,r_2)}_{\overline{\mathbf {C}}}\). This colored operad depends on two different kinds of input data, which control the spacetime category of interest and the type of field theory. More precisely, the first datum is an orthogonal category \(\overline{\mathbf {C}} = (\mathbf {C},\perp )\) (see Definition 3.1), and the second is a bipointed singlecolored operad \(\mathcal {P}^{(r_1,r_2)} = (\mathcal {P}, r_1,r_2 : I[2]\rightrightarrows \mathcal {P})\) (see Definition 3.14). The orthogonality relation \(\perp \) and the two pointings \(r_1,r_2\) are required to formalize a suitable generalization of the Einstein causality axiom. We prove that the assignment \((\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}) \mapsto \mathcal {P}^{(r_1,r_2)}_{\overline{\mathbf {C}}}\) of field theory operads is in a suitable sense functorial.
In Sect. 4, we harness this functorial behavior in order to study adjunctions between the categories of field theories corresponding to different \(\overline{\mathbf {C}}\) and \(\mathcal {P}^{(r_1,r_2)}\). This includes generalizations of the timeslicification and localtoglobal adjunctions from [8], which have already found interesting applications to quantum field theory on spacetimes with boundaries [4]. A novel feature of our framework, which is not captured by [8], is a second kind of functorial assignment \(\mathcal {P}^{(r_1,r_2)} \mapsto \mathcal {P}^{(r_1,r_2)}_{\overline{\mathbf {C}}}\) of our colored operads to bipointed singlecolored operads. This results in adjunctions between the categories of field theories of different types. We shall investigate in detail the interplay of such adjunctions with the timeslice axiom and localtoglobal property of field theories. A particularly interesting example, which we study in detail in Sect. 5, is given by an adjunction whose left adjoint describes the quantization of linear field theories.
In Sect. 6, we extend our results to the case of \(\mathbf {Ch}(\mathbb {K})\)valued field theories, i.e. gauge theories, by using techniques from model category theory [17, 30]. Our reformulation in Sect. 5 of the usual quantization of linear field theories in terms of (the left adjoint of) an adjunction is very valuable for studying the quantization of linear gauge theories. In particular, it allows us to construct a derived linear quantization functor which provides a homotopically meaningful quantization prescription for linear gauge theories in the sense that it maps weakly equivalent linear gauge theories to weakly equivalent quantum gauge theories. A deeper homotopical study of the building blocks of the derived linear quantization functor (see Appendix A) reveals that it can be modeled (up to a natural weak equivalence) by the underived linear quantization functor. From a computational point of view, this is a very pleasing result because it allows us to write down explicit formulas for the quantization of linear gauge theories. This will be illustrated by studying linear Chern–Simons theory on oriented surfaces. We conclude by analyzing in some detail the interplay between our (derived) linear quantization functor and suitable homotopical generalizations of the timeslice axiom and localtoglobal property of gauge theories.
2 Preliminaries
Example 2.1
A simple example of a bicomplete closed symmetric monoidal category is the Cartesian closed category \(\mathbf {Set}\) of sets. Here \(\otimes = \times \) is the Cartesian product, \(I =\{*\}\) is any singleton set and \([S,T] = \text {Map}(S,T)\) is the set of maps from S to T. The symmetric braiding is given by the flip map \(\tau : S\times T\rightarrow T\times S\,,~(s,t)\mapsto (t,s)\).
Example 2.2
Another standard example of a bicomplete closed symmetric monoidal category is the category \(\mathbf {Vec}_\mathbb {K}\) of vector spaces over a field \(\mathbb {K}\). Here \(\otimes \) is the usual tensor product of vector spaces, \(I=\mathbb {K}\) is the 1dimensional vector space and \([V,W] = \text {Hom}_\mathbb {K}(V,W)\) is the vector space of linear maps from V to W. The symmetric braiding is given by the flip map \(\tau : V\otimes W \rightarrow W\otimes V\,,~ v\otimes w\mapsto w\otimes v\).
2.1 Colored operads
We provide a brief review of those aspects of the theory of colored operads that are relevant for this work. We refer to [8, 10, 34] for a more detailed presentation.
Let \(\mathfrak {C}\in \mathbf {Set}\) be a nonempty set, which we shall call the ‘set of colors’. We will use the notation \(\underline{c} {:=} (c_1, \ldots ,c_n) \in \mathfrak {C}^{n}\) for elements of the nfold product set.
Definition 2.3

for each \(n\ge 0\) and \((\underline{c},t) \in \mathfrak {C}^{n+1}\), an object \(\mathcal {O}\big (\begin{array}{c} t \\ \underline{c} \end{array}\big )\in \mathbf {M}\) (called the object of operations from \(\underline{c}\) to t);

for each \(n\ge 0\), \((\underline{c},t) \in \mathfrak {C}^{n+1}\) and permutation \(\sigma \in \Sigma _n\), an \(\mathbf {M}\)morphism \(\mathcal {O}(\sigma ):\,\mathcal {O}\big (\begin{array}{c} t \\ \underline{c} \end{array}\big ) {\rightarrow } \mathcal {O}\big (\begin{array}{c} t\\ \underline{c}\sigma \end{array}\big )\) (called the permutation action), where \(\underline{c} \sigma {:=}\, (c_{\sigma (1)},\ldots ,c_{\sigma (n)})\);

for each \(n>0\), \(k_1,\ldots ,k_n \ge 0\), \((\underline{a},t)\in \mathfrak {C}^{n+1}\) and \((\underline{b}_i,a_i)\in \mathfrak {C}^{k_i+1}\), for \(i=1,\ldots ,n\), an \(\mathbf {M}\)morphism Open image in new window (called the operadic composition), where \(\underline{b} := (\underline{b}_1,\ldots ,\underline{b}_n)\) is defined by concatenation;

for each \(c\in \mathfrak {C}\), an \(\mathbf {M}\)morphism \(\mathbb {1}: I\rightarrow \mathcal {O}\big (\begin{array}{c} c\\ c \end{array}\big )\) (called the operadic unit).
Colored operads generalize the concept of (enriched) categories in the following sense. In contrast to allowing only for 1to1 operations, such as the morphisms \(\mathbf {C}(c,c^\prime )\) in a category \(\mathbf {C}\), colored operads also describe nto1 operations in terms of the objects of operations \(\mathcal {O}\big (\begin{array}{c} t \\ \underline{c} \end{array}\big )\). The operadic composition generalizes the usual categorical composition to operations of higher arity and the operadic unit is analogous to the identity morphisms in a category. Permutation actions are a new feature for operations of arity \(\ge 2\) and they have no analog in ordinary category theory. The following example clarifies how every category defines a colored operad with only 1ary operations.
Example 2.4
Definition 2.5
Example 2.6
Let us note that the associative operad can be defined in any bicomplete closed symmetric monoidal category \(\mathbf {M}\). Using the \(\mathbf {Set}\)tensoring (2.1) and the unit object \(I\in \mathbf {M}\), we define generators \(G\otimes I\in \mathbf {Seq}_{\{*\}}(\mathbf {M})\) and relations \(r_1\otimes I, r_2\otimes I : R\otimes I \rightarrow UF(G)\otimes I \cong UF(G\otimes I)\) in the category of \(\mathbf {M}\)valued sequences \(\mathbf {Seq}_{\{*\}}(\mathbf {M})\). The corresponding coequalizer then defines the \(\mathbf {M}\)valued associative operad \(\mathsf {As}:= F(G\otimes I)/ ( r_1\otimes I=r_2\otimes I)\in \mathbf {Op}_{\{*\}}(\mathbf {M})\).
Example 2.7
Note that for defining the Lie relations we had to use the natural Abelian group structure on the \(\text {Hom}\)sets of \(\mathbf {Vec}_\mathbb {K}\), i.e. addition of linear maps between vector spaces. Hence, the Lie operad can not be defined in a generic bicomplete closed symmetric monoidal category \(\mathbf {M}\). If, however, \(\mathbf {M}\) is an additive category, then one can define the Lie operad \(\mathsf {Lie}\in \mathbf {Op}_{\{*\}}(\mathbf {M})\) with values in \(\mathbf {M}\) along the same lines as above.
Example 2.8
Example 2.9
We shall often require a generalization of the concept of colored operad morphisms from Definition 2.3 to morphisms that do not necessarily preserve the underlying sets of colors. As a preparation for the relevant definition, note that for every \(\mathfrak {D}\)colored operad \(\mathcal {P}\in \mathbf {Op}_{\mathfrak {D}}(\mathbf {M})\) and every map of sets \(f : \mathfrak {C}\rightarrow \mathfrak {D}\), one may define the pullback\(\mathfrak {C}\)colored operad\(f^*(\mathcal {P})\in \mathbf {Op}_\mathfrak {C}(\mathbf {M})\). Concretely, it is defined by setting \(f^*(\mathcal {P})\big (\begin{array}{c} t \\ \underline{c} \end{array}\big ) := \mathcal {P}\big (\begin{array}{c} f(t)\\ f(\underline{c}) \end{array}\big )\), for all \(n\ge 0\) and \((\underline{c},t)\in \mathfrak {C}^{n+1}\), and restricting the permutation action, operadic composition and operadic unit in the evident way.
Definition 2.10
The category \(\mathbf {Op}(\mathbf {M})\) of operads with varying colors with values in \(\mathbf {M}\) has as objects all pairs \((\mathfrak {C},\mathcal {O})\) consisting of a nonempty set \(\mathfrak {C}\) and a \(\mathfrak {C}\)colored operad \(\mathcal {O}\in \mathbf {Op}_\mathfrak {C}(\mathbf {M})\). A morphism is a pair \((f,\phi ) : (\mathfrak {C},\mathcal {O})\rightarrow (\mathfrak {D},\mathcal {P})\) consisting of a map of sets \(f : \mathfrak {C}\rightarrow \mathfrak {D}\) and an \(\mathbf {Op}_\mathfrak {C}(\mathbf {M})\)morphism \(\phi : \mathcal {O}\rightarrow f^*(\mathcal {P})\) to the pullback \(\mathfrak {C}\)colored operad.
2.2 Algebras over colored operads
We have seen above that a colored operad \(\mathcal {O}\) describes abstract nto1 operations, for all \(n\ge 0\), together with a composition law \(\gamma \), specified identities \(\mathbb {1}\) and a permutation action \(\mathcal {O}(\sigma )\) that allows us to permute the inputs of operations. Forming concrete realizations/representations of these abstract operations leads to the concept of algebras over colored operads.
Definition 2.11

for each \(c\in \mathfrak {C}\), an object \(A_c\in \mathbf {M}\);

for each \(n\ge 0\) and \((\underline{c},t)\in \mathfrak {C}^{n+1}\), an \(\mathbf {M}\)morphism \(\alpha : \mathcal {O}\big (\begin{array}{c} t \\ \underline{c} \end{array}\big )\otimes A_{\underline{c}} \rightarrow A_t\) (called \(\mathcal {O}\)action), where \(A_{\underline{c}} := \bigotimes _{i=1}^n A_{c_i}\) with the convention that \(A_\emptyset =I\) for \(n=0\).
Example 2.12
Consider the diagram operad \(\text {Diag}_{\mathbf {C}}\in \mathbf {Op}_{\mathbf {C}_0}(\mathbf {Set})\) from Example 2.4. A \(\text {Diag}_{\mathbf {C}}\)algebra is a family of sets \(A_c\in \mathbf {Set}\), for all objects \(c\in \mathbf {C}_0\) in the category \(\mathbf {C}\), together with maps \(\alpha : \text {Diag}_{\mathbf {C}}\big (\begin{array}{c} t\\ c \end{array}\big ) \times A_c\rightarrow A_t\), for all \(c,t\in \mathbf {C}_0\). (Here we already used that \(\text {Diag}_{\mathbf {C}}\) only contains 1ary operations.) Because \(\text {Diag}_{\mathbf {C}}\big (\begin{array}{c} t\\ c \end{array}\big ) = \mathbf {C}(c,t)\) is the \(\text {Hom}\)set, the latter data is equivalent to specifying for each \(\mathbf {C}\)morphism \(f:c\rightarrow t\) a map of sets \(A(f):= \alpha (f,) : A_c\rightarrow A_t\). The axioms for \(\mathcal {O}\)algebras imply that \(A(g\,f) = A(g) \, A(f)\), for all composable \(\mathbf {C}\)morphism, and \(A(\text {id}) = \text {id}\) for the identities. Hence, a \(\text {Diag}_{\mathbf {C}}\)algebra is precisely a functor \(\mathbf {C}\rightarrow \mathbf {Set}\), i.e. a diagram of shape \(\mathbf {C}\). One observes that morphisms between \(\text {Diag}_\mathbf {C}\)algebras are precisely natural transformations between the corresponding functors.
Example 2.13
Consider for the moment \(\mathbf {M}=\mathbf {Set}\) and the associative operad \(\mathsf {As}\in \mathbf {Op}_{\{*\}}(\mathbf {Set})\) from Example 2.6. An \(\mathsf {As}\)algebra is a single set \(A = A_*\in \mathbf {Set}\) together with an \(\mathsf {As}\)action. The latter is equivalent to providing a family of maps \(\alpha : \mathsf {As}(n) \rightarrow \text {Map}(A^{\times n},A)\), for all \(n\ge 0\), which define an \(\mathbf {Op}_{\{*\}}(\mathbf {Set})\)morphism to the endomorphism operad \(\text {End}(A)\), see e.g. [34, Definition 13.8.1]. Because \(\mathsf {As}\) is presented by generators and relations (see Example 2.6), this is equivalent to defining \(\alpha \) on the generators such that the relations hold true. This yields two maps \(\mu _A := \alpha (\mu ) : A\times A\rightarrow A\) and \(\eta _A:=\alpha (\eta ) : \{*\} \rightarrow A\), which because of the relations have to satisfy the axioms of an associative and unital algebra in \(\mathbf {Set}\). One finds that morphisms of \(\text {As}\)algebras are precisely morphisms of associative and unital algebras.
For a general bicomplete closed symmetric monoidal category \(\mathbf {M}\), one obtains that the category \(\mathbf {Alg}_{\mathsf {As}}\) of algebras over \(\mathsf {As}\in \mathbf {Op}_{\{*\}}(\mathbf {M})\) is the category of associative and unital algebras in \(\mathbf {M}\). In particular, for \(\mathbf {M}=\mathbf {Vec}_\mathbb {K}\), this is the category of associative and unital \(\mathbb {K}\)algebras.
Example 2.14
A similar argument as in Example 2.13 shows that the category \(\mathbf {Alg}_{\mathsf {Lie}}\) of algebras over the Lie operad \(\mathsf {Lie}\in \mathbf {Op}_{\{*\}}(\mathbf {M})\) (see Example 2.7) is the category of Lie algebras in \(\mathbf {M}\) and that the category \(\mathbf {Alg}_{\mathsf {Pois}}\) of algebras over the Poisson operad \(\mathsf {Pois}\in \mathbf {Op}_{\{*\}}(\mathbf {M})\) (see Example 2.8) is the category of Poisson algebras in \(\mathbf {M}\).
Theorem 2.15
Example 2.16
Every functor \(F: \mathbf {C}\rightarrow \mathbf {D}\) defines an evident \(\mathbf {Op}(\mathbf {Set})\)morphism \((F_0, F) : (\mathbf {C}_0,\text {Diag}_\mathbf {C})\rightarrow (\mathbf {D}_0,\text {Diag}_\mathbf {D})\) between the corresponding diagram operads (see Example 2.4). Recalling from Example 2.12 that \(\mathbf {Alg}_{\text {Diag}_\mathbf {C}}\cong \mathbf {Set}^\mathbf {C}\) is the category of functors from \(\mathbf {C}\) to \(\mathbf {Set}\) (and similarly that \(\mathbf {Alg}_{\text {Diag}_\mathbf {D}}\cong \mathbf {Set}^\mathbf {D}\)), one shows that the pullback functor \((F_0,F)^*\) is the usual pullback functor \(F^*:=()\circ F : \mathbf {Set}^\mathbf {D}\rightarrow \mathbf {Set}^\mathbf {C}\) for functor categories. Its left adjoint \((F_0,F)_!\) is therefore the ordinary categorical left Kan extension \({\text {Lan}}_F : \mathbf {Set}^\mathbf {C}\rightarrow \mathbf {Set}^\mathbf {D}\).
3 Field theory operads
3.1 Orthogonal categories and field theories
For the purpose of this paper, we consider the following generalization of quantum field theories satisfying the Einstein causality axiom. (Examples which justify this generalization are presented at the end of this subsection.) Let \(\mathbf {C}\) be a small category which we interpret as a category of spacetimes. Instead of associative and unital algebras, let us take any singlecolored operad \(\mathcal {P}\in \mathbf {Op}_{\{*\}}(\mathbf {M})\) and consider the functor category \({\mathbf {Alg}_{\mathcal {P}}}^\mathbf {C}\). An object in this category is a functor \(\mathfrak {A}: \mathbf {C}\rightarrow \mathbf {Alg}_{\mathcal {P}}\), i.e. an assignment of \(\mathcal {P}\)algebras to spacetimes, and the morphisms are natural transformations between such functors. To encode physical axioms which generalize the Einstein causality axiom above, we recall the concept of orthogonal categories from [8].
Definition 3.1
 (i)
Symmetry: If \((f_1,f_2)\in \perp \), then \((f_2,f_1)\in \perp \).
 (ii)
Stability under compositions: If \((f_1,f_2)\in \perp \), then \((g\, f_1\, h_1 ,g\, f_2\, h_2) \in \perp \) for all composable \(\mathbf {C}\)morphisms g, \(h_1\) and \(h_2\).
Example 3.2
Let \(\mathbf {Loc}\) be any small category that is equivalent to the usual category of oriented, timeoriented and globally hyperbolic Lorentzian spacetimes of a fixed dimension \(\ge 2\), see [13, 18]. We define \(\perp _\mathbf {Loc}\) as the subset of all pairs \((f_1 : M_1\rightarrow M, f_2: M_2\rightarrow M)\) of causally disjoint \(\mathbf {Loc}\)morphisms, i.e. pairs of morphisms such that the images \(f_1(M_1)\) and \(f_2(M_2)\) are causally disjoint subsets in M. The pair \(\overline{\mathbf {Loc}} := (\mathbf {Loc},\perp _{\mathbf {Loc}})\) defines an orthogonal category.
Definition 3.3
Remark 3.4
Our concept of field theories in Definition 3.3 is based on the idea that there exist two distinguished arity 2 operations in \(\mathcal {P}\), which act in the same way when precomposed with an orthogonal pair \(f_1 \perp f_2\) of \(\mathbf {C}\)morphisms. There exists an obvious generalization of this scenario to nary operations in \(\mathcal {P}\) and orthogonal ntuples of \(\mathbf {C}\)morphisms. We however decided not to introduce this more general framework for field theories, because all examples of interest to us are field theories in the sense of Definition 3.3.
Example 3.5
(Quantum field theories) Consider the associative operad \(\mathsf {As}\in \mathbf {Op}_{\{*\}}(\mathbf {M})\) from Example 2.6 and the two \(\mathbf {Seq}_{\{*\}}(\mathbf {M})\)morphisms \(\mu ,\mu ^\text {op}: I[2] \rightrightarrows U(\mathsf {As})\) which select the multiplication and opposite multiplication operations. A field theory of type \(\mathsf {As}^{(\mu ,\mu ^\text {op})}\) on \(\overline{\mathbf {C}}\) is a functor \(\mathfrak {A}: \mathbf {C}\rightarrow \mathbf {Alg}_{\mathsf {As}}\) to the category of associative and unital algebras which satisfies the analog of (3.1). For \(\overline{\mathbf {C}} = \overline{\mathbf {Loc}}\) (see Example 3.2), this is a locally covariant quantum field theory [13, 18] that satisfies the Einstein causality axiom but not necessarily the timeslice axiom. The timeslice axiom will be discussed in Sect. 4.2.
Remark 3.6
Example 3.7
Example 3.8
(Linear field theories) In the usual construction of linear quantum field theories, see e.g. [1, 2, 3] for reviews, one first defines a functor \(\mathfrak {L}: \mathbf {Loc}\rightarrow \mathbf {PSymp}\) to the category of presymplectic vector spaces, which is then quantized by forming CCRalgebras (CCR stands for canonical commutation relations). Recall that a presymplectic vector space \((V,\omega )\) is a pair consisting of a vector space V and an antisymmetric linear map \(\omega : V\otimes V\rightarrow \mathbb {K}\). Notice that this is not an operation of arity 2 in the sense of operads because the target is the ground field and not V. Hence, \(\mathbf {PSymp}\) is not the category of algebras over an operad and, as a consequence, functors \(\mathfrak {L}: \mathbf {Loc}\rightarrow \mathbf {PSymp}\) do not define field theories in the sense of Definition 3.3.
3.2 Operadic description
In this section, we show that the category of field theories from Definition 3.3 is the category of algebras over a suitable colored operad. This generalizes previous results in [8] and it is the key insight that allows us to study a large family of universal constructions for field theories in Sect. 4. As a preparation for the relevant definition, we define an auxiliary colored operad that describes functors from a small category \(\mathbf {C}\) to the category of \(\mathcal {P}\)algebras.
Definition 3.9
 for \(n\ge 0\) and \((\underline{c},t)\in \mathbf {C}_0^{n+1}\), the object of operations iswhere \(\otimes \) is the \(\mathbf {Set}\)tensoring (2.1) and \(\mathbf {C}(\underline{\text {c}},t):=\prod _{i=1}^n \mathbf {C}(c_i,t)\) is the product of \(\text {Hom}\)sets;$$\begin{aligned} \mathcal {P}_\mathbf {C}\big (\begin{array}{c} t \\ \underline{c} \end{array}\big )\,:=\, \mathbf {C}(\underline{c},t) \otimes \mathcal {P}(n) \,\in \,\mathbf {M}, \end{aligned}$$(3.11)
 for \(n\ge 0\), \((\underline{c},t)\in \mathbf {C}_0^{n+1}\) and \(\sigma \in \Sigma _n\), the permutation action \(\mathcal {P}_\mathbf {C}(\sigma )\) is defined by(3.12)
 for \(n>0\), \(k_1,\ldots ,k_n \ge 0\), \((\underline{a},t)\in \mathbf {C}_0^{n+1}\) and \((\underline{b}_i,a_i)\in \mathbf {C}_0^{k_i+1}\), for \(i=1,\ldots ,n\), the operadic composition \(\gamma ^{\mathcal {P}_\mathbf {C}}\) is defined by(3.13)
 for \(c\in \mathbf {C}_0\), the operadic unit \(\mathbb {1}^{\mathcal {P}_\mathbf {C}}\) is(3.14)
Lemma 3.10
Proof
Definition 3.11
The importance of this operad is evidenced by the following theorem.
Theorem 3.12
Proof
Example 3.13
Recalling Examples 3.5, 3.7 and 3.8, our construction defines colored operads for quantum field theory \(\mathsf {As}^{(\mu ,\mu ^\text {op})}_{\overline{\mathbf {C}}}\) (or equivalently \(\mathsf {As}^{([\cdot ,\cdot ],0)}_{\overline{\mathbf {C}}}\) provided that \(\mathbf {M}\) is additive, see Remark 3.6), for classical field theory \(\mathsf {Pois}^{(\{\cdot ,\cdot \},0)}_{\overline{\mathbf {C}}}\) and for linear field theory \(\mathsf {uLie}^{([\cdot ,\cdot ],0)}_{\overline{\mathbf {C}}}\) formalized in terms of Heisenberg Lie algebras.
3.3 Functoriality
Note that the field theory operad \(\mathcal {P}^{(r_1,r_2)}_{\overline{\mathbf {C}}}\in \mathbf {Op}_{\mathbf {C}_0}(\mathbf {M})\) from Definition 3.11 depends on the choice of two kinds of data: (1) An orthogonal category \(\overline{\mathbf {C}} = (\mathbf {C},\perp )\) and (2) a bipointed singlecolored operad \(\mathcal {P}^{(r_1,r_2)}= (\mathcal {P}, r_1,r_2 : I[2] \rightrightarrows U(\mathcal {P}) )\). We will see that both of these dependencies are functorial. Recall from Definition 3.1 that orthogonal categories are the objects of the category \(\mathbf {OrthCat}\). The second kind of data may be arranged in terms of a category as follows.
Definition 3.14
Proposition 3.15
The assignment \((\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}) \longmapsto (\mathbf {C}_0, \mathcal {P}^{(r_1,r_2)}_{\overline{\mathbf {C}}})\) of the field theory operads from Definition 3.11 naturally extends to a functor \(\mathbf {OrthCat}\times \mathbf {Op}_{\{*\}}^{2\text {pt}}(\mathbf {M}) \rightarrow \mathbf {Op}(\mathbf {M})\) with values in the category of operads with varying colors (see Definition 2.10).
Proof
4 Universal constructions for field theories
 1.Adjunctions induced by orthogonal functors \(F : \overline{\mathbf {C}} \rightarrow \overline{\mathbf {D}}\)(4.1)
 2.Adjunctions induced by \(\mathbf {Op}_{\{*\}}^{2\text {pt}}(\mathbf {M})\)morphisms \(\phi : \mathcal {P}^{(r_1,r_2)}\rightarrow \mathcal {Q}^{(s_1,s_2)}\)(4.2)
 3.The interplay between these two cases via the diagram of categories and functors(4.3)
4.1 Full orthogonal subcategories
Recall from [8] that a full orthogonal subcategory of an orthogonal category \(\overline{\mathbf {D}} = (\mathbf {D},\perp _\mathbf {D})\) is a full subcategory \(\mathbf {C}\subseteq \mathbf {D}\) that is endowed with the pullback orthogonality relation, i.e. \(f_1\perp _{\mathbf {C}} f_2\) if and only if \(f_1\perp _\mathbf {D}f_2\). The embedding functor \(j : \mathbf {C}\rightarrow \mathbf {D}\) defines an orthogonal functor \(j : \overline{\mathbf {C}} \rightarrow \overline{\mathbf {D}}\).
Proposition 4.1
Proof
The proof is analogous to the corresponding one in [8] and will not be repeated. \(\square \)
Example 4.2
A nontrivial application of a similar universal extension functor for quantum field theories on spacetimes with boundaries has been studied in [4]. It has been shown that the ideals of the universal extension \(j_!(\mathfrak {B})\) of a theory \(\mathfrak {B}\) that is defined only on the interior of a spacetime with boundary are related to boundary conditions.
Remark 4.3
The result in Proposition 4.1 that \(j_!\) exhibits \(\mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )\) as a full coreflective subcategory of \(\mathbf {FT}\big (\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)}\big )\) is crucial for a proper interpretation of \(j_!\) as a universal extension functor and \(j^*\) as a restriction functor in the spirit of Example 4.2. Given any field theory \(\mathfrak {B}\in \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )\) on the full orthogonal subcategory \(\overline{\mathbf {C}}\subseteq \overline{\mathbf {D}}\), one may apply the left and then the right adjoint functor in (4.4) to obtain another field theory \(j^*j_!(\mathfrak {B})\in \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )\) on \(\overline{\mathbf {C}}\subseteq \overline{\mathbf {D}}\). The latter is interpreted as the restriction of the universal extension of \(\mathfrak {B}\). By Proposition 4.1, the unit \(\eta _\mathfrak {B}^{} :\mathfrak {B}\rightarrow j^*j_!(\mathfrak {B})\) defines an isomorphism between these two theories, which means that \(j_!\) extends field theories from \(\overline{\mathbf {C}}\subseteq \overline{\mathbf {D}}\) to all of \(\overline{\mathbf {D}}\) without altering their values on the subcategory \(\overline{\mathbf {C}}\).
With this observation in mind, we would like to comment on existing criticisms that universal constructions, such as Fredenhagen’s universal algebra or our universal extension \(j_!\), may fail to provide a nontrivial result, see e.g. [32] and also [31]. (We would like to thank the anonymous referee for bringing this to our attention.) It is indeed true that the algebra \(j_!(\mathfrak {B})(d)\in \mathbf {Alg}_\mathcal {P}\) associated with an object \(d\in \mathbf {D}\) that is not in the subcategory \(\mathbf {C}\subseteq \mathbf {D}\), i.e. \(d\not \in \mathbf {C}\), might be trivial. However, for every nontrivial \(\mathfrak {B}\in \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )\), the universally extended field theory \(j_!(\mathfrak {B})\in \mathbf {FT}\big (\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)}\big )\) as a whole is nontrivial because, as we explained above, its restriction \(j^*j_!(\mathfrak {B})\) to \(\overline{\mathbf {C}}\) is isomorphic to the input \(\mathfrak {B}\) of the construction. We expect that one can construct physical examples of such theories that are nontrivial on simple spacetimes in \(\overline{\mathbf {C}}\), but might be trivial on certain complicated spacetimes in \(\overline{\mathbf {D}}\), by considering field equations that admit only local solutions.
An interesting application of the class of adjunctions in (4.4) is that they allow us to formalize a kind of localtoglobal (i.e. descent) condition for field theories. Given a field theory \(\mathfrak {A}\in \mathbf {FT}\big (\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)}\big )\) on the bigger category \(\overline{\mathbf {D}}\), one may ask whether it is already completely determined by its values on the full orthogonal subcategory \(\overline{\mathbf {C}}\subseteq \overline{\mathbf {D}}\). In the context of Example 4.2, this means asking if the value of a field theory on a general spacetime \(M\in \overline{\mathbf {Loc}}\) is completely determined by its behavior on spacetimes diffeomorphic to \(\mathbb {R}^m\), which is a typical question of descent. The following definition provides a formalization of this idea.
Definition 4.4
A field theory \(\mathfrak {A}\in \mathbf {FT}\big (\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)}\big )\) on \(\overline{\mathbf {D}}\) is called jlocal if the corresponding component of the counit \(\epsilon _\mathfrak {A}^{} : j_! \,j^*(\mathfrak {A})\rightarrow \mathfrak {A}\) is an isomorphism. The full subcategory of jlocal field theories is denoted by \(\mathbf {FT}\big (\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)}\big )^{j\text {{}loc}}\subseteq \,\mathbf {FT}\big (\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)}\big )\).
The following result, which extends earlier results from [8] to our more general framework, shows that jlocal field theories on the bigger category \(\overline{\mathbf {D}}\) may be equivalently described by field theories on the full orthogonal subcategory \(\overline{\mathbf {C}}\subseteq \overline{\mathbf {D}}\).
Corollary 4.5
Proof
This is an immediate consequence of Proposition 4.1. \(\square \)
Example 4.6
Being a powerful localtoglobal property, it is in general not easy to prove that a given field theory \(\mathfrak {A}\in \mathbf {FT}\big (\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)}\big )\) on \(\overline{\mathbf {D}}\) is jlocal for some full orthogonal subcategory embedding \(j:\overline{\mathbf {C}}\rightarrow \overline{\mathbf {D}}\). Positive results are known for the usual Klein–Gordon quantum field theory and Open image in new window , see [31] and [8, Section 5]. We expect that this proof can be adapted to cover all vector bundle valued linear quantum field theories in the sense of [1, 2, 3].
4.2 Orthogonal localizations
Recall from [8] that the orthogonal localization of an orthogonal category \(\overline{\mathbf {C}}\) at a subset \(W \subseteq \text {Mor}\,\mathbf {C}\) of the set of morphisms is given by the localized category \(\mathbf {C}[W^{1}]\) endowed with the pushforward orthogonality relation \(\perp _{\mathbf {C}[W^{1}]} \,:= L_*(\perp _\mathbf {C})\) along the localization functor \(L : \mathbf {C}\rightarrow \mathbf {C}[W^{1}]\), i.e. \(\perp _{\mathbf {C}[W^{1}]}\) is the smallest orthogonality relation such that \(L(f_1)\perp _{\mathbf {C}[W^{1}]} L(f_2)\) for all \(f_1\perp _\mathbf {C}f_2\). The localization functor defines an orthogonal functor \(L : \overline{\mathbf {C}}\rightarrow \overline{\mathbf {C}[W^{1}]}\).
Proposition 4.7
Proof
The proof is analogous to the corresponding one in [8] and will not be repeated. \(\square \)
Let us now explain in some detail the relationship between the adjunction (4.7) and a suitable generalization of the ‘timeslice axiom’ that we shall call Wconstancy.
Definition 4.8
A field theory \(\mathfrak {A}\in \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )\) is called Wconstant if the \(\mathbf {Alg}_\mathcal {P}\)morphism \(\mathfrak {A}(f) : \mathfrak {A}(c)\rightarrow \mathfrak {A}(c^\prime )\) is an isomorphism, for all \((f:c\rightarrow c^\prime )\in W\). The full subcategory of Wconstant field theories is denoted by \(\mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )^{W\text {{}const}} \subseteq \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )\).
Proposition 4.9
Proof
By Proposition 4.7, we already know that the right adjoint functor \(L^*\) is fully faithful, and hence it remains to prove that its essential image is \(\mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )^{W\text {{}const}} \). The image of \(L^*\) lies in \(\mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )^{W\text {{}const}} \) because, for every \(\mathfrak {B}\in \mathbf {FT}\big (\overline{\mathbf {C}[W^{1}]},\mathcal {P}^{(r_1,r_2)}\big )\), the field theory \(L^*(\mathfrak {B})\) is Wconstant since \(L^*= ()\circ L\) is given by restricting the pullback functor for functor categories and the localization functor \(L:\mathbf {C}\rightarrow \mathbf {C}[W^{1}]\) maps morphisms in W to isomorphisms. To prove essential surjectivity, let \(\mathfrak {A}\in \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )^{W\text {{}const}} \) and consider its underlying functor \(\mathfrak {A}:\mathbf {C}\rightarrow \mathbf {Alg}_\mathcal {P}\). By definition of localization, there exists a functor \(\mathfrak {B}: \mathbf {C}[W^{1}]\rightarrow \mathbf {Alg}_\mathcal {P}\) and a natural isomorphism \(\mathfrak {A}\cong L^*(\mathfrak {B})\). Using that the orthogonality relation \(\perp _{\mathbf {C}[W^{1}]}\) is generated by \(L(f_1)\perp _{\mathbf {C}[W^{1}]} L(f_2)\), for all \(f_1\perp _\mathbf {C}f_2\), one easily checks that \(\mathfrak {B}\in \mathbf {FT}\big (\overline{\mathbf {C}[W^{1}]},\mathcal {P}^{(r_1,r_2)}\big )\). \(\square \)
Example 4.10
The right adjoint \(L^*\) of the adjunction (4.9) can be interpreted as the functor that forgets that a field theory \(\mathfrak {B}\in \mathbf {FT}\big (\overline{\mathbf {Loc}[W^{1}]},\mathcal {P}^{(r_1,r_2)}\big )\) satisfies the timeslice axiom. More interestingly, the left adjoint \(L_!\) assigns to a field theory \(\mathfrak {A}\in \mathbf {FT}\big (\overline{\mathbf {Loc}},\mathcal {P}^{(r_1,r_2)}\big )\) that does not necessarily satisfy the timeslice axiom a theory that does. Hence, one may call the left adjoint functor \(L_!\) a ‘timeslicification’ functor. Notice that the result in Proposition 4.7 that \(L^*\) exhibits \( \mathbf {FT}\big (\overline{\mathbf {Loc}[W^{1}]},\mathcal {P}^{(r_1,r_2)}\big )\) as a full reflective subcategory of \(\mathbf {FT}\big (\overline{\mathbf {Loc}},\mathcal {P}^{(r_1,r_2)}\big )\) has a concrete meaning. The isomorphisms \(\epsilon _{\mathfrak {B}}^{} : L_!\, L^*(\mathfrak {B})\rightarrow \mathfrak {B}\) given by the counit say that timeslicification does not alter those field theories that already do satisfy the timeslice axiom, which is of course a very reasonable property.
4.3 Change of bipointed singlecolored operad
We observe the following preservation results for jlocal field theories (see Definition 4.4) and for Wconstant field theories (see Definition 4.8) under the adjunctions (4.12).
Proposition 4.11
 (a)
The left adjoint functor \((\phi ^*)^! : \mathbf {FT}\big (\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)}\big ) \rightarrow \mathbf {FT}\big (\overline{\mathbf {D}},\mathcal {Q}^{(s_1,s_2)}\big )\) preserves jlocal field theories.
 (b)
The right adjoint functor \((\phi ^*)_*: \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {Q}^{(s_1,s_2)}\big ) \rightarrow \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )\) preserves Wconstant field theories.
Proof
Let us emphasize that the result in Proposition 4.11 is asymmetric in the sense that jlocal field theories are preserved by the left adjoints \((\phi ^*)^!\) and Wconstant field theories are preserved by the right adjoints \((\phi ^*)_*\). The opposite preservation properties do not hold true in general; however, we would like to note the following special case in which there exists a further preservation result. This will become relevant in Sect. 5 below.
Proposition 4.12
Proof
This is immediate because by hypothesis there is a natural isomorphism \((\phi ^*)^! \cong \phi _!\circ ()\) and every functor \(\phi _!\) preserves isomorphisms. \(\square \)
Lemma 4.13
If the functor \((\phi _!)_*\,\pi ^*: \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big ) \rightarrow {\mathbf {Alg}_\mathcal {Q}}^\mathbf {C}\) factors through the full reflective subcategory \(\mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {Q}^{(s_1,s_2)}\big ) \subseteq {\mathbf {Alg}_\mathcal {Q}}^\mathbf {C}\), then the left adjoint \((\phi ^*)^! : \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big ) \rightarrow \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {Q}^{(s_1,s_2)}\big )\) is (naturally isomorphic to) the restriction to the categories of field theories of the pushforward functor \((\phi _!)_*: {\mathbf {Alg}_\mathcal {P}}^\mathbf {C}\rightarrow {\mathbf {Alg}_\mathcal {Q}}^\mathbf {C}\).
5 Linear quantization adjunction
Let us first provide an explicit description of the right adjoint functor \(\mathfrak {U}_\mathsf {lin}^{~}= (\phi ^*)_*\). Note that the functor \(\phi ^*: \mathbf {Alg}_{\mathsf {As}} \rightarrow \mathbf {Alg}_{\mathsf {uLie}}\) from associative and unital algebras to unital Lie algebras is very explicit. It assigns to any \((A,\mu _A,\eta _A)\in \mathbf {Alg}_{\mathsf {As}}\) the unital Lie algebra \(\phi ^*(A,\mu _A,\eta _A) = (A, \mu _A\mu _A^\text {op},\eta _A)\in \mathbf {Alg}_{\mathsf {uLie}}\), where the Lie bracket is given by the commutator. The corresponding pushforward functor \(\mathfrak {U}_\mathsf {lin}^{~}= (\phi ^*)_*: \mathbf {QFT}(\overline{\mathbf {C}}) \rightarrow \mathbf {LFT}(\overline{\mathbf {C}})\) carries out this construction objectwise on \(\mathbf {C}\). Concretely, for \(\big (\mathfrak {A}: \mathbf {C}\rightarrow \mathbf {Alg}_{\mathsf {As}}\big ) \in \mathbf {QFT}(\overline{\mathbf {C}})\), the functor underlying \(\mathfrak {U}_\mathsf {lin}^{~}(\mathfrak {A})\in \mathbf {LFT}(\overline{\mathbf {C}})\) is given by \(\mathfrak {U}_\mathsf {lin}^{~}(\mathfrak {A})(c) = \phi ^*\big (\mathfrak {A}(c)\big ) \in \mathbf {Alg}_{\mathsf {uLie}}\), for all \(c\in \mathbf {C}\).
We now provide an explicit description of the left adjoint functor \(\mathfrak {Q}_\mathsf {lin}^{~}\) in (5.4). Our strategy is to analyze the pushforward functor \((\phi _!)_*: {\mathbf {Alg}_{\mathsf {uLie}}}^\mathbf {C}\rightarrow {\mathbf {Alg}_{\mathsf {As}}}^\mathbf {C}\) for the functor categories and to prove that it satisfies the criterion of Lemma 4.13. As a consequence of this lemma, the restriction to the categories of field theories of the pushforward functor \((\phi _!)_*\) defines a model for the left adjoint functor \(\mathfrak {Q}_\mathsf {lin}^{~}\).
Lemma 5.1
The functor \(\phi _! : \mathbf {Alg}_{\mathsf {uLie}}\rightarrow \mathbf {Alg}_{\mathsf {As}}\) described above is left adjoint to the functor \(\phi ^*: \mathbf {Alg}_{\mathsf {As}} \rightarrow \mathbf {Alg}_{\mathsf {uLie}}\).
Proof
It is easy to construct a natural bijection \(\text {Hom}_{\mathbf {Alg}_{\mathsf {As}}}(\phi _!(V),A)\cong \text {Hom}_{\mathbf {Alg}_{\mathsf {uLie}}}(V,\phi ^*(A))\), for all \(V\in \mathbf {Alg}_{\mathsf {uLie}}\) and \(A\in \mathbf {Alg}_{\mathsf {As}}\). Concretely, given \(\kappa : \phi _!(V)\rightarrow A\) in \(\mathbf {Alg}_{\mathsf {As}}\), then \(\kappa \, \pi ^\prime \,\pi \,\iota _1 : V\rightarrow \phi ^*(A)\) defines an \(\mathbf {Alg}_{\mathsf {uLie}}\)morphism. On the other hand, given \(\rho : V\rightarrow \phi ^*(A)\) in \(\mathbf {Alg}_{\mathsf {uLie}}\), then the canonical extension to an \(\mathbf {Alg}_{\mathsf {As}}\)morphism \(\rho : T^\otimes V\rightarrow A\) on the tensor algebra descends to the quotients in (5.5) and (5.6). \(\square \)
Proposition 5.2
For every linear field theory \((\mathfrak {B}: \mathbf {C}\rightarrow \mathbf {Alg}_{\mathsf {uLie}})\in \mathbf {LFT}(\overline{\mathbf {C}})\), the functor \((\phi _!)_*(\mathfrak {B}) : \mathbf {C}\rightarrow \mathbf {Alg}_{\mathsf {As}}\) defines a quantum field theory, i.e. \((\phi _!)_*(\mathfrak {B})\in \mathbf {QFT}(\overline{\mathbf {C}})\).
Proof
By hypothesis, given any orthogonal pair \((f_1 :c_1\rightarrow c)\perp (f_2:c_2\rightarrow c)\) in \(\overline{\mathbf {C}}\), the induced Lie bracket \( [ \mathfrak {B}(f_1)(), \mathfrak {B}(f_2)() ]_c : \mathfrak {B}(c_1)\otimes \mathfrak {B}(c_2) \rightarrow \mathfrak {B}(c)\) is the zero map. We have to prove that the induced commutator \([ \phi _!\,\mathfrak {B}(f_1)(), \phi _!\,\mathfrak {B}(f_2)() ]_c : \phi _!\,\mathfrak {B}(c_1)\otimes \phi _!\, \mathfrak {B}(c_2) \rightarrow \phi _!\, \mathfrak {B}(c)\) associated to the functor \((\phi _!)_*(\mathfrak {B}) = \phi _!\, \mathfrak {B}: \mathbf {C}\rightarrow \mathbf {Alg}_{\mathsf {As}}\) is the zero map too. This is an immediate consequence of our definition of the unital universal enveloping algebra [see (5.5) and (5.6)] and the fact that the commutator bracket satisfies the Leibniz rule in both entries. The latter property is used to expand the commutator of polynomials to a sum of terms containing as a factor the commutator of generators, which is identified via (5.5) with the Lie bracket. \(\square \)
As a consequence of Lemma 4.13, we obtain
Corollary 5.3
The restriction to the categories of field theories of \((\phi _!)_*: {\mathbf {Alg}_{\mathsf {uLie}}}^\mathbf {C}\rightarrow {\mathbf {Alg}_{\mathsf {As}}}^{\mathbf {C}}\) is a model for the left adjoint functor \(\mathfrak {Q}_\mathsf {lin}^{~}: \mathbf {LFT}(\overline{\mathbf {C}})\rightarrow \mathbf {QFT}(\overline{\mathbf {C}})\) in (5.4).
Remark 5.4
Let us explain why \(\mathfrak {Q}_\mathsf {lin}^{~}: \mathbf {LFT}(\overline{\mathbf {C}})\rightarrow \mathbf {QFT}(\overline{\mathbf {C}})\) deserves the name quantization functor. Suppose that \(\mathfrak {B}= H\, \mathfrak {L}\in \mathbf {LFT}(\overline{\mathbf {C}})\) is the composition of a functor \(\mathfrak {L}: \mathbf {C}\rightarrow \mathbf {PSymp}\) to the category of presymplectic vector spaces with the Heisenberg Lie algebra functor \(H : \mathbf {PSymp}\rightarrow \mathbf {Alg}_{\mathsf {uLie}}\) as described in Example 3.8. It is easy to check that the composition \(\phi _!\, H : \mathbf {PSymp}\rightarrow \mathbf {Alg}_{\mathsf {As}}\) of the Heisenberg Lie algebra functor and the unital universal enveloping algebra functor is naturally isomorphic to the usual (polynomial) CCRalgebra functor \(\mathfrak {CCR} : \mathbf {PSymp}\rightarrow \mathbf {Alg}_{\mathsf {As}}\) that is used in the quantization of linear field theories, see e.g. [1, 2, 3]. In particular, we obtain a natural isomorphism \(\mathfrak {Q}_\mathsf {lin}^{~}\big (H\,\mathfrak {L}\big ) \cong \mathfrak {CCR}~ \mathfrak {L}: \mathbf {C}\rightarrow \mathbf {Alg}_{\mathsf {As}}\), which means that our quantization prescription via \(\mathfrak {Q}_\mathsf {lin}^{~}\) is in this case equivalent to the ordinary CCRalgebra quantization of linear field theories.
We would like to emphasize that our linear quantization functor preserves both jlocality and Wconstancy, i.e. it preserves descent and the timeslice axiom of field theories.
Corollary 5.5
 (a)
Let \(j :\overline{\mathbf {C}}\rightarrow \overline{\mathbf {D}}\) be a full orthogonal subcategory. Then the linear quantization functor \(\mathfrak {Q}_\mathsf {lin}^{~}: \mathbf {LFT}(\overline{\mathbf {D}})\rightarrow \mathbf {QFT}(\overline{\mathbf {D}})\) maps jlocal linear field theories to jlocal quantum field theories, (see Definition 4.4).
 (b)
Let \(\overline{\mathbf {C}}\) be an orthogonal category and \(W\subseteq \text {Mor}\,\mathbf {C}\) a subset. Then the linear quantization functor \(\mathfrak {Q}_\mathsf {lin}^{~}: \mathbf {LFT}(\overline{\mathbf {C}})\rightarrow \mathbf {QFT}(\overline{\mathbf {C}})\) maps Wconstant linear field theories to Wconstant quantum field theories, (see Definition 4.8).
6 Toward the quantization of linear gauge theories
The techniques we developed in this paper can be refined to the case where \(\mathbf {M}\) is a suitable symmetric monoidal model category. Let us recall that a model category is a category that comes equipped with three distinguished classes of morphisms—called weak equivalences, fibrations, and cofibrations—that satisfy a list of axioms going back to Quillen; see, e.g. [17] for a concise introduction. The main role is played by the weak equivalences, which introduce a consistent concept of “two things being the same” that is weaker than the usual concept of categorical isomorphism. For example, the category \(\mathbf {M}= \mathbf {Ch}(\mathbb {K})\) of (possibly unbounded) chain complexes of vector spaces over a field \(\mathbb {K}\) may be endowed with a symmetric monoidal model category structure in which the weak equivalences are quasiisomorphisms; see e.g. [30].
Model category theory plays an important role in the mathematical formulation of (quantum) gauge theories. In particular, the ‘spaces’ of fields in a gauge theory are actually higher spaces called stacks, which may be formalized within model category theory. We refer to [33] for the general framework and also to [7] for the example of Yang–Mills theory. Consequently, the observable ‘algebras’ in a quantum gauge theory are actually higher algebras, e.g. the differential graded algebras arising in the BRST/BV formalism. We refer to [22, 23, 29] for concrete constructions within the BRST/BV formalism in algebraic quantum field theory, to [9] for the relevant model categorical perspective and to [26] for a related deformation theoretic point of view.
The aim of this last section is to refine our results for the linear quantization adjunction from Sect. 5 to the framework of model category theory. This will provide a mathematically solid setup to quantize linear gauge theories to quantum gauge theories in a way that is consistent with the concept of weak equivalences. As an explicit example, we discuss the quantization of linear Chern–Simons theory on oriented surfaces. In order to simplify our analysis, we restrict ourselves to the case where \(\mathbf {M}=\mathbf {Ch}(\mathbb {K})\) is the symmetric monoidal model category of chain complexes of vector spaces over a field \(\mathbb {K}\) of characteristic zero, e.g. \(\mathbb {K}=\mathbb {C}\) or \(\mathbb {K}=\mathbb {R}\). In this section, we shall freely use terminology and results from general model category theory [17, 30] and more specifically the model structures for colored operads and their algebras [27, 28]. We refer to [6, 9] for a more gentle presentation of how these techniques can be applied to \(\mathbf {Ch}(\mathbb {K})\)valued algebraic quantum field theory.
6.1 Model structures on field theory categories
Our first (immediate) result is that the categories \(\mathbf {FT}(\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)})\) of field theories with values in \(\mathbf {M}= \mathbf {Ch}(\mathbb {K})\) from Definition 3.3 are model categories, i.e. there exists a consistent concept of weak equivalences for \(\mathbf {Ch}(\mathbb {K})\)valued field theories. Furthermore, the adjunctions in (3.25) are compatible with these model category structures in the sense that they are Quillen adjunctions.
Proposition 6.1
 (i)
a weak equivalence if the underlying \(\mathbf {Ch}(\mathbb {K})\)morphism of each component \(\zeta _c : \mathfrak {A}(c)\rightarrow \mathfrak {B}(c)\) is a quasiisomorphism,
 (ii)
a fibration if the underlying \(\mathbf {Ch}(\mathbb {K})\)morphism of each component \(\zeta _c : \mathfrak {A}(c)\rightarrow \mathfrak {B}(c)\) is degreewise surjective, and
 (iii)
a cofibration if it has the leftlifting property with respect to all acyclic fibrations.
Proof
This is a consequence of Theorem 3.12 and Hinich’s results [27, 28], which show that all colored operads in \(\mathbf {Ch}(\mathbb {K})\) are admissible for \(\mathbb {K}\) a field of characteristic zero. \(\square \)
Proposition 6.2
Let \(F : \overline{\mathbf {C}}\rightarrow \overline{\mathbf {D}}\) be any orthogonal functor and \(\phi : \mathcal {P}^{(r_1,r_2)}\rightarrow \mathcal {Q}^{(s_1,s_2)}\) any \(\mathbf {Op}_{\{*\}}^{2\text {pt}}(\mathbf {Ch}(\mathbb {K}))\)morphism. Then the adjunction in (3.25) is a Quillen adjunction with respect to the model structures from Proposition 6.1.
As a specific instance of the general result of Proposition 6.1, we obtain that both the category of \(\mathbf {Ch}(\mathbb {K})\)valued linear field theories \(\mathbf {LFT}(\overline{\mathbf {C}})\) and the category of \(\mathbf {Ch}(\mathbb {K})\)valued quantum field theories \(\mathbf {QFT}(\overline{\mathbf {C}})\) carry a canonical model structure. In order to develop a better intuition for \(\mathbf {Ch}(\mathbb {K})\)valued field theories and their relation to gauge theories, let us introduce a simple example of a \(\mathbf {Ch}(\mathbb {K})\)valued linear field theory.
Example 6.3
Precisely as in Example 3.8, we can assign to the presymplectic chain complex \((\mathfrak {L}(M),\omega )\) its Heisenberg Lie algebra, which we shall denote by \(\mathfrak {B}^{}_{\text {CS}}(M):= \mathfrak {L}(M)\oplus \mathbb {K}\in \mathbf {Alg}_{\mathsf {uLie}}\). Using pushforwards of compactly supported forms along \(\mathbf {Man}_2\)morphisms \(f:M\rightarrow N\) and observing that (6.4) are the components of a natural transformation, we can promote the assignment \(M\mapsto \mathfrak {B}^{}_{\text {CS}}(M)\) to a functor \(\mathfrak {B}^{}_{\text {CS}} : \mathbf {Man}_2 \rightarrow \mathbf {Alg}_{\mathsf {uLie}}\). Because the integration of any product of forms with disjoint support yields zero, this functor defines a \(\mathbf {Ch}(\mathbb {K})\)valued linear field theory \(\mathfrak {B}^{}_{\text {CS}}\in \mathbf {LFT}(\overline{\mathbf {Man}_2})\) on the orthogonal category \(\overline{\mathbf {Man}_2}\). By construction, this linear field theory describes linear Chern–Simons theory on oriented surfaces.
6.2 Homotopical properties of linear quantization
For practical applications, it is crucial to find simple models for derived functors that can be computed explicitly. The goal of this subsection is to obtain such simple models for the derived functors of the linear quantization adjunction (5.4). For the right derived functor \(\mathbb {R}\mathfrak {U}_\mathsf {lin}^{~}\), this problem is easy to solve because every object in the model category \(\mathbf {QFT}(\overline{\mathbf {C}})\) is fibrant, hence the identity functor \(R=\text {id}\) defines a fibrant replacement functor. This immediately implies
Proposition 6.4
The underived functor \(\mathfrak {U}_\mathsf {lin}^{~}: \mathbf {QFT}(\overline{\mathbf {C}})\rightarrow \mathbf {LFT}(\overline{\mathbf {C}})\) is a model for the right derived functor \(\mathbb {R}\mathfrak {U}_\mathsf {lin}^{~}\) in (6.5).
For the left derived functor \(\mathbb {L}\mathfrak {Q}_\mathsf {lin}^{~}\), i.e. the derived linear quantization functor, the situation gets more complicated because not every object in \(\mathbf {LFT}(\overline{\mathbf {C}})\) is cofibrant. However, a more detailed study of \(\mathfrak {Q}_\mathsf {lin}^{~}\) reveals the following pleasing result.
Proposition 6.5
The underived functor \(\mathfrak {Q}_\mathsf {lin}^{~}: \mathbf {LFT}(\overline{\mathbf {C}}) \rightarrow \mathbf {QFT}(\overline{\mathbf {C}})\) preserves weak equivalences. As a consequence, it is a model for the left derived functor \(\mathbb {L}\mathfrak {Q}_\mathsf {lin}^{~}\) in (6.6).
Proof
Recall from Corollary 5.3 that \(\mathfrak {Q}_\mathsf {lin}^{~}= (\phi _!)_*= \phi _!\circ ()\) is given by postcomposing with the left adjoint functor \(\phi _! : \mathbf {Alg}_{\mathsf {uLie}}\rightarrow \mathbf {Alg}_{\mathsf {As}}\). It is shown in Lemma A.1 that \(\phi _!\) preserves weak equivalences, hence \(\mathfrak {Q}_\mathsf {lin}^{~}\) preserves weak equivalences as these are defined componentwise (see Proposition 6.1).
Example 6.6

\(\mathbb {K}\)linearity: \(\widehat{C}(k\,\chi + k^\prime \,\chi ^\prime ) = k\,\widehat{C}(\chi ) + k^\prime \,\widehat{C}(\chi ^\prime )\), for all \(\chi ,\chi ^\prime \in \Omega ^2_\text {c}(M)\) and \(k,k^\prime \in \mathbb {K}\), and similarly for \(\widehat{A}\) and \(\widehat{B}\);
 Commutation relations: The nonvanishing graded commutators are$$\begin{aligned} \big [\widehat{A}(\alpha ),\widehat{A}(\alpha ^\prime )\big ]\,&=\,\omega (\alpha ,\alpha ^\prime ) \,=\,\int _M \alpha \wedge \alpha ^\prime ,\end{aligned}$$(6.9a)$$\begin{aligned} \big [\widehat{C}(\chi ),\widehat{B}(\beta )\big ] \,&=\, \omega (\chi ,\beta ) \,=\,  \int _M\chi \wedge \beta ,\end{aligned}$$(6.9b)$$\begin{aligned} \big [\widehat{B}(\beta ),\widehat{C}(\chi )\big ] \,&=\, \omega (\beta ,\chi ) \,=\,  \int _M\beta \wedge \chi . \end{aligned}$$(6.9c)
6.3 Homotopy jlocality and homotopy Wconstancy
We would like to conclude by introducing natural homotopical generalizations of the jlocality property (see Definition 4.4) and the Wconstancy property (see Definition 4.8) in the context of model category theory. It will be shown that these properties are preserved by linear quantization.
Homotopy jlocality: Let \(j : \overline{\mathbf {C}}\rightarrow \overline{\mathbf {D}}\) be a full orthogonal subcategory and \(\mathcal {P}^{(r_1,r_2)}\) a bipointed singlecolored operad. From Proposition 6.2, we obtain a Quillen adjunction \(j_! : \mathbf {FT}(\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}) \rightleftarrows \mathbf {FT}(\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)}) :j^*\). For the right derived functor, we can choose again the underived functor \(\mathbb {R}j^*:= j^*\), because every object in \(\mathbf {FT}(\overline{\mathbf {D}},\mathcal {P}^{(r_1,r_2)})\) is fibrant. However, in contrast to the linear quantization functor from the previous subsection, the left adjoint functor \(j_!\) in general does not preserve weak equivalences and hence it has to be derived \(\mathbb {L}j_!:= j_!\,Q\). (See [9, Appendix A] for concrete examples illustrating this fact.) As a consequence, our previous concept of jlocality from Definition 4.4 has to be derived as well in order to be homotopically meaningful. In what follows, we denote by \(q : Q\rightarrow \text {id}\) the natural weak equivalence corresponding to our choice of cofibrant replacement functor Q.
Definition 6.7
Proposition 6.8
The linear quantization functor \(\mathfrak {Q}_\mathsf {lin}^{~}: \mathbf {LFT}(\overline{\mathbf {D}}) \rightarrow \mathbf {QFT}(\overline{\mathbf {D}})\) (see Proposition 6.5) maps homotopy jlocal linear field theories to homotopy jlocal quantum field theories.
Proof
Let \(\mathfrak {B}\in \mathbf {LFT}(\overline{\mathbf {D}})\) be a homotopy jlocal linear field theory, i.e. \(\widetilde{\epsilon }_{\mathfrak {B}}^{} : j_! \,Q\, j^*(\mathfrak {B})\rightarrow \mathfrak {B}\) is a weak equivalence. We have to prove that the derived counit \(\widetilde{\epsilon }_{\mathfrak {Q}_\mathsf {lin}^{~}(\mathfrak {B})}^{} : j_!\,Q\,j^*\,\mathfrak {Q}_\mathsf {lin}^{~}(\mathfrak {B}) \rightarrow \mathfrak {Q}_\mathsf {lin}^{~}(\mathfrak {B})\) corresponding to the quantum field theory \(\mathfrak {Q}_\mathsf {lin}^{~}(\mathfrak {B})\in \mathbf {QFT}(\overline{\mathbf {D}})\) is a weak equivalence too.
Example 6.9
Let \(j: \overline{\mathbf {Disk}_2}\rightarrow \overline{\mathbf {Man}_2}\) be the full orthogonal subcategory of all oriented 2manifolds M that are diffeomorphic to \(\mathbb {R}^2\). It is an interesting question whether the linear Chern–Simons quantum field theory \(\mathfrak {A}_{\text {CS}}^{}\in \mathbf {QFT}(\overline{\mathbf {Man}_2})\) from Example 6.6 is homotopy jlocal with respect to this j. In particular, homotopy jlocality would imply that its value \(\mathfrak {A}_{\text {CS}}^{}(M)\) on a topologically nontrivial oriented 2manifold M such as the torus is already encoded in the restriction \(j^*(\mathfrak {A}_{\text {CS}}^{})\in \mathbf {QFT}(\overline{\mathbf {Disk}_2})\) of the quantum field theory to disks. Unfortunately, proving homotopy jlocality of a given theory is a complicated task and hence we can not yet provide an answer to the question whether \(\mathfrak {A}_{\text {CS}}^{}\in \mathbf {QFT}(\overline{\mathbf {Man}_2})\) is homotopy jlocal or not. We however would like to mention that positive results are already available for simple toymodels which do not involve quantization, see [9] for details. We expect that Proposition 6.8 will be very useful for investigating homotopy jlocality of \(\mathfrak {A}_{\text {CS}}^{}=\mathfrak {Q}_\mathsf {lin}^{~}(\mathfrak {B}_{\text {CS}}^{})\in \mathbf {QFT}(\overline{\mathbf {Man}_2})\) because it allows us to replace this question by the (slightly) simpler question whether the linear field theory \(\mathfrak {B}_{\text {CS}}^{}\in \mathbf {LFT}(\overline{\mathbf {Man}_2})\) from Example 6.3 is homotopy jlocal. We hope to come back to this issue in a future work.
Homotopy Wconstancy: Let \(\overline{\mathbf {C}}\) be an orthogonal category, \(W\subseteq \text {Mor}\,\mathbf {C}\) a subset and \(\mathcal {P}^{(r_1,r_2)}\) a bipointed singlecolored operad. Similarly to locally constant factorization algebras [15], we propose a homotopical generalization of the Wconstancy property from Definition 4.8.
Definition 6.10
A field theory \(\mathfrak {A}\in \mathbf {FT}\big (\overline{\mathbf {C}},\mathcal {P}^{(r_1,r_2)}\big )\) is called homotopyWconstant if the \(\mathbf {Ch}(\mathbb {K})\)morphism underlying the \(\mathbf {Alg}_\mathcal {P}\)morphism \(\mathfrak {A}(f) : \mathfrak {A}(c)\rightarrow \mathfrak {A}(c^\prime )\) is a quasiisomorphism for all \((f:c\rightarrow c^\prime )\in W\).
Proposition 6.11
The linear quantization functor \(\mathfrak {Q}_\mathsf {lin}^{~}: \mathbf {LFT}(\overline{\mathbf {C}}) \rightarrow \mathbf {QFT}(\overline{\mathbf {C}})\) (see Proposition 6.5) maps homotopy Wconstant linear field theories to homotopy Wconstant quantum field theories.
Proof
Recall from Corollary 5.3 that \(\mathfrak {Q}_\mathsf {lin}^{~}= (\phi _!)_*= \phi _!\circ ()\) is given by postcomposing with the left adjoint functor \(\phi _! : \mathbf {Alg}_{\mathsf {uLie}}\rightarrow \mathbf {Alg}_{\mathsf {As}}\). By Lemma A.1, the latter preserves weak equivalences and hence it preserves the homotopy Wconstancy property. \(\square \)
Example 6.12
Notes
Acknowledgements
We would like to thank the anonymous referee for useful comments that helped us to improve this manuscript. We also would like to thank Marco Benini and Lukas Woike for useful discussions. S.B. is supported by a Ph.D. scholarship of the Royal Society (UK). A.S. gratefully acknowledges the financial support of the Royal Society (UK) through a Royal Society University Research Fellowship, a Research Grant and an Enhancement Award (Grant Nos. UF150099, RG160517, RGF\({\backslash }\)EA\({\backslash }\)180270).
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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