A Cohomological Perspective on Algebraic Quantum Field Theory

Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model. This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited. To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory.

At present, the best known description of fundamental physics is given by the Standard Model, which is an interacting quantum field theory (QFT) in 4 dimensions. Unfortunately, a mathematically consistent description of the Standard Model is not yet known. It is a fundamental problem of mathematical physics to construct mathematical models of interacting quantum field theories such as the Standard Model or whatever may supplant it. Algebraic Quantum Field Theory (AQFT) [19] does provide a framework for describing QFT, but thus far, interacting models have only been constructed in dimensions less than 4. More tools are needed for constructing and understanding quantum field theories.
A possible approach to constructing interacting QFTs is by deformation -either deforming a free QFT into an interacting one or deforming an interacting classical field theory into a quantum one. This is analogous to deforming a commutative algebra into a noncommutative one, as is done in formal [2,23] or strict [29] deformation quantization.
Formal deformation quantization is part of the theory of algebraic deformations [15], which is based upon Hochschild cohomology and the algebraic structure of the Hochschild complex. The purpose of this paper is to consider AQFT from the perspective of the relevant generalization of Hochschild cohomology. This is a necessary step toward a theory of deformation quantization of field theories and thus an approach to building interacting QFT models.
This perspective provides a unified framework for three seemingly disparate concepts: the symmetries of a QFT, the transition from classical to quantum field theory, and the transition from free to interacting QFT. It also leads to a more general definition of symmetry and a generalization of AQFT.
1.1. Algebraic quantum field theory. The fundamental difference between quantum field theory and other models of quantum physics is locality. Consistency with relativity means that only some observables can be measured in a given region, O, of spacetime. Observables regarding processes spacelike separated from O cannot be measured in O. This is a manifestation of the principle that no signal can travel faster than light.
Any sum or product of observables that can be measured in O can also be measured there, therefore the set of observables measurable in O is an algebra, A(O). If O 1 ⊂ O 2 , then any observable that can be measured in O 1 can a fortiori be measured This correspondence between regions and algebras completely encodes the structure of a quantum field theory. This is the fundamental idea of Algebraic Quantum Field Theory [19].
In fact, only Einstein causality will be needed in this paper. The category Loc is monoidal under the operation of disjoint union of spacetimes. Einstein causality is almost equivalent to requiring A to be a monoidal functor (see [4]).
If X ⊂ Loc is a small category whose inclusion is an equivalence of categories, then an LCQFT can equivalently be described as a functor A : X → Alg. For example, X could be the subcategory of spacetimes whose underlying manifolds are submanifolds of R 2n+1 . The results of [28] imply that Loc is equivalent to the small subcategory of globally hyperbolic submanifolds of Minkowski spacetime (of sufficiently large dimension).
A QFT on a fixed manifold, M ∈ Obj(Loc), can also be encoded as a functor. If X ⊂ Loc is the subcategory of spacetimes that happen to be open subsets of M, then a QFT on M can be encoded as a functor A : X → Alg. Einstein causality and the time slice axiom are perfectly meaningful conditions on such a functor. Note that this encodes the action of any oriented, time-oriented isometries of M. In particular, if M is Minkowski spacetime, then this functor encodes the action of the Poincaré group.
A similar approach can be taken to conformal field theory. A conformal net is a functor from a category of open intervals in S 1 to von Neumann algebras (satisfying further axioms). See, e.g., [18].
A cruder description of quantum physics on a fixed spacetime, ignoring locality, can also be described in this way. Given M ∈ Obj(Loc), the full subcategory of Loc with the single object M is the group of oriented, time-oriented isometries of M. A functor from this group (as a category) to Alg encodes the algebra of observables on M and the action of this group on that algebra. For most of this paper, I will talk about an arbitrary small category, X. I have in mind any of the examples above. In the later sections, this will be limited to a subcategory X ⊂ Loc for which Einstein causality is a meaningful condition.

Hochschild cohomology.
The continuous functions on a topological space and the smooth functions on a manifold form commutative algebras. Many geometrical constructions can be expressed algebraically in terms of these commutative algebras and extend easily to noncommutative algebras. It is often useful to view a noncommutative algebra as if it comes from a topological space and to apply geometrical ideas and intuition. This is the fundamental idea of Noncommutative Geometry.
For example, let M be a compact, smooth manifold, and X • (M) the space of smooth, antisymmetric multivector fields. This is a Gerstenhaber algebra with both a graded commutative, associative product (the exterior product) and a graded Lie bracket (the Schouten-Nijenhuis bracket). The Hochschild cohomology H • (A, A) of the commutative algebra A = C ∞ (M) is naturally identified with X • (M) as a graded vector space.
Moreover, Gerstenhaber constructed a graded Lie bracket and an associative product on the Hochschild complex C • (A, A) of any algebra, which give the cohomology the structure of a Gerstenhaber algebra (hence the name) and for C ∞ (M) this is the natural structure mentioned in the last paragraph. This means that Hochschild cohomology should be thought of as a noncommutative generalization of the Gerstenhaber algebra of multivector fields. This -and the detailed structure on the complex -play a central role in the theory of formal deformation quantization.
1.3. Algebraic quantum field theory as noncommutative geometry. Definition 1.3. A diagram of algebras is a covariant functor from a small category to Alg.
As described above, an AQFT can be expressed as a functor A : X → Alg, where X ⊂ Loc is a small subcategory, thus an AQFT is a diagram of algebras.
Remark. In [16,17] a diagram is defined as a presheaf (contravariant functor) but an AQFT is covariant, and the difference is just a matter of replacing X with its opposite category.
Let 1 be the category with one object and one morphism. A single algebra is trivially equivalent to a functor 1 → Alg. Thus: • An AQFT is in particular a diagram of algebras.
• A diagram of algebras is a generalization of an algebra.
• An algebra is a generalization of an algebra of functions on a space. In this way, QFT is a generalization of geometry. This is the perspective that I will pursue here.
1.4. Notation and terminology. Vec and Alg will denote the categories of vector spaces and algebras over the field of complex numbers, C. * -Alg will denote the category of * -algebras. , φ * f is defined by the conditions that φ * φ * f = f and Supp(φ * f) ⊆ Im φ. If we define D[φ] := φ * , then D : Loc → Vec is a covariant functor.
Given two points x, y ∈ M ∈ Obj(Loc), denote [1,20]: This is really a locality condition, but note that a linear map is automatically additive.
The bracket notation [ · , · ] will be used for both the Gerstenhaber bracket and the commutator in an associative algebra. I hope that this will be clear in context. 1.5. Outline. In Section 2, I review the definitions of Hochschild cohomology and the Gerstenhaber algebra structure for a single algebra and for a diagram of algebras.
In Section 3, I discuss the relationship of "asimplicial" Hochschild cohomology to the deformations and automorphisms of a diagram of algebras. Seeking a similar interpretation of full Hochschild cohomology leads me to define skew diagrams of algebras and their morphisms. This gives the first main results: a generalization of AQFT and a more general definition of global symmetries of an AQFT. The category of skew diagrams is shown to be a 2-category of functors between 2-categories.
The first main calculation is in Section 4, where I compute the characteristic class in Hochschild cohomology of an interaction term for an AQFT. This involves defining a smoothed-out analogue of Cauchy surfaces.
In Section 5, I discuss the construction of perturbative AQFT by the algebraic adiabatic limit. The next main result is an alternative, more concrete construction; this is motivated by my computation of the characteristic class. This construction leads to the last main calculation -a direct proof that the characteristic class satisfies the appropriate Maurer-Cartan equation.

HOCHSCHILD COHOMOLOGY
The definition of Hochschild cohomology H • (A, A) for an algebra extends to diagrams of algebras 1 . This is referred to as Yoneda cohomology by Gerstenhaber and Schack in [16] because they were working with algebras over a ground ring that was not necessarily a field; that degree of generality is irrelevant here.
Hochschild cohomology of a diagram of algebras is still a Gerstenhaber algebra. This cohomology governs deformations of diagrams, just as it does for a single algebra.
This does not perfectly characterize deformations of LCQFTs, because a LCQFT might be deformed to a diagram of algebras that violates Einstein causality. Nevertheless, this does describe a lot of the relevant structure, and an infinitesimal deformation will have a characteristic class in Hochschild cohomology.
The category of algebras in AQFT is most often taken to be C * -algebras or von Neumann algebras. These are not well suited for studying infinitesimal deformations. To construct multivector fields via Hochschild cohomology, we use not the C *algebra of continuous functions but the dense subalgebra of smooth functions. This suggests that studying infinitesimal deformations of an LCQFT may require identifying analogous dense subalgebras.
The main explicit calculation here will be in the setting of perturbative LCQFT, which does not use C * -algebras.
Let's begin by recalling the definition and properties of Hochschild cohomology for an algebra.
2.1.2. The Gerstenhaber bracket. [14] Now consider the case that B = A.
Definition 2.5. From this, define and the Gerstenhaber bracket This bracket is a graded Lie bracket of degree −1. Equivalently, C • (A, A) with this bracket is a graded Lie algebra with the shifted grading in which Γ ∈ C q (A, A) has degree q − 1. Note that δΓ = [m, Γ ]. From this, it is a simple exercise to deduce that C • (A, A) is a differential graded Lie algebra. The defining property, follows from the Jacobi identity. This implies that the Gerstenhaber bracket induces a well defined graded Lie bracket on the Hochschild cohomology H • (A, A).
There is also an associative product. for a 1 , . . . , a q+q ′ ∈ A.
This is obviously associative but is not commutative. Less obviously, this descends to an associative product on cohomology, where: • The product is commutative.
• The bracket is a derivation of the product (in each argument). A deformation is trivial if A with the deformed product is isomorphic to A with the undeformed product. If α ∈ C 1 (A, A) is such an isomorphism, then the deformed product of a and b is Suppose that there is a 1-parameter family of such isomorphisms, starting from the identity. Differentiating this expression and then setting α = id giveṡ In other words, trivial infinitesimal deformations correspond to exact cocycles. This means that the Hochschild cohomology class of an infinitesimal deformation describes it modulo trivial deformations.
Similarly, if there is a 1-parameter family of automorphisms, starting from the identity, then 0 =ṁ = δα. (This means precisely thatα is a derivation.) So, an infinitesimal automorphism is a 1-cocycle.
If A is unital, an invertible element b ∈ C 0 (A, A) = A determines an inner automorphism, α(a) = b −1 ab. Suppose that b is part of a 1-parameter family, starting from the unit. Differentiating giveṡ α(a) = aḃ −ḃa =⇒α = δḃ , so infinitesimal inner automorphisms are exact 1-cocycles. This means that H 1 (A, A) describes infinitesimal automorphisms modulo inner ones.
Finally, the equation 0 = δb is the condition that b be central, so H 0 (A, A) = Z(A), the center of A.
Note that here there are various structures -algebra elements, automorphisms, multiplication -that are elements of C • (A, A) in various degrees. These satisfy properties that are most naturally expressed as the vanishing of elements of C • (A, A) in other degrees.

2.2.
A diagram of algebras. [26,16,17] Let X be a small category and A : X → Alg a covariant functor, i.e., a diagram of algebras over X. Because I mainly have in mind X ⊂ Loc, I will denote elements of X as M, N, et cetera.
Such a functor consists of 3 types of information: Every object determines a vector space; every object also determines an associative product on that vector space; and every morphism determines a homomorphism of algebras. A vector space cannot be deformed, but the other two structures can. This is in contrast to a single algebra, where there is only one deformable structure.
Let φ : M → N and ψ : N → P be morphisms in X. These two structures satisfy three properties. Associativity means that for every M, the map A(M) ⊗3 → A(M), a ⊗ b ⊗ c → a(bc) − (ab)c vanishes. Being a homomorphism means that for every φ, the map A(M) ⊗2 → A(N), a ⊗ b → A(φ; ab) − A(φ; a)A(φ; b) vanishes. Functoriality means that for every pair of composable morphisms, φ and ψ, the map A(M) → A(P), a → A(ψ; A(φ; a)) − A(ψ • φ; a) vanishes. 2 A symmetry 3 of the functor A is a natural automorphism α : A→ A. This is given by, for Each of these structures and conditions depends upon an element of the nerve, B • X, of the category X. B 0 X = Obj X is the set of objects. B 1 X = Mor X is the set of morphisms. B 2 X is the set of composable pairs of morphisms. In general, B p X is the set of composable p-tuples of morphisms. Each element of B p X begins at an object and ends at an object; for example, M ∈ B 0 X begins and ends at M, but M φ → N ψ → P begins at M and ends at P. Each structure or condition consists of -for every chain of a given length in B • X -a multilinear map from the algebra at the beginning to the algebra at the end. This suggests that the generalization of C • (A, A) is bigraded. One degree is (again) the multilinearity and the other degree is the chain length (the degree in B • X).  Remark. I am writing morphisms as arrows from right to left. This is consistent with the usual convention for writing compositions.
Any chain in B p X can be composed to a single morphism. Applying A to this morphism gives a homomorphism from A(M p ) to A(M 0 ), which makes A(M 0 ) a bimodule of A(M p ).
The nerve, B • X is a simplicial set. In particular, there are face maps ∂ i : B p X → B p−1 X, for 0 ≤ i ≤ p. For φ : M → N, these are the source and target, ∂ 0 (φ) = M and ∂ 1 (φ) = N. For p ≥ 2, ∂ 0 (φ 1 , . . . , φ p ) = (φ 2 , . . . , φ p ) The face maps correspond to injective maps in the simplicial category. Specifically, ∂ i corresponds to the inclusion of {0, . . . , p − 1} into {0, . . . , p} that skips i. Other injective maps can be specified by the numbers that they skip, and the corresponding face maps will be useful. Specifically, There are also degeneracy maps, given by inserting identity morphisms, but these will not be needed.
The simplicial coboundary δ S : C p,q (A, A) → C p+1,q (A, A) is dual to this simplicial structure and is defined by with the coboundaries δ S and δ H is a first quadrant bicomplex. Following Gerstenhaber and Schack [17], also define the asimplicial bicomplex 4 Elements Γ ∈ C p,q (A, A) and ∆ ∈ C p ′ ,q ′ (A, A) can be combined by several binary operations.
The cup product, Note that Γ and ∆ are multiplied in a surprising order. These combine to define Γ The analogue of the • operation of the ordinary Hochschild complex is The cup product and bracket give well defined operations on cohomology, and these make H • (A, A) and H • a (A, A) into Gerstenhaber algebras. However, in contrast to the case of a single algebra, this bracket does not make C • (A, A) into a graded Lie algebra.

Involution. If
A : X → * -Alg, then there is also an antilinear involution on this bicomplex.
This operation does not commute with the coboundaries. Instead, This is sufficient to give a well defined involution on cohomology. This is an involution of the ⌣ product, up to a homotopy given by the • operation. For Γ ∈ C p,q (A, A) and ∆ ∈ C p ′ ,q ′ (A, A),

THE SIGNIFICANCE OF HOCHSCHILD COHOMOLOGY
Hochschild cohomology is mainly concerned with infinitesimal things such as derivations and infinitesimal deformations. These concepts are appropriate to purely algebraic AQFT (such as perturbative AQFT) but are not well suited to C * -algebraic AQFT. However, some of these infinitesimal concepts have finite analogues, which are conceptually clearer and directly applicable to C * -algebraic AQFT. These structures satisfy three conditions, which can be expressed as the vanishing of cochains. Associativity is a condition on m, expressed in C 0,3 (A, A). The compatibility between m and µ is that µ must consist of morphisms between the products given by m; this is expressed in C 1,2 (A, A) and is explicitly is part of a 1parameter family of structures satisfying these conditions. Differentiating these conditions gives 3 conditions onṁ andμ. As in the case of a single algebra, differentiating associativity gives the condition 0 = δ Hṁ . Differentiating eq. (3.1) gives thus 0 = δ Sṁ + δ Hμ . Differentiating the functoriality condition (3.2) gives δ S 1μ = δ S 2μ + δ S 0μ , thus 0 = δ Sμ . Together, these mean thatṁ andμ give a cocycleṁ +μ ∈ Z 2 a (A, A). Differentiating twice gives further conditions. As in the case of a single algebra, the second derivative of associativity is 2ṁ •ṁ = [ṁ,ṁ] = −δ Hm . The second derivative of eq. (3.1) is in other words, [ṁ +μ,ṁ +μ] is exact.
The collection of diagrams of algebras, Alg X is a category in the usual way, meaning that a homomorphism of diagrams of algebras is a natural transformation.
Consider a diagram of algebras, A, and let B be another diagram with the same underlying vector spaces and the other structures denoted by m and µ. Now, imagine that α is part of a 1-parameter family of isomorphisms, starting from the identity. Differentiating eq. (3.3) (and then setting α = id) gives Together, this isṁ +μ = δα. In other words, a trivial deformation is given to first order by an exact cocycle.
A class in H 2 a (A, A) is called a Maurer-Cartan element if the bracket with itself is 0. Deformations of A are classified to first order modulo trivial deformations by the Maurer-Cartan elements.
The symmetries of A are its natural automorphisms, so now consider α a natural automorphism of A. This means that for a, b ∈ A(M), α(M; ab) = α(M; a)α(M; b) , (3.5) and for φ : M → N, Suppose that α is part of a 1-parameter family of natural automorphisms, starting from the identity. Differentiating eq. (3.5) (and setting α = id) gives . The set of natural automorphisms of A is a group. For two natural automorphisms, α and β, their product is simply α The Lie algebra of the group of natural automorphisms is the space H 1 a (A, A), with the •-commutator. This operation is the Gerstenhaber bracket in this degree.
For a diagram of * -algebras, we should also require that α(M; a * ) = α(M; a) * . This just means that α ⋆ = α. The condition on the derivative is the same. The Lie algebra of infinitesimal natural * -automorphisms is thus the ⋆-invariant part of H 1 a (A, A).

Full Hochschild cohomology.
3.2.1. Z 2 . A cocycle in the full Hochschild complex contains more structure. This complex should describe the deformations of a diagram of algebras to something more general.
Let m + µ + u be an example of this (yet undetermined) generalization of a diagram of algebras. Suppose that if this is part of an 1-parameter family of such structures (starting from A with u = 1) then the first derivative satisfies 0 = δ(ṁ + µ +u) and The condition in degree (2, 1) is modified by terms involving u. This means that the functoriality condition (3.2) on µ must be modified by u. The second derivative of this condition should be, 0) is new and only involves u. This suggests that u satisfies some nonlinear cocycle condition indexed by B 3 X. The second derivative of this condition should be 0 = δ Sü + 2u •u, which is . This leads to the following definition.
Remark. The penultimate condition can be stated as commutativity of the diagram, where Ad[u] : a → uau −1 .
This definition extends easily to some other categories of algebras.
Remark. Note that a diagram of algebras is precisely a skew diagram of the form (A, 1).
By construction, if a diagram of algebras, A, is deformed to a 1-parameter family of skew diagrams of algebras, then at first order the deformation determines an element of Z 2 (A, A) whose bracket with itself is exact.

H 2 .
By analogy with H 2 a (A, A), the Maurer-Cartan elements of H 2 (A, A) should classify the deformations of A into skew diagrams up to first order, modulo trivial deformations. A trivial deformation should mean one in which all of the skew diagrams are isomorphic to A. This guides us to define morphisms of skew diagrams.
If A is a diagram of algebras, then a trivial deformation of A to skew diagrams can be constructed using a 1-parameter family of isomorphisms. This is given to first order by an element of C 0,1 (A, A) ⊕ C 1,0 (A, A). The first part is not new, and corresponds to a family of homomorphisms (indexed by the objects of X). The second part is a family of algebra elements indexed by morphisms in X. In the case of * -algebras, a ⋆-invariant element of C 1,0 (A, A) is a family of anti-self-adjoint algebra elements. This corresponds to a family of unitary (or generally, invertible) elements of the unitalized algebras.
We don't yet know what a morphism of skew diagrams is, but suppose that (α, v) : (B, u) → (A, 1) is an isomorphism of skew diagrams, where B has the same underlying vector spaces. The structures m, µ, and u of this (B, u) should be determined by some formulae from α, v, and the structures of A. As before, the product is defined by Suppose that (α, v) is part of a 1-parameter family, starting from α = id and v = 1. To first order, µ is given byμ for φ : M → N and a ∈ A(M). This is the derivative of To first order, u is given byu which defines u.
Remark. The orderings in the products in this formula are not obvious. This is linked with the reversed order of multiplication in the definition of •.
In fact, there are 2 possible conventions. The definition of a skew diagram of algebras can be changed so that the multiplication here and in the definition of • is in the obvious order. The repercussion of this choice is that the definition of the cup product would be more awkward. I am using the convention consistent with Gerstenhaber and Schack [16,17].
which is like the diagram defining naturality, but with the lower right corner modified.
For a 1-parameter family of such automorphisms, starting from (id, 1), the first derivatives of these conditions give precisely thatα +v is closed. So, Z 1 (A, A) is the space of infinitesimal skew automorphisms.
The Gerstenhaber bracket actually satisfies the Jacobi identity on the larger subspace of ξ + υ ∈ C 1 (A, A) such that δ H ξ = 0 (i.e., ξ[M] is a derivation). This Lie algebra has subalgebras indexed by the objects and morphisms of X. and Remark. There are two natural ways of identifying the semidirect product of groups with the Cartesian product as a set. The other choice is not compatible with the fact that v(φ) ∈ 1 + A(N). Let ξ+υ ∈ Z 1 (A, A) and ζ ∈ C 0 (A, A). The Gerstenhaber bracket in these degrees reduces to so the action of (α, v) ∈ SAut(A) on ζ is also just α • 1 ζ, i.e., The condition that 0 = δ S ζ is a sort of invariance.
Example. If X = G is a group (viewed as a category with one object, * ∈ Obj(X)) then A : G → Alg is equivalent to a single algebra A = A( * ) with an action of G by automorphisms of A. The condition that 0 = δ S ζ means that ζ is G-invariant, and 0 = δ H ζ means that ζ is central, so For a diagram of * -algebras, ζ ⋆ = ζ precisely if ζ(M) is self-adjoint. For A : Loc → * -Alg a LCQFT, ζ = ζ ⋆ ∈ H 0 (A, A) is an observable whose values are unaffected by the action of other observables. Moreover, it can be measured in any arbitrarily small region of spacetime.

Higher categorical interpretation.
For the definitions of 2-categories and their functors, transformations, and modifications, see [24,25]. Definition 3.6. Let AlgInn be the strict 2-category whose underlying 1-category is Alg and • for any homomorphisms α, β : A → B, a 2-morphism α u =⇒ β is u ∈ 1 + B such that for any a ∈ A, uα(a) = β(a)u ; • the horizontal composition of α As a category, X is in particular a strict 2-category, with only identity 2-morphisms. Both X and AlgInn are in particular weak 2-categories (bicategories).

Theorem 3.2.
The category of skew diagrams of algebras over X is the category whose objects are pseudofunctors from X to AlgInn (written as oplax functors) and whose morphisms are lax transformations.
These are the components of a natural transformation, but because X has only identity 2-morphisms, the naturality condition is trivial. For any (χ, ψ, φ) ∈ B 3 X, there is a commutative diagram of 2-morphisms, that is, (Here, id denotes the identity 2-morphism over a 1-morphism in AlgInn.) The definition of horizontal composition in AlgInn gives that

The definition of vertical composition in AlgInn simplifies this to
This shows that a pseudofunctor (A, u) : X → AlgInn is a skew diagram, and that a skew diagram is a pseudofunctor. Let (A, u) and (B, u ′ ) be diagrams of algebras over X. As we have just seen, these are pseudofunctors X → AlgInn, written as oplax functors. A lax transformation This is precisely eq. (3.7).
that is, By the definitions of horizontal and vertical composition, this is precisely eq.
Remark. There are a few minor variations possible on these definitions, depending upon what is required to be invertible and lax versus oplax versions.
Applying the definition of AlgInn makes this explicit: In particular, for a given diagram A, there is a 2-group of automorphisms, which can be described by a crossed module involving the structures discussed in Sec- 3.4. Generalized AQFT. Definition 3.1 of a skew diagram of algebras suggests a generalization of algebraic quantum field theory. With X = Loc or the category of causally complete regions of a fixed spacetime, we can simply define a generalized AQFT as a skew diagram of * -algebras over X. Einstein causality, the time-slice axiom, and isotony can be required just as before. This sets AQFT models within a larger class of structures.
Given a skew diagram (A, u) of algebras over X, there exists another category Y, a functor π : Y → X, a diagram B : Y → Alg, and a (non-functorial) section σ of π such that A = B • σ.
If, in some attempt to construct an AQFT, it is only possible to construct a generalized AQFT in this sense, then this is an indication that some additional structure is required beyond the globally hyperbolic spacetimes in Loc.
For example, a model with a spin-1 2 field cannot be formulated as a LCQFT over Loc (see remarks following Cor. 13 in [11]) but can be formulated over the category of globally hyperbolic spin-manifolds. It appears likely that such a model can be formulated as a generalized AQFT with a skew diagram over Loc.
This situation is extremely similar to that considered in [3], and the relationship deserves further investigation. Can a quantum field theory on a category Y fibered in groupoids over Loc be described by a skew diagram over Loc? Can a generalized AQFT be described by a QFT over a fibered category? The answers are almost certainly yes in some cases.

INTERACTION
As I have mentioned, an AQFT (or any diagram of algebras) has two deformable structures: the associative products and the maps between algebras. These are elements of the Hochschild bicomplex in degrees (0, 2) and (1, 1), respectively. This means that there are 2 qualitatively different ways of deforming an AQFT. For example: • The transition from a classical to a quantum field theory deforms the associative algebra structures. • The transition from a free to an interacting field theory deforms the maps between algebras. It may not always be possible to disentangle these 2 aspects of deformation, but it is known that the von Neumann algebra associated to any connected, precompact region of spacetime is isomorphic to the (unique) hyperfinite type III 1 factor [6,13,12]. (This does not apply to classical field theories.) This strongly indicates that deformation of maps is far more important to the deformation of a quantum field theory.
If an AQFT, A, is deformed smoothly by changing the interaction, then this should be described to first order by a class in H 2 a (A, A). This suggests that that class can be given by an element of C 1,1 (A, A).
In practice, computing a cohomology class means computing some cocycle in that class. This requires making an additional choice.
In principle, in order to compute the characteristic class of an interaction, we should (for each M) identify the algebras for a family of field theories with a fixed vector space. It may then be possible to differentiate the product and homomorphisms to get a cocycle. Different choices of identifications should give cohomologous cocycles.
To understand what is needed, it is simplest to first consider classical field theories. A classical algebra of observables is a commutative algebra of functionals on the space of solutions. We need to choose a way of identifying solutions of different field theories on a spacetime M. The simplest way to do this is by initial data. Given a Cauchy surface Σ ⊂ M, solutions of field theories on M can be identified with their initial data on Σ. The characteristic class of an interaction can thus be computed by using an arbitrary choice of a Cauchy surface for every spacetime M ∈ Obj(Loc).
To see how this can work, first consider a classical field theory given by some Lagrangian density, L, on a spacetime M ∈ Obj(Loc). Let Σ 1 , Σ 2 ⊂ M be Cauchy surfaces, and suppose for simplicity that 5 Σ 1 Σ 2 and that these are equal outside of some compact set.
Assume for simplicity that this theory has no gauge degeneracies, so that the phase space is just the set of solutions of the equations of motion for L. Such a solution can be identified with initial data along Σ 1 or Σ 2 . If we change the Lagrangian density to L + λV, then the equations of motion change, and evolution from Σ 1 to Σ 2 defines a map from the phase space to itself. Differentiating with respect to λ and then setting λ = 0 gives a vector field, Ξ, on the phase space. Let This is a functional on the phase space, and Ξ is the Hamiltonian vector field given by H. (This is immediate from Peierls' construction of the Poisson structure [27,19].) If we define (and we no longer need to assume Σ 1 Σ 2 ). This is good enough for classical field theory, but for a quantum field theory, V will need to be a distribution, which is best thought of as a linear map, for any test function χ ∈ D(M) such that χ = θ 1 − θ 2 over J(Supp a). This allows us to drop the assumption that θ 1 − θ 2 has compact support. By construction, Supp(θ 1 − θ 2 ) is future compact and past compact, so Supp(θ 1 − θ 2 ) ∩ J(Supp a) is compact, and there exists χ ∈ D(M) equal to θ 1 − θ 2 on J(Supp a).

Equation (4.2) can then be taken as the definition of Ξ.
This was for a classical field theory. For a quantum field theory, Ξ should be a derivation of the algebra A(M) of quantum observables. In the classical limit, the commutator is approximately proportional to the Poisson bracket. This suggests that the quantum version of (4.2) may be . This is indeed a derivation, and we shall see that it is the right answer.
4.1. The character of an interaction. Let X ⊂ Loc be a small subcategory that is closed under pullbacks (i.e., intersections) and such that the inclusion of any causally complete open subset into M ∈ Obj(X) is a morphism in X. Let A be a functor from X to topological C-algebras, satisfying Einstein causality.
In perturbative AQFT, the algebra A(M) is constructed from functionals, which have clearly defined support in M. However, I am trying to be more general here. Instead of defining the support of an observable, I have the following definitions for the subalgebra of observables supported on a given subset and for the subalgebra of compactly supported observables. Therefore, Let V : D→ A be a linear natural transformation. The idea is to use this as an interaction term for a Lagrangian, L+λV, although this works even if A is not given by a Lagrangian, and it certainly doesn't need to be free. In perturbative AQFT, it is normally only assumed that V is additive, rather than linear; however, we are only interested in the first order effect of this interaction, and only the linear part of V will be relevant.
In order to choose a specific cocycle in the character of the interaction V, we need a choice of smoothed-out Cauchy surface on each spacetime, so following Definition 4.1 define: Because this holds for any such ι, the result follows.
A test function f ∈ D(M) in V M (f) serves as an infrared cutoff of the interaction. It can be thought of as varying the coupling constant over M, and thus λV M (f) should really be V M (λf). The characteristic class Ξ V is supposed to describe the first order effect of the interaction. From this perspective, even if V is nonlinear, then only the first order, linear part of V should be used to construct Ξ V .
I have tried to prove the results in this section as generally as possible, but this has led to constructing Ξ V ∈ H 2 a (A c , A c ) rather than in H 2 a (A, A). In order to fix this, we need to use some more specific category of topological algebras such that A c (M) ⊂ A(M) is dense and Ξ V,θ extends uniquely and continuously. This is true in the setting of perturbative AQFT. Remark. These should really also be * -algebras, but for simplicity, I am ignoring the involution here.

PERTURBATIVE AQFT
Suppose that we have made a natural choice of time-ordered product · T on each A(M); naturality, in this case, means that each A[φ] is a homomorphism under the time-ordered products. Let Exp T denote the · T exponential function. The Naturality of V and · T imply that S is natural.

Definition 5.2. The retarded Møller operator
This is a linear map, but not a homomorphism.
Remark. The formal S-matrix includes negative powers ofh, so it is slightly surprising that R f does not. See [8,21].
Remark. I am not concerned with the details of renormalization here, but these are needed in order to actually construct · T and V.
Remark. The interaction V is not actually used directly to construct the interacting theory. We only need a natural time-ordered product and a natural formal S-matrix satisfying eq. (5.1) and (5.2) Note that, for a ∈ A(M; K), Roughly speaking, the interaction to the future of K is irrelevant and the interaction to the past of K only gives a unitary transformation. If we heuristically imagine that f is {0, 1}-valued, and f = 1 on the causal completion K ′′ of K, then R f (a) uses the interacting theory to evolve a back to the past boundary of Supp f, where it is identified with an observable of the free theory. This is good enough to identify interacting observables supported in K with free observables.
The problem is that, as we consider larger subsets of M, we must adjust f and change the identification with free observables. No single choice of f works for all observables, and this is why the algebraic adiabatic limit is needed. 5.1. Adiabatic limit. The following is a formalization of the algebraic adiabatic limit construction [9] as applied to constructing an interacting LCQFT.
is a commutative diagram.

which implies that the restrictions of A[φ] give a natural transformation
The transformation (5.3) and universality of the direct limit lim This A V is the perturbative AQFT given by modifying A with the interaction λV.

Modified construction.
Definition 5.6. For f, g ∈ D(M) and a ∈ A(M), Remark. If we heuristically imagine that f and f + g are {0, 1}-valued, thenR f,g (a) evolves a forward to the future boundary of Supp(f + g) and then back to the past boundary of Supp f, where it is identified with an observable of the free theory. This gives a uniform way of identifying interacting observables with free observables, becauseR f,g (a) does not change if f is changed in the future and g is changed in the past. and so the right side of (5.6) is independent of the choice of f. If g ′ is another possible choice of g, then Supp θ ∩ J − (K ∪ Supp g ∪ Supp g ′ ) is compact, so a choice of f exists that is compatible with both g and g ′ . Now, Supp(g − g ′ ) K, so and therefore the right side of (5.6) is independent of the choice of g, i.e.,R θ (a) is well defined for a ∈ A(M; K).
For any a, b ∈ A c (M), there exists K ⊂ M compact such that a, b ∈ A(M; K). Any larger compact set determines the sameR θ (a), thereforeR θ is well defined.  (φ; a)). First, note that if R θ M (a) is computed with f, g ∈ D(M) as before, then In this formula, φ * f can safely be replaced with a test function on N, as long as the support of the difference is φ(K ∪ Supp g).

Corollary 5.3. For any
Proof. The calculation is the same as in the previous proof, although slightly simplified. More formally, this result follows if we apply the previous theorem to a doubled version of the category X in which every object of X appears as 2 isomorphic copies. The image of A V (M; K, f) is densely generated byR θ M (A(M; K)), but these subalgebras densely generate A V,θ (M), therefore this homomorphism is surjective. It is injective by construction, therefore it is an isomorphism. This shows that A V,θ is completely equivalent to A V . It has the advantage of being more concrete. Whereas A V (M) is defined abstractly as a limit, 5.3. Maurer-Cartan. The computations in the previous section make it possible to explicitly show that the characteristic class Ξ V of an interaction is a Maurer-Cartan element of H 2 a (A, A), provided that a natural time-ordered product exists. In this section A : X → Alg[[h]] satisfies Einstein causality and has a natural time-ordered product as in the previous section.
If V is nonlinear, then only its linear part is relevant at first order in λ, so in this section I will assume that V : D→ A is a natural linear transformation.
For θ ∈ Θ(M), the mapR θ is a formal power series in λ, so denote this expansion explicitly as • g 1 = 1 − θ M on J + K, by construction.
Finally, consider any φ : M → N and K ⊂ M compact. The sets J + K ∩ Supp(1 − θ M ) and J − K ∩ Supp θ M are compact, because Supp(1 − θ M ) and Supp θ M are (respectively) future-compact and past-compact. Therefore, is compact. Because M is locally compact, every point has a precompact neighborhood. By compactness, K 2 has a finite cover by precompact open sets, therefore K 2 has a precompact open neighborhood, O ⊂ M.
The functionθ M − θ N is defined over J N M, where it has future and past-compact support. Because the closure O is compact, there exists χ ∈ D(N) with compact support in J N M and χ =θ M − θ N over JO.
Finally, because Supp χ and O are compact and Supp θ N is past-compact, there exists f 2 ∈ D(N) such that f 2 = θ N over J − N (O ∪ Supp χ). Remark. I began by assuming the existence of a natural time-ordered product, · T . However, in the construction of Ψ V,θ it is only used on the image of V, not on arbitrary elements of the algebra. This suggests that it may be sufficient to only define · T on this subspace. Proof. Both terms have components in degrees (2, 1) and (1, 2), so this is really two equations.
Suppose that a, b ∈ A(M; K), for some compact K ⊂ M, and let χ be as in Lemma 5.6. The last term of eq. (5.12) is manifestly a derivation, so it doesn't contribute to δ H Ψ V,θ . Note that in any associative algebra, the commutator satisfies In terms of the cohomology class Ξ V ∈ H 2 a (A, A) of Ξ V,θ , this means simply that it satisfies the Maurer-Cartan equation, (5.14) This is the analogue of the Jacobi identity satisfied by a Poisson structure.

CONCLUSIONS
This paper has presented several new ideas and perspectives on constructing models in algebraic quantum field theory. The first is that the algebraic structures and properties of a locally covariant quantum field theory are naturally organized in the Hochschild bicomplex. This makes it possible to consider deformations of a quantum field theory as a generalization of deformation quantization of a Poisson manifold.
This perspective leads to the definition of skew diagrams of algebras (Def. 3.1) as a generalization of functors X → Alg. Because the other axioms of AQFT are still meaningful for a skew diagram, this is a more general framework for building physical models.
An AQFT model is a functor A : X → Alg, and the global symmetries of the model (e.g., the U(1)-symmetry associated to charge conservation) are described as the natural automorphisms of A. This cohomological perspective leads to a more general definition of symmetry as skew automorphisms (Def. 3.3) or outer skew automorphisms (Def. 3.5) of A. This means that some models may have symmetries that were not previously recognized.
From this perspective, an interaction is analogous to a bivector field on a manifold. Just as bivectors can be constructed by multiplying and adding vectors, interactions may be constructed by multiplying and adding infinitesimal skew automorphisms, using the Gerstenhaber algebra structure on Hochschild cohomology. For an interaction of this form, the full interacting model may then be constructed directly by methods similar to those in [7], generalizing the strict deformation quantization construction in [29].
One way of deforming AQFT is to begin with a model defined by a Lagrangian and then perturb it by adding an interaction term, V, to that Lagrangian. I have explicitly constructed (Thm. 4.8) the characteristic Hochschild cohomology class Ξ V of such an interaction term. However, this construction does not require the initial theory to be defined by a Lagrangian. This makes it possible to perturb a non-Lagrangian model with a Lagrangian interaction.
The construction of this characteristic class required introducing a notion (Def. 4.1) of a smoothed-out Cauchy surface as a function that vanishes in the distant past and equals 1 in the distant future. In order to compare this characteristic class with the construction of perturbative AQFT, I introduced an alternative version (Thm. 5.2) of that construction, which uses smoothed-out Cauchy surfaces instead of the algebraic adiabatic limit. This is more concrete than the standard construction, so this is likely to be advantageous for many calculations.
Finally, assuming the existence of a time-ordered product, I showed that the characteristic class of an interaction satisfies the appropriate Maurer-Cartan equation (5.14).
Much remains to be done. In particular, nontrivial examples of skew automorphisms would be very useful, because they can be used to construct interacting models. This cohomology is defined in a purely algebraic setting, so it needs to be extended or adapted to apply to C * -algebras or von Neumann algebras. This is not the first cohomological structure to be associated with quantum field theory. Rejzner used a BV-bicomplex to construct gauge theories in perturbative AQFT [30]. Hochschild cohomology needs to be extended to gauge theories, and combining the Hochschild complex with the BV-bicomplex may lead to further new perspectives.