Supergeometry in locally covariant quantum field theory

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor A : SLoc -->S*Alg to the category of super-*-algebras which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors eA : eSLoc -->eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess-Zumino model in 3|2-dimensions.


Introduction and summary
Over the past decades, supersymmetry and supergravity have been strongly vital research areas in theoretical and mathematical physics. On the one hand, supersymmetric extensions of the standard model provide interesting perspectives on particle physics and, on the other hand, supergravity arises as a low-energy limit of string theory and it might have potential applications to e.g. early universe cosmology. Regarded from the perspective of a quantum field theorist, interest in supersymmetry arises because of the well-known fact that certain supersymmetric quantum field theories enjoy unexpected renormalization properties, which collectively go under the name 'non-renormalization theorems', see e.g. [GSR79,Sei93].
In contrast to the immense progress which theoretical physics has made during the past decades, mathematically rigorous developments of supersymmetric quantum field theories are quite rare. There are however some notable exceptions: In [BG06], Buchholz and Grundling address the nontrivial problems of implementing supersymmetry transformations into the C *algebraic framework of algebraic quantum field theory and constructing super-KMS states. The study of superconformal nets in two spacetime dimensions has been initiated by Capri, Kawahigashi and Longo in [CKL08]. Since then superconformal nets have been intensively developed, also with a focus on extended supersymmetry [CHKLX12]. Perturbative superconformal quantum field theories on a special class of (curved) spacetimes have been discussed quite recently by de Medeiros and Hollands [dMH13], where also a perturbative non-renormalization theorem is rigorously proven. A formulation of (Euclidean) supersymmetric quantum field theories within the Atiyah-Segal approach and their connection to elliptic cohomology has been investigated by Stolz and Teichner, see e.g. the survey article [ST11].
In our work we shall study supersymmetric quantum field theories from the perspective of locally covariant quantum field theory [BFV03], which is a relatively modern extension of algebraic quantum field theory to curved spacetimes. In locally covariant quantum field theory, the focus is on the construction and analysis of functors from a category of spacetimes to a category of algebras, which are supposed to describe the assignment of observable algebras to spacetimes. Besides establishing a mathematical foundation for quantum field theory on curved spacetimes, locally covariant quantum field theory is essential for constructing perturbatively interacting models [HW01,HW02,BDF09]. Our aim is to extend carefully the formalism of locally covariant quantum field theory to the realm of supergeometry, focusing in the present work only on the case of non-interacting models. On the one hand, a solid understanding of non-interacting super-quantum field theories (super-QFTs) is a necessary prerequisite for constructing perturbative models and analyzing their renormalization behavior, especially concerning potential non-renormalization theorems. On the other hand, already simple examples of non-interacting super-QFTs indicate that the basic framework of locally covariant quantum field theory has to be generalized in order to be able to cope with the concept of supersymmetry transformations. In more detail, as we will show in this work, the framework of ordinary category theory, on which locally covariant quantum field theory is based, is insufficient to capture supersymmetry transformations on the level of the super-QFT functor. In particular, we observe that both the fermionic and the bosonic component fields are locally covariant quantum fields in the sense of [BFV03] (i.e. natural transformations to the super-QFT functor), which indicates that supersymmetry transformations are not appropriately described in this framework. Using techniques from enriched category theory, we shall propose a generalization of the framework in [BFV03] which is general enough to capture supersymmetry transformations. Loosely speaking, we shall develop suitable categories of superspacetimes eSLoc and superalgebras eS * Alg that are enriched over the monoidal category of supersets and consider super-QFTs as enriched functors eA : eSLoc → eS * Alg between these enriched categories. Supersymmetry transformations are captured in terms of the higher superpoints of the morphism supersets in eSLoc and their action on the superalgebras of observables is dictated by the enriched functor eA : eSLoc → eS * Alg. Our results will therefore clarify the structure of supersymmetry transformations in locally covariant quantum field theory, which will be essential for analyzing perturbative super-QFTs and their renormalization behavior in future works.
Let us outline the content of this paper: In Section 2 we shall give a self-contained and rather detailed introduction to those techniques of super-linear algebra and supergeometry that will be used in our work. This should allow readers who do not have a solid background in those fields to follow our constructions in the main part of this paper. In Section 3 we introduce super-Cartan structures on supermanifolds and study their properties. These structures have their origin in the superspace formulation of supergravity [WZ77] and they are used in our work in order to describe 'superspacetimes', which generalize Lorentz manifolds in ordinary quantum field theory. The dimension and the 'amount of supersymmetry' of a super-Cartan supermanifold is captured in its local model space, which we describe by using representation theoretic data corresponding to some spin group. We define a suitable category of super-Cartan supermanifolds and show that to any super-Cartan supermanifold there is a functorially associated oriented and time-oriented ordinary Lorentz manifold. This allows us to introduce a natural notion of the chronological and causal future/past in a super-Cartan supermanifold and therewith the concept of globally hyperbolic super-Cartan supermanifolds. In Section 4 we formulate a set of axioms to describe non-interacting super-field theories in an abstract way. According to our Definition 4.2, a super-field theory is specified by the following data: 1.) A choice of representation theoretic data that fixes the local model space of the super-Cartan supermanifolds. 2.) A full subcategory SLoc of the category of globally hyperbolic super-Cartan supermanifolds, which allows us later to implement constraints on the super-Cartan structures, e.g. the supergravity supertorsion constraints [WZ77]. 3.) A suitable natural super-differential operator which encodes the dynamics of the super-field theory. We show in Section 5 that given any super-field theory as described above, one can construct a functor A : SLoc → S * Alg to the category of super- * -algebras which satisfies the axioms of locally covariant quantum field theory [BFV03] adapted to our supergeometric setting, cf. Theorem 5.11. In other words, any super-field theory gives rise to a super-QFT. As in the case of ordinary quantum field theory, we first construct a functor L : SLoc → X to the category of super-symplectic spaces or the category of super-inner product spaces (depending on the representation theoretic data), which is then quantized by a quantization functor Q : X → S * Alg that constructs super-canonical (anti)commutation relation algebras. We analyze the functor A : SLoc → S * Alg and show that, in addition to the locally covariant quantum field which describes the linear superfield operators, the bosonic and fermionic component fields are also natural transformations in this framework. This is an undesirable feature which indicates that the framework developed in the Sections 4 and 5 does not capture supersymmetry transformations as those would mix the bosonic and fermionic components. We then solve this problem by making use of techniques from enriched category theory. In Section 6 we provide a stronger axiomatic framework for super-field theories by generalizing the category SLoc to a suitable category eSLoc which is enriched over the monoidal category of supersets SSet. The category SSet is defined as the functor category Fun(SPt op , Set), where SPt is the category of superpoints and op denotes the opposite category. Hence, a superset is a functor SPt op → Set, which means that, in addition to its ordinary points, a superset has further content that is captured by its 'higher superpoints'. Loosely speaking, enriching the morphism sets in SLoc to the morphism supersets in eSLoc we obtain in addition to ordinary supermanifold morphisms M → M ′ also supermanifold morphisms between the 'fattened' supermanifolds pt n × M → pt n × M ′ , where pt n is any superpoint, that are able to capture supersymmetry transformations; indeed, the odd parameters which are used in the physics literature in order to parametrize supersymmetry transformations are elements in the structure sheaf Λ n = • R n (the Grassmann algebra) of pt n . It is important to notice that in this functorial approach we do not have to fix a superpoint (or equivalently a Grassmann algebra) from the outside, as it is typically done in the physics literature, but we are working functorially over the category of all superpoints. Similar techniques have been used before in order to describe super-mapping spaces between supermanifolds, see e.g. [Sac09,SW11,Han14]. In the enriched setting, the super-differential operators which govern the dynamics of the super-field theory should form an enriched natural transformation. We explicitly characterize these enriched natural transformations and show that they are in bijective correspondence to ordinary natural transformations (as used in Section 4) satisfying further conditions, which one may interpret as covariance conditions under supersymmetry transformations. This allows us to give a simple axiomatic characterization of enriched super-field theories in Definition 6.15. In Section 7 we show that any enriched super-field theory gives rise to an enriched super-QFT that we describe by an enriched functor eA : eSLoc → eS * Alg to a suitable enriched category of super- * -algebras. We show that this enriched functor satisfies a generalization of the axioms of locally covariant quantum field the-ory, cf. Theorem 7.11. We further show that the enriched super-QFT has an enriched locally covariant quantum field (given by an enriched natural transformation) which describes the linear superfield operators. In contrast to the non-enriched theory studied in Section 5, our enriched natural transformation does not decompose into the bosonic and fermionic component fields, which indicates that supersymmetry transformations are appropriately described within our enriched categorical framework. This is confirmed and illustrated in Section 8 by constructing and analyzing explicit examples of 1|1 and 3|2-dimensional enriched super-QFTs, together with the structure of supersymmetry transformations. Our 1|1-dimensional example is the usual superparticle and our 3|2-dimensional example is the free Wess-Zumino model on a class of curved super-Cartan supermanifolds. In Appendix A we collect some elementary definitions from enriched category theory which are needed in our work.

Preliminaries on supergeometry
We give a self-contained review of those aspects of super-linear algebra and supergeometry which we shall need for our work. For more details see e.g. [CCF11] and [DM99]. In the following the ground field K will be either R or C and we set Z 2 := {0, 1}. Whenever there is no need to distinguish between the real and complex case, we shall drop the field K from our notations.
Super-vector spaces: A super-vector space is a Z 2 -graded vector space V = V 0 ⊕ V 1 . We assign to the non-zero homogeneous elements 0 = v ∈ V i the Z 2 -parity |v| := i ∈ Z 2 , for i = 0, 1, and call elements in V 0 even and elements in V 1 odd. The superdimension (or simply dimension) of a super-vector space V is denoted by dim(V ) := dim(V 0 )|dim(V 1 ). An example of an n|m-dimensional super-vector space is K n|m := K n ⊕ K m , with n, m ∈ N 0 . For simplicity, we shall denote K 1|0 simply by K. A super-vector space morphism L : V → V ′ is a linear map which preserves the Z 2 -parity, i.e. L(V i ) ⊆ V ′ i for i = 0, 1. The category SVec of super-vector spaces has as objects all super-vector spaces and as morphisms all super-vector space morphisms. Recall that SVec is a monoidal category with tensor product functor ⊗ : SVec × SVec → SVec and unit object K = K 1|0 . Explicitly, the tensor product V ⊗ W of two super-vector spaces V and W is the ordinary tensor product V ⊗ W of vector spaces equipped with the Z 2 -grading The tensor product of two SVec-morphisms is simply given by the tensor product of linear maps. The monoidal category SVec is symmetric with respect to the commutativity constraints Moreover, it is closed with internal hom-objects given by the vector space Hom(V, W ) of all linear maps L : V → W equipped with the obvious Z 2 -grading; L ∈ Hom(V, W ) is even/odd if it preserves/reverses the Z 2 -parity.
Given an object V = V 0 ⊕ V 1 in SVec, a super-vector subspace is a vector subspace W ⊆ V together with a Z 2 -grading W = W 0 ⊕ W 1 such that W i is a vector subspace of V i , for i = 0, 1. We then may form the quotient super-vector space V /W := V 0 /W 0 ⊕ V 1 /W 1 , which comes together with a canonical SVec-morphism V → V /W assigning equivalence classes.
Superalgebras: A (unital and associative) superalgebra is an algebra object in SVec. Explicitly, this means that a superalgebra is an object A in SVec together with two SVec-morphisms µ A : A ⊗ A → A (called product) and η A : K → A (called unit), such that the diagrams in SVec commute. We shall often denote the products by juxtaposition, i.e. µ A (a 1 ⊗a 2 ) = a 1 a 2 , and the unit element by η A (1) = ½. An example of a (real) superalgebra is the Grassmann algebra Λ n := • R n , for n ∈ N 0 . A superalgebra morphism κ : A → A ′ is a SVec-morphism which preserves products and units, i.e.
We denote the category of superalgebras by SAlg and notice that it is a monoidal category: The tensor product A ⊗ B of two superalgebras is the super-vector space A ⊗ B equipped with the following product and unit Explicitly, we have for the product ( homogeneous a 1 , a 2 ∈ A and b 1 , b 2 ∈ B, and for the unit ½ A⊗B = ½ A ⊗ ½ B . The tensor product of two SAlg-morphisms is simply given by the tensor product of linear maps.
We shall require some special classes of superalgebras. A superalgebra A is called supercommutative if the product is compatible with the commutativity constraint, i.e. µ A • σ A,A = µ A . Notice that supercommutative superalgebras form a monoidal subcategory of SAlg, which is symmetric with respect to the commutativity constraints induced by SVec. Moreover, for a supercommutative superalgebra A the product µ A : A ⊗ A → A is a SAlg-morphism with respect to the tensor product superalgebra structure on A ⊗ A.
Let us now consider superalgebras over C. A super- * -algebra is a superalgebra A over C together with an even C-antilinear map * A : Explicitly, these conditions read as ½ * = ½ and (a 1 a 2 ) * = (−1) |a 1 | |a 2 | a * 2 a * 1 , for homogeneous elements a 1 , a 2 ∈ A. A super- * -algebra morphism κ : A → A ′ is a SAlg-morphism satisfying κ • * A = * A ′ • κ. We denote the category of super- *algebras by S * Alg and notice that it is a monoidal category; the superinvolution on the tensor Super-Lie algebras: A super-Lie algebra is a Lie algebra object in SVec. Explicitly, a super-Lie algebra is an object g in SVec together with a SVec-morphism [ · , · ] g : g ⊗ g → g (called super-Lie bracket) which satisfies the super-skew symmetry condition [ · , · ] g • id g⊗g + σ g,g = 0 (2.5a) and the super-Jacobi identity [ · , [ · , · ] g ] g • id g⊗g⊗g + σ g,g⊗g + σ g⊗g,g = 0 . (2.5b) A super-Lie algebra morphism L : g → g ′ is a SVec-morphism which preserves the super-Lie brackets, i.e.
Supermodules and the Berezinian: Let A be a superalgebra. A left A-supermodule is a left module object in SVec. Explicitly, a left A-supermodule is an object V in SVec together with a SVec-morphism l V : A ⊗ V → V (called left A-action), such that the diagrams in SVec commute. A right A-supermodule is defined similarly and an A-bisupermodule is a left and right A-supermodule with commuting left and right A-actions. If A is a supercommutative superalgebra, then any left A-supermodule V is also a right A-supermodule with right A- Notice that these left and right A-actions are compatible, hence V is an A-bisupermodule. We shall often denote the left and right A-actions simply by juxtaposition, i.e.
We denote the category of left A-supermodules by A-SMod. In the case of A being supercommutative, A-SMod is a monoidal category with tensor product functor ⊗ A : A-SMod × A-SMod → A-SMod (taking tensor products over A) and unit object A (regarded as a left A-supermodule with left A-action given by the product µ A ). Again for A being supercommutative, the monoidal category A-SMod is also symmetric with commutativity constraints induced by those in SVec and closed with internal hom-objects given by the left A-supermodules Hom A (V, W ) of all right A-linear maps L : V → W equipped with the obvious Z 2 -grading; L ∈ Hom A (V, W ) is even/odd if it preserves/reverses the Z 2 -parity.
A free left A-supermodule of dimension n|m is a left A-supermodule V for which there exists a basis of n ∈ N 0 even elements {e 1 , . . . , e n } and m ∈ N 0 odd elements {ǫ 1 , . . . , ǫ m }, such that (2.7a) The collection {e 1 , . . . , e n+m } := {e 1 , . . . , e n , ǫ 1 , . . . , ǫ m } of elements in V is called an adapted basis for V . Notice that any free left A-supermodule of dimension n|m is isomorphic (in the category A-SMod) to the standard free left A-supermodule A n|m := A ⊗ K n|m with the obvious left A-action. The A-SMod-morphisms between two free left A-supermodules can be represented in terms of matrices with entries in A. Explicitly, let L : V → V ′ be any A-SModmorphism between an n|m-dimensional free left A-supermodule V and an n ′ |m ′ -dimensional free left A-supermodule V ′ . Making use of any adapted bases for V and V ′ we define the elements {L j i ∈ A : i = 1, . . . , n + m , j = 1, . . . , n ′ + m ′ } via L(e i ) = n ′ +m ′ j=1 L j i e ′ j , which can be arranged in an (n + m) × (n ′ + m ′ )-matrix of the form where L 1 is an n × n ′ -matrix with entries in A 0 , L 2 is an n × m ′ -matrix with entries in A 1 , L 3 is an m × n ′ -matrix with entries in A 1 and L 4 is an m × m ′ -matrix with entries in A 0 .
Let now A be a supercommutative superalgebra and V any free left A-supermodule. Denoting the group of A-SMod-automorphisms of V by GL(V ), there exists a group homomorphism (called the Berezinian) to the group of invertible elements in A For notational simplicity we shall denote the SAlg-morphism of global sections by , for n, m ∈ N 0 , which we denote with the usual abuse of notation by the same symbol as the standard super-vector space above.
A supermanifold of dimension n|m is a superspace M = ( M , O M ) that is locally isomorphic to R n|m , with n, m ∈ N 0 fixed. In more detail, given any p ∈ M , there exists an open neighborhood V ⊆ M of p, such that V is homeomorphic to an open subset U ⊆ R n and such that the restricted sheaves of superalgebras O M | V and C ∞ R n | U ⊗ • R m are isomorphic. Taking the standard coordinates (x 1 , . . . , x n ) of R n and the standard coordinates (θ 1 , . . . ,  Explicitly, in local coordinates (x 1 , . . . , x n+m ) of M in V an adapted basis for Ω 1 M (V ) is given by the differentials {dx 1 , . . . , dx n+m }, which can also be characterized by the duality relations ∂ i , dx j = δ j i , for all i, j = 1, . . . , n + m. Notice that the first n differentials are even and that the last m differentials are odd.
Super-one-forms can be pulled back along SMan-morphisms χ : M → M ′ . In our work we shall only need the pull-back for the special case where ) is a SAlg-isomorphism and we can define a push-forward of superderivations by for all open U ⊆ M ′ . The pull-back of super-one-forms (denoted with a slight abuse of notation also by χ * is then defined by the duality relations for all ω ∈ Ω 1 M ′ (U ) and X ∈ Der M ( χ −1 (U )). In the case where we only have that χ : M → M ′ | χ( M ) is a SMan-isomorphism we define the pull-back of super-one-forms by  Given any SMan-isomorphism χ : M → M ′ , there exists a pull-back χ * : Ber(Ω 1 M ′ ) → Ber(Ω 1 M ). In local coordinates (x ′1 , . . . , x ′n+m ) of M ′ in V ′ and (x 1 , . . . , x n+m ) of M in V = χ −1 (V ′ ), the pull-back is given by (2.24) where J(χ) is the super-Jacobi matrix with entries defined by dχ * and all open U ⊆ M ′ . Given now two oriented supermanifolds M and M ′ together with an orientation preserving SMan-isomorphism χ : M → M ′ , the global Berezin integral transforms as . This follows in the usual way from the aforementioned local change of variables formula.

Super-Cartan supermanifolds
We introduce the concept of super-Cartan structures on supermanifolds. These structures have their origin in the superspace formulation of supergravity, see e.g. [WZ77]. In our work they are required for constructing natural super-differential operators which govern the dynamics of super-field theories at the classical and quantum level. In this paper we shall only focus on the case where the super-principal bundle underlying the super-Cartan structure is globally trivial (and trivialized), which considerably simplifies our discussion as we do not have to deal with super-principal bundles and their associated super-vector bundles. Notice that the latter concepts are essentially well understood, see e.g. [BBHR91], however, they are cumbersome to work with. We hope to come back in a future work to super-field theories which are also defined on globally non-trivial super-Cartan supermanifolds.

Representation theoretic data and super-Poincaré super-Lie algebras
We shall briefly review super-Poincaré super-Lie algebras in various dimensions. See e.g. [Del99] and [Fre99, Lecture 3] for details.
Let W be a finite-dimensional real vector space and g : W ⊗ W → R a Lorentz metric of signature (+, −, · · · , −). Let further S be a real spin representation of the associated spin group Spin(W, g) and a symmetric and Spin(W, g)-equivariant pairing. We denote the spin group actions on W and S by, respectively, ρ W : Spin(W, g) × W → W and ρ S : Spin(W, g) × S → S. Moreover, we simply write spin for the Lie algebra of Spin(W, g) and recall that the spin group actions above induce Lie algebra actions, which we denote by ρ W * : spin ⊗ W → W and ρ S * : spin ⊗ S → S.
Definition 3.1. Let us fix any choice of the data (W, g, S, Γ).
(i) The super-Poincaré super-Lie algebra sp (corresponding to this data) is given by the super-vector space (with (spin ⊕ W ) even and S odd), together with the super-Lie bracket defined by (ii) The supertranslation super-Lie algebra t (corresponding to this data) is given by the super-vector space (with W even and S odd), together with the super-Lie bracket defined by Remark 3.2. Notice that the bracket defined in (3.2b) is indeed a super-Lie bracket. The super-skew symmetry is evident from its definition and the super-Jacobi identity is a straightforward check using the Spin(W, g)-equivariance of Γ, which implies that for all L ∈ spin and s 1 , s 2 ∈ S.
The following statement is easily shown. We therefore can omit the proof. with the obvious definition of the arrows, is a short exact sequence of super-Lie algebras.
The data (W, g, S, Γ) we have introduced above is sufficient in order to construct the super-Poincaré and supertranslation super-Lie algebras (corresponding to this choice of data). For our applications to super-Cartan supermanifolds and super-field theories we require some additional data. First, let us fix a positive cone C ⊂ W of timelike vectors and assume that Γ : S ⊗S → W is positive in the sense that Γ(s, s) ∈ C, for all s ∈ S, with Γ(s, s) = 0 only for s = 0. Here C denotes the closure of the cone C ⊂ W . The existence of such Γ has been shown in [Del99]. The positive cone C ⊂ W will later play the role of a time-orientation. Next, we assume that we have given a Spin(W, g)-invariant linear map which is either a metric (of positive signature) or a symplectic structure. Such linear maps exist if dim(W ) is not equal to 2 or 6 modulo 8, see [DF99]. Finally, we take as part of the data a choice of orientations o W of W and o S of S. These orientations and also ǫ will be used define a canonical Berezinian density on any super-Cartan supermanifold and therefore a notation of integration. In summary, we will always assume as a starting point for our constructions that the data (W, g, S, Γ, C, ǫ, o W , o S ) are given.

Super-Cartan structures
In order to simplify our studies on super-Cartan supermanifolds we shall restrict our attention to super-Cartan structures which are based on globally trivial (and also trivialized) superprincipal Spin(W, g)-bundles. Let us fix any choice of the data (W, g, S, Γ, C, ǫ, o W , o S ).
Definition 3.4. Let M be a dim(W )|dim(S)-dimensional supermanifold. A (globally trivial) super-Cartan structure on M is a pair (Ω, E) consisting of an even super-one-form Ω ∈ Ω 1 (M, spin) (called the super-spin connection) and an even and non-degenerate super-one-form E ∈ Ω 1 (M, t) (called the supervielbein). The triple M := (M, Ω, E) is called a super-Cartan supermanifold.
Remark 3.5. Notice that the requirement that E is non-degenerate fixes the dimension of M to be the dimension dim(W )|dim(S) of the supertranslation super-Lie algebra t.
To any super-Cartan supermanifold M = (M, Ω, E) we can assign its supercurvature and supertorsion, which play an important role in supergravity. They are defined by (3.7a) where the brackets are those induced by the super-Lie bracket in sp via We now shall study integration on super-Cartan supermanifolds. Let us recall that the super-vector space t has an adapted basis {p 0 , . . . , p dim(M )−1 , q 1 , . . . , q dim(S) }, i.e. p α ∈ W and q a ∈ S, for all α = 0, . . . , dim(M ) − 1 and a ∈ 1, . . . , dim(S). Making use of the Lorentz metric g on W and the metric (or symplectic structure) ǫ on S, we can demand that {p α } is an orthonormal basis for (W, g) and that {q a } is an orthonormal (or symplectic/Darboux) basis for (S, ǫ). Making further use of the orientations o W of W and o S of S we demand that these bases are oriented and finally by using the positive cone C ⊂ W we demand that the basis for W is time-oriented, i.e. the timelike basis vector p 0 lies in C. We shall call any adapted basis for t which is of this kind an orthonormal (or orthosymplectic) time-oriented and oriented adapted basis for t. Notice that any two orthonormal (or orthosymplectic) timeoriented and oriented adapted bases for t are related by a SVec-automorphism L ∈ GL(t), whose block-matrix components (cf. (2.8)) are L 1 ∈ SO 0 (1, dim(W ) − 1), L 2 = L 3 = 0 and L 4 ∈ SO(dim(S)) (or L 4 ∈ Sp(dim(S), R)). Because of det(L 1 ) = det(L 4 ) = 1 we find that Ber(L) = 1, cf. (2.9). We now may expand the supervielbein E ∈ Ω 1 (M, t) in terms of any orthonormal (or orthosymplectic) time-oriented and oriented adapted basis for t, which yields where all e α ∈ Ω 1 (M ) are even and all ξ a ∈ Ω 1 (M ) are odd. Notice that the collection {e 0 , . . . , e dim(W )−1 , ξ 1 , . . . , ξ dim(S) } is an adapted basis for Ω 1 (M ) since E was assumed to be non-degenerate. We hence can define an element of the Berezinian supermodule of Ω 1 (M ) by Ber(E) := [e 0 , . . . , e dim(M )−1 , ξ 1 , . . . , ξ dim(S) ] ∈ Ber(Ω 1 (M )) . (3.10) Recalling (2.10), we find that this definition does not depend on the choice of the orthonormal (or orthosymplectic) time-oriented and oriented adapted basis for t, since, as we have explained above, any two such bases are related by an L ∈ GL(t) with Ber(L) = 1.
Using the Berezinian density (3.10) we can define a pairing on the compactly supported sections of the structure sheaf of M by (3.11) Notice that the Z 2 -parity of the linear map · , · M is dim(S) mod 2 and that · , · M can be extended to all F 1 , F 2 ∈ O(M ) with compactly overlapping support. Notice further that for all homogeneous F 1 , F 2 ∈ O(M ) with compactly overlapping support.
We finish this subsection by defining a suitable category of super-Cartan supermanifolds.
Definition 3.6. The category SCart consists of the following objects and morphisms: • The objects are all super-Cartan supermanifolds M = (M, Ω, E).
• The morphisms χ : M → M ′ are all SMan-morphisms (denoted by the same symbol)   Proof. We have to prove that the reduced morphism χ : M → M ′ is isometric and that it preserves the orientations and time-orientations. All these properties follow from the fact that χ * ( E ′ ) = E, which is shown by the calculation where in the third equality we have used the commutative diagram (2.13).
Due to this proposition we can define the chronological and causal future/past of a subset

The category of globally hyperbolic super-Cartan supermanifolds
For our studies on super-field theories the category of globally hyperbolic super-Cartan supermanifolds will play a major role. It can be defined as a certain subcategory of SCart.
Definition 3.10. The category ghSCart consists of the following objects and morphisms: • The objects in ghSCart are all objects M in SCart such that the reduced oriented and time-oriented Lorentz manifold M is globally hyperbolic.

Axiomatic definition of super-field theories
Motivated by the examples we will discuss in Section 8, we shall give an axiomatic characterization of super-field theories by representation theoretic and geometric data. This is a reasonable and useful approach since all of our statements concerning the construction of super-QFTs in Section 5 can be made at this abstract level, so there is no need to focus on explicit models at this point. Moreover, the problem of constructing models of super-QFTs is thereby reduced to finding explicit realizations of the assumed representation theoretic and geometric data. It will be instructive to first provide some motivations explaining our choice of data.
Representation theoretic data: Motivated by Section 3, our first choice of data is given by an eight-tuple (W, g, S, Γ, C, ǫ, o W , o S ) consisting of a finite-dimensional real vector space W , a Lorentz metric g : W ⊗ W → R, a real spin representation S of Spin(W, g), a symmetric and Spin(W, g)-equivariant pairing Γ : S ⊗S → W which is positive with respect a choice of positive cone C ⊂ W of timelike vectors, a Spin(W, g)-invariant linear map ǫ : S ⊗S → R which is either a metric (of positive signature) or a symplectic structure, and orientations o W on W and o S on S. It becomes evident from Section 3 that this data is required, on the one hand, to specify a super-Poincaré and supertranslation super-Lie algebra and, on the other hand, to describe super-Cartan structures on supermanifolds together with time-orientation and integration. In other words, the representation theoretic data fixes the local model space of the super-Cartan supermanifold and in particular its dimension to dim(W )|dim(S). Physically speaking, this means that the representation theoretic data fixes the dimension of the reduced spacetime and the amount of supersymmetry.
Admissible super-Cartan supermanifolds: As we will show in Section 8 by studying explicit examples, one should not expect that the super-field theory will be defined on the whole category ghSCart of globally hyperbolic super-Cartan supermanifolds, see Definition 3.10. A common feature in many super-field theories (especially in supergravity) is that one has to impose constraints on the superfields in order arrive at a reasonable theory. Such constraints may in particular include the supergravity supertorsion constraints [WZ77], which restrict the class of super-Cartan structures. In our axiomatic approach, which is summed up in Definition 4.2 below, we implement possible constraints by taking a full subcategory SLoc of ghSCart as a part of the data. Objects in SLoc will be called admissible super-Cartan supermanifolds.
Super-differential operators: We shall only consider super-field theories whose configurations on any object M in SLoc can be described   2 We additionally demand that, for any object M in SLoc, the super-differential operator P M is formally super-self adjoint with respect to the pairing (3.11), i.e.
for all homogeneous F 1 , F 2 ∈ O(M ) with compactly overlapping support. For many of our constructions we also have to assign retarded/advanced super-Green's operators to the superdifferential operators P M , for all objects M in SLoc. We therefore demand that, for any object M in SLoc, the super-differential operator P M is super-Green's hyperbolic in the following sense: Definition 4.1. Let M be any object in SLoc. A homogeneous super-differential operator Motivated by the discussion above, we can now define abstractly a notion of super-field theories. Our present axiomatic framework is supposed to cover all super-field theories which in the physics literature would be called 'real superfields'. The typical examples in 1|1 and 3|2-dimensions are discussed in Section 8, where it is shown that they comply with our axioms. In contrast, 'super-gauge theories' and 'chiral superfields' will require a more sophisticated set of axioms, which should include aspects of gauge invariance and the chirality constraints (see [HS13] for an axiomatic approach to ordinary gauge theories). We shall leave these problems for future work and consider in the present work the case of (real) super-field theories which we characterize by the following axioms: 3. A natural transformation P : O ⇒ O of functors from SLoc op to SVec, such that P M is a formally super-self adjoint and super-Green's hyperbolic super-differential operator of Z 2 -parity dim(S) mod 2, for any object M in SLoc.

Construction of super-quantum field theories
We show that given any super-field theory according to Definition 4.2, one can construct a functor A : SLoc → S * Alg which satisfies the locality, causality and time-slice axiom of locally covariant quantum field theory [BFV03]. In other words, any super-field theory gives rise to a super-QFT. We establish a connection between the super-field theory and its associated super-QFT by showing that the latter has a locally covariant quantum field which satisfies (in a weak sense) the equations of motion given by the super-differential operators P . As usual our construction will be done in two steps. First, we assign to a super-field theory a functor L : SLoc → X, where X is the category of super-symplectic spaces in the case of dim(S) even and the category of super-inner product spaces in the case of dim(S) odd. In the spirit of [BFV03] this functor should be interpreted as a locally covariant classical field theory. The locally covariant classical field theory is then quantized by a quantization functor Q : X → S * Alg, which implements super-canonical commutation relations (SCCR) in the case of dim(S) even and super-canonical anticommutation relations (SCAR) in the case of dim(S) odd.
which induces a SVec-isomorphism on compactly supported sections Making use of the sheaf properties of O M ′ , we can define a SVec-morphism which extends compactly supported sections by zero. This SVec-morphism is a monomorphism since We define the push-forward of compactly supported sections by the composition and notice that it is a SVec-monomorphism. The following lemma collects important properties of the push-forward of compactly supported sections, which shall be frequently used in our work.
Lemma 5.1. Let χ : M → M ′ be any SLoc-morphism. Then the following properties hold true: where the pairing is defined in (3.11).
Proof. Item (i) is shown by two simple calculations. The first part follows from where in the first equality we have used the diagram (2.12) characterizing sheaf morphisms (applied to U = M ′ and V = χ( M )) and in the second equality we have used (5.4). To show the second part, notice that if By using the same argument as above we find that where the last equality holds true by direct inspection.
Item (ii) holds true as a consequence of the transformation formula for the Berezin integral (2.25) and the property (3.13); explicitly, we have In the second equality we have used that the support of χ * (F 2 ) is contained in χ( M ) and in the last equality item (i) of the present lemma.
The first part of item (iii) follows immediately from the definition (5.5) and the second part from the following calculation where we have made frequent use of standard properties of sheaf morphisms. Proof. This is a direct consequence of Lemma 5.1 (iii).

Properties of the super-Green's operators
for all homogeneous F 1 , F 2 ∈ O c (M ).
Proof. The proof follows from a short calculation The first equality holds because of property (i) of Definition 4.1. The integral on the right-hand side is well-defined because of property (iii) of the same Definition and the fact that the reduced Lorentz manifold M is by assumption globally hyperbolic. The second equality follows from formal super-self adjointness of P M , cf. (4.2). The last equality is a consequence of property (i) of Definition 4.1 and |G ∓ M | = |P M |. We define the retarded-minus-advanced super-Green's operator is a complex which is exact everywhere.
Proof. The proof follows easily by adapting the steps in the proofs of [BG11, Theorem 3.5] or [BGP07, Theorem 3.4.7] to our supergeometric setting. We therefore can omit the details.
We shall now show that the retarded/advanced super-Green's operators are natural in the following sense.
Lemma 5.6. Let χ : M → M ′ be any morphism in SLoc. Then We will show that G ± M is a retarded/advanced super-Green's operator for P M , which due to the uniqueness result in Corollary 5.5 implies that G ± M = G ± M . We have to show that G ± M satisfies the three conditions of Definition 4.1. Item (i) is satisfied because of where in the second equality we have used (4.1) and in the last equality Lemma 5.1 (i). Item (ii) is satisfied because of In the second equality we have used (4.1) and in the first, third and last equality Lemma 5.1 (i). To show that item (iii) is satisfied we use the same argument as in [BG11, Lemma 3.2], which is based on the causal compatibility of the image of the reduced morphism χ : M is the retarded-minus-advanced super-Green's operator and · , · M is the pairing (3.11). Since, by definition, the Z 2 -parity of G M agrees with that of the pairing, the linear map τ M is even and hence a SVec-morphism. As a consequence of (3.12) and Lemma 5.3 we find that  is therefore a super-symplectic space if dim(S) is even and a super-inner product space if dim(S) is odd.
We shall now show that the assignment (5.22) is functorial. For this we introduce the following category which depends on the choice of super-field theory via dim(S) mod 2.
Definition 5.7. The category X consists of the following objects and morphisms: • The objects are all pairs V := (V, τ ) consisting of a real super-vector space V and a weakly non-degenerate SVec-morphism τ : V ⊗ V → R, which is super-skew symmetric if dim(S) is even and super-symmetric if dim(S) is odd, i.e.
• The morphisms L : V → V ′ are all SVec-morphisms (denoted by the same symbol) Proposition 5.8. The following assignment is a functor L : SLoc → X: To any object M in SLoc we assign the object L(M ) in X given by (5.22) and to any SLoc-morphism χ : M → M ′ we assign the X-morphism Proof. We have already seen above that L(M ) is an object in X. It remains to show that (5.24) is well-defined and an X-morphism. It is well-defined since We finish this subsection by proving some properties of the functor L : SLoc → X, which are the axioms of locally covariant quantum field theory [BFV03] applied to classical theories.
Theorem 5.9. For any super-field theory according to Definition 4.2 the associated functor L : SLoc → X satisfies the following properties: • Locality: For any SLoc-morphism χ : is monic.

The quantization functor Q : X → S * Alg
The quantization is performed by assigning to objects V in X SCCR superalgebras in the case of dim(S) even and SCAR superalgebras in the case of dim(S) odd. This reflects the fact that the objects in X are super-symplectic spaces if dim(S) is even and super-inner product spaces if dim(S) is odd. We can perform this construction in one step by using suitable sign and imaginary unit i ∈ C factors (depending on dim(S) mod 2) in the definitions below.
Let V = (V, τ ) be any object in X. We consider the complexified tensor superalgebra and denote its product simply by juxtaposition. Notice that T C (V ) is generated (over C) by the unit ½ := 1 ∈ T 0 We equip T C (V ) with the superinvolution which is defined on the generators by ½ * = ½ and v * = v, for all v ∈ V , and extended to all of T C (V ) by C-antilinearity and (a 1 a 2 ) * = (−1) |a 1 | |a 2 | a * 2 a * 1 , (5.31) for all Z 2 -parity homogeneous a 1 , a 2 ∈ T C (V ). Using the SVec-morphism τ : V ⊗ V → R, we define I(V ) to be the two-sided super- * -ideal in T C (V ) that is generated by the elements Proposition 5.10. The following assignment is a functor Q : X → S * Alg: To any object V in X we assign the object Q(V ) in S * Alg given by (5.34) and to any X-morphism L : V → V ′ we assign the S * Alg-morphism Q(L) : Q(V ) → Q(V ′ ) that is specified by defining on the generators Q(L)(v) := L(v), for all v ∈ V .
Proof. By our constructions above, we already know that Q(V ) is an object in S * Alg, for all objects V in X. It remains to show that Q(L) : Q(V ) → Q(V ′ ) specified above is well-defined, i.e. that it preserves the two-sided super- * -ideals. This is a consequence of the explicit form of the generators of these ideals (5.32) and the fact that L : V → V ′ satisfies τ ′ • (L ⊗ L) = τ . for all homogeneous a 1 ∈ A(χ 1 )(A(M 1 )) and a 2 ∈ A(χ 2 )(A(M 2 )).

The locally covariant quantum field theory
• Time-slice axiom: Given any Cauchy SLoc-morphism χ : M → M ′ , then A(χ) : Proof. All properties listed above follow by standard arguments from the corresponding properties of the classical theory given in Theorem 5.9. Let us briefly give a sketch or reference: The locality property follows by using the techniques summarized in [FV12, Appendix A]. The causality property for a 1 and a 2 being two generators follows explicitly from the form of the two-sided super- * -ideal (5.32) and for generic a 1 and a 2 by expressing these elements in terms of generators and using iteratively the causality property for the generators. The time-slice axiom for A = Q • L follows since functors (here Q) preserve isomorphisms.
We conclude this section by showing that the functor A : SLoc → S * Alg has a locally covariant quantum field, which establishes a connection to the data specifying a super-field theory in Definition 4.2. Let us consider the functor O c : SLoc → SVec and regard A also as a functor to SVec (denoted by the same symbol) by composing with the forgetful functor. There is a natural transformation Φ : O c ⇒ A of functors from SLoc to SVec with components given by the SVec-morphisms  ⇒ A, which describe within our physical interpretation the even and odd component quantum fields of the superfield Φ. The appearance of the even and odd quantum fields is an undesirable feature, which indicates that our formulation does not appropriately capture supersymmetry transformations. In fact, supersymmetry transformations are supposed to mix the even and odd component fields, hence allowing neither of them to be a natural transformation, i.e. a locally covariant quantum field. It is the goal of the next section to 'enrich' (in a mathematically precise way) the categories and functors appearing in our construction in order to capture also supersymmetry transformations.

Axiomatic definition of enriched super-field theories
Motivated by the shortcomings of our present theory, which have been summarized in Remark 5.12, we shall now systematically 'enrich' all categories, functors and natural transformations appearing in the Definition 4.2 of super-field theories. A suitable mathematical framework is that of enriched category theory, see e.g. [Kel82,BS00] and also Appendix A for a brief introduction to the basic concepts. Loosely speaking, in an ordinary category the morphisms between two objects have to form a set and in an enriched category the morphisms between two objects are allowed to be an object in another (monoidal) category. Enriched functors and natural transformations are then defined by a suitable generalization of the standard concepts of functors and natural transformations in ordinary category theory. In our definition of enriched super-field theories, as well as in the construction of the corresponding enriched super-QFTs in Section 7, we shall consider enriched categories over the monoidal category SSet of supersets, which we also call SSet-categories. Again loosely speaking, while an ordinary set is determined by its points, a superset is determined by its superpoints. To make precise the notion of supersets, we shall use the category theoretical approach to supergeometry proposed by Schwarz [Shv84] and Molotkov [Mol84], and developed in detail by Sachse [Sac08], see also [SW11,AL12].

The monoidal category SSet of supersets
For better understanding the concept of supersets, it will be helpful to view ordinary sets from a categorical perspective. Let A be any set. Then A is determined by its points, which can be described by maps x : pt → A from a (once and for all fixed) singleton set pt := {⋆} to the set A. In other words, the points of A are described by the morphism set Hom Set (pt, A). Using the usual composition of maps, any map between two sets f : A → B induces a map between the morphism sets Hom Set (pt, A) −→ Hom Set (pt, B) , x −→ f • x .
(6.1) Let Pt be the category consisting of the single object pt and the single morphism id pt . Then the morphism set above can be regarded as a functor Hom Set ( · , A) : Pt op → Set (in foresight we use here the opposite category Pt op ) and the map (6.1) as a natural transformation Hom Set ( · , A) ⇒ Hom Set ( · , B) between functors from Pt op to Set. What this means is that we have constructed a functor Set −→ Fun(Pt op , Set) (6.2) from the category of sets to the category of functors from Pt op to Set. Notice that the functor (6.2) is fully faithful and essentially surjective, which implies that it induces an equivalence between the categories Set and Fun(Pt op , Set). In other words, we can choose freely if we want to work with the usual category Set of sets or with the functor category Fun(Pt op , Set).
Motivated by this functorial point of view, we shall define the category of supersets as the functor category Fun(SPt op , Set), where SPt is the following category of superpoints: Definition 6.1. The category SPt consists of the following objects and morphisms: • The objects are given by the supermanifolds pt n := (pt, Λ n ), where Λ n := • R n is the real Grassmann algebra over R n and n ∈ N 0 .
• The morphisms λ : pt n → pt m are all supermanifold morphisms.

The category SSet of supersets is defined as the functor category
SSet := Fun(SPt op , Set) . (6.3) Remark 6.2. In [Sac08], the category of superpoints is defined as the full subcategory of SMan with objects given by all supermanifolds whose underlying topological space is a singleton. This category is equivalent to our category SPt and moreover we have that SPt op is equivalent to the category of finite-dimensional real Grassmann algebras Gr. Hence, our category of supersets (6.3) is equivalent to the functor category Fun(Gr, Set), which is used for example in [Sac08].
Recall that the category Set of ordinary sets is a monoidal category with bifunctor × : Set × Set → Set given by the Cartesian product and unit object given by the singleton set pt. By a general construction, the monoidal structure on Set induces a monoidal structure on the functor category Fun(SPt op , Set) and hence on the category SSet of supersets. Let us briefly recall this construction and give explicit formulas. We define a bifunctor (denoted with a slight abuse of notation also by ×) × : SSet × SSet −→ SSet (6.4) by assigning to any object (F : SPt op → Set, F ′ : SPt op → Set) in SSet × SSet the object F × F ′ : SPt op → Set in SSet, which is the functor specified on objects pt n in SPt op by and on SPt op -morphisms λ op : pt n → pt m by To any morphism (η : F ⇒ G, η ′ : F ′ ⇒ G ′ ) in SSet × SSet we assign the morphism η × η ′ : F × F ′ ⇒ G × G ′ in SSet which is given by the natural transformation with components In summary, we have Proposition 6.3. The category SSet of supersets is a monoidal category with bifunctor × : SSet × SSet → SSet defined by (6.5) and unit object I defined by (6.6).

The SSet-category eSLoc
Let us choose as in Definition 4.2 any full subcategory SLoc of ghSCart. The goal of this subsection is to define a SSet-category eSLoc, such that the objects in eSLoc coincide with those in SLoc and that the morphism supersets in eSLoc enrich (in a suitable way) the ordinary morphism sets in SLoc. The main feature of this enrichment will be that supersymmetry transformations appear as superpoints of the morphism supersets, see Section 8 for explicit examples.
Before we can define the SSet-category eSLoc we need some preparations. A supermanifold M can be described in the framework of supersets (6.3) by the functor Hom SMan ( · , M ) : SPt op → Set. We will not describe the details of this approach (see [Shv84,Mol84,Sac08,Sac09]), but we make use of an equivalent picture: Hom SMan (pt n , M ) clearly coincides with the set sections of the trivial super-fibre bundle pt n × M → pt n and natural transformations Hom SMan ( · , M ) → Hom SMan ( · , M ′ ) correspond to super-fibre bundle morphisms. We will discuss the basic properties of this "family point of view" and refer to the literature [Sac09, in SMan commutes, where pr pt n ×N,pt n : pt n × N → pt n denote the projection SMan-morphisms on the first factor, for all objects N in SMan. Explicitly, a SMan-morphism χ : pt n × M → pt n ×M ′ is a SMan/pt n -morphism if and only if χ * (ζ ⊗½) = ζ ⊗½, for all ζ ∈ Λ n . We shall write M/pt n whenever we regard pt n × M as a pt n -relative supermanifold and indicate SMan/pt nmorphisms by χ : M/pt n → M ′ /pt n . Notice that the identity id pt n ×M : M/pt n → M/pt n is a SMan/pt n -morphism and that any two SMan/pt n -morphisms χ : M/pt n → M ′ /pt n and χ ′ : M ′ /pt n → M ′′ /pt n can be composed, i.e. χ ′ • χ : M/pt n → M ′′ /pt n is a SMan/pt n -morphism. Using the defining property (6.7), the set of all SMan/pt n -morphisms χ : M/pt n → M ′ /pt n can be easily characterized. We next show that the assignment pt n → Hom SMan/pt n (M/pt n , M ′ /pt n ) defines a functor SPt op → Set, which is basically the functor used in [Sac09,Han14] to define super-mapping spaces. Given any two objects M and M ′ in SMan and any SPt op -morphism λ op : pt n → pt m (i.e. a SPt-morphism λ : pt m → pt n ) we can define a map of sets (6.10) Using also Lemma 6.4 we obtain a map of sets which describes how relative SMan-morphisms behave under the exchange of superpoints. The following properties can be easily derived from (6.11). We therefore can omit the proof.
Lemma 6.5. (i) For any identity SPt op -morphism λ op = id pt n : pt n → pt n the map λ op * is the identity. For any two SPt op -morphisms λ op : pt n → pt m and λ ′op : (ii) λ op * preserves identities and compositions, i.e. We shall also require a relative notion of differential geometry on pt n -relative supermanifolds M/pt n . Let Der pt n ×M be the superderivation sheaf of the product supermanifold pt n × M and U ⊆ M be any open subset. A superderivation X ∈ Der pt n ×M (U ) is called a pt n -relative superderivation provided that X(ζ ⊗ ½) = 0, for all ζ ∈ Λ n . The pt n -relative superderivations form a subsheaf Der M/pt n of left O pt n ×M -supermodules of Der pt n ×M , which is isomorphic to Λ n ⊗ Der M . The dual Ω 1 M/pt n := Hom O pt n ×M (Der M/pt n , O pt n ×M ) of the pt n -relative superderivation sheaf Der M/pt n is called the pt n -relative super-one-form sheaf and it is isomorphic to Λ n ⊗ Ω 1 M . The pt n -relative differential d M/pt n : O pt n ×M → Ω 1 M/pt n is defined as in the nonrelative case and it can be identified with id Λn ⊗ d : M is the usual differential. Loosely speaking, we obtain the pt n -relative geometric objects on M/pt n by Λ n -superlinear extension of the ones on M . As a consequence, given any object M = (M, Ω, E) in SLoc and any object pt n in SPt op , we can assign a pt n -relative supermanifold M/pt n together with pt n -relative super-one-forms ½⊗Ω ∈ Λ n ⊗Ω 1 (M, spin) ≃ Ω 1 (M/pt n , spin) and ½ ⊗ E ∈ Λ n ⊗ Ω 1 (M, t) ≃ Ω 1 (M/pt n , t).
With these preparations we can now define the SSet-category eSLoc.
Definition 6.6. The SSet-category eSLoc is given by the following data: where id pt n ×M is the identity SMan/pt n -morphism.
Remark 6.7. Using Lemma 6.5 one can easily see that eSLoc is a SSet-category according to Definition A.1: Lemma 6.5 (iii) implies that the map of sets eSLoc(M , M ′ )(λ op ) is welldefined, i.e. that it has the claimed codomain, and Lemma 6.5 (i) implies that eSLoc(M , M ′ ) : SPt op → Set is a functor. The composition • and identity 1 are natural transformations because of Lemma 6.5 (ii). Finally, the diagrams in Definition A.1 commute because of the associativity and identity property of the composition • of SMan/pt n -morphisms.

The SSet-functor eO : eSLoc op → eSVec
Our next goal is to show that the ordinary global section functor O : SLoc op → SVec can be promoted to a SSet-functor eO : eSLoc op → eSVec with values in the SSet-category eSVec of super-vector spaces. For defining the latter SSet-category we will first discuss how to promote super-vector spaces to (left) supermodules over Grassmann algebras. This procedure is known as "extension of ring of scalars" and discussed in detail in [Bou89, Chapter II.5].
Given any object V in SVec and any object pt n in SPt op , we can consider the left Λ nsupermodule Λ n ⊗ V . Given any two objects V and V ′ in SVec, the set of Λ n -SMod-morphisms L : Λ n ⊗ V → Λ n ⊗ V ′ can be easily characterized.
Lemma 6.8. Let V and V ′ be any two objects in SVec and pt n any object in SPt op . Then the map is a bijection of sets, where η n : R → Λ n denotes the unit in Λ n .
Proof. The map β pt n is invertible by assigning to any SVec-morphism K : , for all ζ ⊗ v ∈ Λ n ⊗ V , and R-linear extension.
Given any two objects V and V ′ in SVec and any SPt op -morphism λ op : pt n → pt m (i.e. a SPt-morphism λ : pt m → pt n ) we can define a map of sets (6.16) Using also Lemma 6.8 we obtain a map of sets The following properties can be easily derived from (6.17), see also [Bou89]. We therefore can omit the proof.
Lemma 6.9. (i) For any identity SPt op -morphism λ op = id pt n : pt n → pt n the map λ op * is the identity. For any two SPt op -morphisms λ op : pt n → pt m and λ ′op : (ii) λ op * preserves identities and compositions, i.e.
for all objects V in SVec and all Λ n -SMod-morphisms L : With these preparations we can now define the SSet-category eSVec.
Definition 6.10. The SSet-category eSVec is given by the following data: • The objects are all objects V in SVec.
• For any two objects V and V ′ in eSVec, the object of morphisms from V to V ′ is given by the following functor eSVec(V, V ′ ) : SPt op → Set: For any object pt n in SPt op we define eSVec(V, V ′ )(pt n ) to be the set of all Λ n -SMod-morphisms L : Λ n ⊗ V → Λ n ⊗ V ′ . For any SPt op -morphism λ op : pt n → pt m we define the map of sets where λ op * is given in (6.17).
• For any three objects V , V ′ and V ′′ in eSVec, we define the composition morphism to be the natural transformation with components where • is the composition of Λ n -SMod-morphisms.
• For any object V in eSVec, we define the identity on V morphism 1 : I → eSVec(V, V ) to be the natural transformation with components Moreover, it can be shown [Sac08, Corollary 3.4.2] that the assignment · : SVec → Mod(SSet) defines a fully faithful functor; its image SVec consists of representable modules. Comparing with Definition 6.10, we see that eSVec and SVec have isomorphic classes of objects but our enriched category eSVec contains more morphisms. In fact, since · is full, we have . Thus, all additional information contained in eSVec(V, W )(pt n ) for n > 0 is not seen in SVec and our definition provides a proper enrichment of the latter category. It may be possible to give a natural meaning to our enrichment constructions inside the functor categories Mod(SSet) (or SMod(SSet)), but this discussion is beyond the scope of the present publication. We Naturality of these components is easily checked. We obtain Proposition 6.12. The assignment eO : eSLoc op → eSVec given above is a SSet-functor.
Proof. We have to prove that this assignment is compatible with the composition and identity, see Definition A.3. Let pt n be any object in SPt op and M , M ′ and M ′′ any three objects in eSLoc op . We obtain that, for all χ ∈ eSLoc op (M , M ′ )(pt n ) and χ ′ ∈ eSLoc op (M ′ , M ′′ )(pt n ), Remark 6.13. As in the ordinary case, we can enlarge the morphisms in the SSet-category eSVec by replacing in Definition 6.10 all appearances of the sets Hom Λn-SMod (Λ n ⊗V, Λ n ⊗V ′ ) of Λ n -SMod-morphisms by the sets underlying the internal hom-objects Hom Λn (Λ n ⊗ V, Λ n ⊗ V ′ ) in the category Λ n -SMod, i.e. all right Λ n -linear maps L : Λ n ⊗ V → Λ n ⊗ V ′ . We denote the resulting SSet-category by eSVec and note that there is an obvious SSet-functor eSVec → eSVec. Consequently, we can regard the SSet-functor eO : eSLoc op → eSVec also as a SSetfunctor (denoted with a slight abuse of notation by the same symbol) eO : eSLoc op → eSVec.
As explained in Footnote 2, this generalization will be needed in Section 8 to describe 1|1dimensional examples (i.e. superparticles), which are somewhat peculiar.

Structure of the SSet-natural transformations eO ⇒ eO
In super-field theories, cf. Definition 4.2, we have described the dynamics by a suitable natural transformation P : O ⇒ O of functors from SLoc op to SVec. In the enriched setting, we shall use suitable SSet-natural transformations P : eO ⇒ eO of SSet-functors from eSLoc op to eSVec.

The definition
With this preparation we can now give a simple definition of enriched super-field theories. In particular, using Proposition 6.14 we can define an enriched super-field theory to be a superfield theory (cf. Definition 4.2) together with extra conditions which ensure that the ordinary natural transformation P : O ⇒ O extends to a SSet-natural transformation P : eO ⇒ eO.
Definition 6.15. An enriched super-field theory is a super-field theory according to Definition 4.2, such that the diagram (6.30) of right Λ n -linear maps commutes, for all objects M and M ′ in eSLoc op , all objects pt n in SPt op and all χ ∈ eSLoc op (M ′ , M )(pt n ).

Construction of enriched super-quantum field theories
We show that given any enriched super-field theory according to Definition 6.15 one can construct a SSet-functor eA : eSLoc → eS * Alg, i.e. an enriched super-QFT. As in Section 5 we decompose our construction into two steps: First, we construct a SSet-functor eL : eSLoc → eX which describes the enriched classical theory. Second, we construct a SSet-functor eQ : eX → eS * Alg describing the enriched quantization. We shall also study properties of the enriched functors and establish a connection between the enriched super-field theory and the enriched super-QFT by constructing an enriched locally covariant quantum field. In contrast to ordinary super-QFTs, our enriched approach captures also supersymmetry transformations. It should be emphasized that the SSet-categories eSLoc, eX and eS * Alg are defined using Z 2 -parity preserving morphisms. The appearance of supersymmetry transformations in the enriched setting is due to the higher superpoints of the morphism supersets in eSLoc, eX and eS * Alg. Let M = (M, Ω, E) be any object in eSLoc and pt n any object in SPt op . Making use of the pt n -relative differential geometry on M/pt n , together with the pt n -relative supervielbein ½ ⊗ E ∈ Ω 1 (M/pt n , t), we can define a pt n -relative version of the pairing (3.11) by where M/pt n 7.3 The SSet-functor eL : eSLoc → eX Given any object V = (V, τ ) in X and any object pt n in SPt op , we can consider the object Λ n ⊗ V in Λ n -SMod and define a Λ n -SMod-morphism Let us now enrich the category X given in Definition 5.7.
Definition 7.4. The SSet-category eX is given by the following data: • The objects are all objects V = (V, τ ) in X.
• For any two objects V and V ′ in eX, the object of morphisms from V to V ′ is the following functor eX(V , V ′ ) : SPt op → Set: For any object pt n in SPt op we define eX(V , V ′ )(pt n ) to be the set of all Λ n -SMod-morphisms L : Λ n ⊗V → Λ n ⊗V ′ satisfying τ ′ pt n •(L⊗ Λn L) = τ pt n . For any SPt op -morphism λ op : pt n → pt m we define the map of sets where λ op * is given in (6.17).
• The composition and identity morphisms are defined as in the SSet-category eSVec, see Definition 6.10.
We can now define the SSet-functor eL : eSLoc → eX as follows: for all χ ∈ eSLoc(M , M ′ )(pt n ). Notice that (7.10) is well-defined since Proposition 7.5. The assignment eL : eSLoc → eX given above is a SSet-functor.
We finish this subsection by observing that the SSet-functor eL : eSLoc → eX satisfies an enriched version of the properties in Theorem 5.9 for ordinary super-field theories, which can be proven in exactly the same way: Theorem 7.6. For any enriched super-field theory according to Definition 6.15 the associated SSet-functor eL : eSLoc → eX satisfies the following properties, for all objects pt n in SPt op : • Enriched locality: For any χ ∈ eSLoc(M , M ′ )(pt n ), we have that (eL M ,M ′ ) pt n (χ) ∈ eX(L(M ), L(M ′ ))(pt n ) is monic.

7.4
The SSet-quantization functor eQ : eX → eS * Alg We define an enriched version of the category S * Alg using "extension of scalars" for algebras (cf. [Bou89, Chapter III.1.5] for the general concept) and adapt the results obtained in Subsection 6.3 to the category of super- * -algebras. For this let us denote by Λ C n := Λ n ⊗ C the complexification of the Grassmann algebra, for all n ∈ N 0 , and notice that Λ C n is an object in S * Alg when equipped with the superinvolution * := id Λn ⊗ · : Λ C n → Λ C n . We shall denote the product in Λ C n by µ C n and the unit by η C n . Let A and A ′ be any two objects in S * Alg and pt n any object in SPt op . A S * Alg-morphism κ : Λ C n ⊗ C A → Λ C n ⊗ C A ′ is called a Λ C n -relative S * Alg-morphism (in short Λ C n -S * Alg-morphism) provided that κ(ζ ⊗ C ½) = ζ ⊗ C ½ for all ζ ∈ Λ C n , i.e. κ is Λ n -superlinear. Notice that the identity id Λ C n ⊗ C A is a Λ C n -S * Alg-morphism and that any two Λ C n -S * Alg-morphisms κ : In analogy to Lemma 6.8, the Λ C n -S * Alg-morphisms κ : Λ C n ⊗ C A → Λ C n ⊗ C A ′ can be easily characterized. Lemma 7.7. Let A and A ′ be any two objects in S * Alg and pt n any object in SPt op . Then the map is a bijection of sets.
Given any two objects A and A ′ in S * Alg and any SPt op -morphism λ op : pt n → pt m (i.e. a SPt-morphism λ : pt m → pt n ) we can define a map of sets where the extension of λ * : Λ n → Λ m to the complexifications is implicitly understood. Using also Lemma 7.7 we obtain a map of sets The following properties can be easily derived from (7.17). We therefore can omit the proof.
Lemma 7.8. (i) For any identity SPt op -morphism λ op = id pt n : pt n → pt n the map λ op * is the identity. For any two SPt op -morphisms λ op : pt n → pt m and λ ′op : pt m → pt l we have that (λ ′op • op λ op ) * = λ ′op * • λ op * .
for all objects A in S * Alg and all Λ C n -S * Alg-morphisms κ : With these preparations we can now define the SSet-category eS * Alg.
Definition 7.9. The SSet-category eS * Alg is given by the following data: • The objects are all objects A in S * Alg.
• For any two objects A and A ′ in eS * Alg, the object of morphisms from A to A ′ is given by the following functor eS * Alg(A, A ′ ) : SPt op → Set: For any object pt n in SPt op we define eS * Alg(A, A ′ )(pt n ) to be the set of all Λ C n -S * Alg-morphisms κ : For any SPt op -morphism λ op : pt n → pt m we define the map of sets where λ op * is given in (7.17).
• For any three objects A, A ′ and A ′′ in eS * Alg, we define the composition morphism where • is the composition of Λ C n -S * Alg-morphisms.
• For any object A in eS * Alg, we define the identity on A morphism 1 : I → eS * Alg(A, A) to be the natural transformation with components where id Λ C n ⊗ C A is the identity Λ C n -S * Alg-morphism.
The quantization SSet-functor eQ : eX → eS * Alg is constructed as follows: To any object V in eX we assign the object eQ(V ) := Q(V ) in eS * Alg that has been constructed in (5.34). To any two objects V and V ′ in eX we assign the SSet-morphism eQ V ,V ′ : eX(V , V ′ ) → eS * Alg(Q(V ), Q(V ′ )) given by the natural transformation (of functors from SPt op to Set) with components where (eQ V ,V ′ ) pt n (L) is the Λ C n -S * Alg-morphism which is specified by defining on the generators (eQ V ,V ′ ) pt n (L)(ζ ⊗ C v) := L(ζ ⊗ v), for all v ∈ V and ζ ∈ Λ n . It remains to show that (eQ V ,V ′ ) pt n (L) is well-defined, i.e. that it preserves the two-sided super- * -ideals (5.32). Written in terms of the tensor product superalgebra Λ C n ⊗ C T C (V ) the super-canonical (anti)commutation relation super- * -ideal is generated by the elements w 1 w 2 + (−1) dim(S)+1 (−1) |w 1 | |w 2 | w 2 w 1 − β τ pt n (w 1 , w 2 ) ⊗ C ½ , (7.23) for all homogeneous w 1 , w 2 ∈ Λ n ⊗V . Using now that by definition of eX, τ ′ pt n •(L⊗ Λn L) = τ pt n , we obtain that (eQ V ,V ′ ) pt n (L) is a well-defined Λ C n -S * Alg-morphism. By direct inspection one further observes that eQ V ,V ′ is compatible with composition and identity. We therefore have shown Proposition 7.10. The assignment eQ : eX → eS * Alg given above is a SSet-functor. 7.5 The enriched locally covariant quantum field theory eA : eSLoc → eS * Alg Recalling Remark A.4, we can compose the two SSet-functors eL : eSLoc → eX and eQ : eX → eS * Alg in order to define the SSet-functor eA := eQ • eL : eSLoc −→ eS * Alg .
(7.24) By using the same arguments as in the proof of Theorem 5.11, the results of Theorem 7.6 imply Theorem 7.11. For any enriched super-field theory according to Definition 6.15 the associated SSet-functor eA : eSLoc → eS * Alg satisfies the following properties, for all objects pt n in SPt op : • Enriched locality: For any χ ∈ eSLoc(M , • Enriched causality: Given any χ 1 ∈ eSLoc(M 1 , M )(pt n ) and χ 2 ∈ eSLoc(M 2 , M )(pt n ), such that the images of the reduced otLor-morphisms M 1 Remark 7.12. We would like to emphasize that the presence of the supersymmetry transformations is due to the enrichment of the involved categories over SSet. The morphism χ ∈ eSLoc(M , M ′ )(pt n ) ⊆ Hom SMan/pt n (M/pt n , M ′ /pt n ) appearing in (7.30) respects Z 2parity by definition. However, due to the presence of odd sections of the structure sheaf Λ n of pt n , the action of χ * need not preserve the splittings O c (M (′) ) = O even c (M (′) ) ⊕ O odd c (M (′) ) of the second tensor factor. Moreover, these odd parameters appearing in the structure sheaf Λ n of pt n are exactly the odd quantities used in the physics literature to parametrize supersymmetry transformations.

1|1-dimensions
As our first example we shall study a super-field theory in 1|1-dimensions, i.e. a superparticle. For defining this theory we have to provide all the data listed in Definition 4.2.
Representation theoretic data: We take W = R together with the standard 1-dimensional (Lorentz) metric (8.1) The corresponding spin group is Spin(W, g) ≃ {+1, −1} and we take S = R the 1-dimensional spin representation Notice that ρ W : Spin(W, g) × W → W , (z, w) → w is the trivial representation. As Γ we shall take the following Spin(W, g)-equivariant symmetric pairing As positive cone we take C := R + ⊂ W and we notice that Γ(s, s) ∈ C, for all s ∈ S, and Γ(s, s) = 0 only for s = 0. For ǫ we take and we notice that it is symmetric and Spin(W, g)-invariant. We define orientations on W and S by using the normalized standard bases p = 1 ∈ W = R and q = 1 ∈ S = R. In 1|1dimensions, the super-Poincaré super-Lie algebra coincides with the supertranslation super-Lie algebra (since spin is trivial), and we obtain for the super-Lie bracket relations in the normalized adapted basis {p, q} for The objects in ghSCart: Let us characterize explicitly the objects in the category ghSCart for the case of 1|1-dimensions. To simplify our studies we shall further demand that the underlying topological spaces are connected, which is also physically reasonable as they describe a time interval. Given any such object M = (M, Ω, E) we first notice that Ω = 0 since spin is trivial in 1|1-dimensions. Moreover, the reduced 1-dimensional manifold M is diffeomorphic to the real line R as the reduced Lorentz manifold M is assumed to be globally hyperbolic. The structure sheaf O M is isomorphic to C ∞ R ⊗ • R and the supervielbein can be expanded as E = (γ dt + α θ dθ) ⊗ p + (δ dθ + β θ dt) ⊗ q , (8.6) where α, β, γ, δ ∈ C ∞ (R) are coefficient functions and t, θ ∈ O(M ) are any global even/odd coordinate functions. As E is by assumption non-degenerate, the functions γ and δ have to be invertible and we may choose new coordinates (denoted with abuse of notation by the same symbols) t ∈ (t 0 , t 1 ) ⊆ R and θ, such that where now α, β ∈ C ∞ (t 0 , t 1 ). Coordinate functions in which E takes the form (8.7) are called geometric coordinates. The supercurvature R M = 0 vanishes in 1|1-dimensions and the supertorsion is given by where ∂ t α denotes the time derivative of α. The pairing (3.11) reads as where we have used the expansion F = f + θ h ∈ O(M ) with f, h ∈ C ∞ (t 0 , t 1 ).
Super-differential operators: Using the dual superderivations corresponding to E, we define an odd super-differential operator Notice that due to the last term in (8.11), the super-differential operator P M is i.g. not formally super-self adjoint with respect to the pairing (8.9). If however β = 0, then P M is formally super-self adjoint. Comparing with (8.8), the constraint β = 0 can be regarded as a supertorsion constraint which demands that the odd part of the supertorsion vanishes. Such constraints also appear in supergravity, see e.g. [WZ77, Eqns. (10) and (11)].
The category SLoc: We define a full category SLoc of ghSCart by the conditions that 1.) the underlying topological spaces are connected and 2.) all supergravity supertorsion constraints given in [WZ77, Eqns. (10) and (11)] hold true, which implies that β = 0 and that α is a constant which we fix to α = 1. Then the admissible super-Cartan supermanifolds M = (M, Ω = 0, E) are such that M = (t 0 , t 1 ) ⊆ R is an open interval (or R itself) and in geometric coordinates. The morphisms in SLoc can be explicitly characterized: Let M and M ′ be any two objects in SLoc. A SMan-morphism χ : M → M ′ is specified by its action on the (geometric) coordinates t ′ , θ ′ ∈ O(M ′ ), which we can parametrize by the ansatz where a, b ∈ C ∞ (t 0 , t 1 ). In order to qualify as a SLoc-morphism, χ has to preserve the supervielbeins which implies that a(t) = t + c, with c ∈ R, and b(t) ≡ 1. Furthermore, for the reduced morphism χ : M = (t 0 , t 1 ) → M ′ = (t ′ 0 , t ′ 1 ) to exist, the constant c ∈ R has to be such that It remains to show that (8.11) are the components of a natural transformation of formally super-self adjoint and super-Green's hyperbolic superdifferential operators. For any object M in SLoc the super-differential operator (8.11) takes the form for all F = f + θ h ∈ O(M ), from which it is clear that it is formally super-self adjoint and super-Green's hyperbolic with super-Green's operators given by and G ± ∂t denote the retarded/advanced Green's operators for the component differential operators ∂ 2 t and ∂ t , respectively. The super-differential operators (8.16) are the components of a natural transformation since they are translation invariant, hence we have constructed an example of a super-field theory according to Definition 4.2. Applying Theorem 5.11 we further obtain a super-QFT, which in the 1|1-dimensional case describes a quantized superparticle.
Enriched super-field theory: We shall now show that the super-field theory defined above is also an enriched super-field theory according to Definition 6.15. Let us take any two objects M and M ′ in SLoc op and any object pt n in SPt op . We characterize explicitly the set eSLoc op (M ′ , M )(pt n ). Any SMan/pt n -morphism χ : M/pt n → M ′ /pt n is specified by its action on the (geometric) coordinates ½ ⊗ t ′ , ½ ⊗ θ ′ ∈ Λ n ⊗ O(M ′ ), which we can parametrize by the ansatz where ζ I is any adapted basis for the super-vector space underlying the Grassmann algebra Λ n and a I , b I , c I , d I ∈ C ∞ (t 0 , t 1 ). For χ ∈ eSLoc op (M ′ , M )(pt n ) it has to preserve the pt n -relative supervielbeins, i.e. χ * (½ ⊗ E ′ ) = ½ ⊗ E. The odd part of this condition reads as and it implies that χ * (½ ⊗ θ ′ ) = ½ ⊗ θ + ζ ⊗ ½, for some odd element ζ ∈ Λ n . Using this result, the even part of the above condition reads as which implies that χ * (½ ⊗ t ′ ) = ½ ⊗ (t + c) − ζ ⊗ θ. As in the ordinary case, the constant c ∈ R has to satisfy t ′ 0 − t 0 ≤ c ≤ t ′ 1 − t 1 for the reduced morphism χ : M = (t 0 , t 1 ) → M ′ = (t ′ 0 , t ′ 1 ) to exist. Hence, we have shown that where (Λ n ) 1 is the odd part of the Grassmann algebra Λ n = (Λ n ) 0 ⊕ (Λ n ) 1 . In particular, the pt n -relative automorphisms eSLoc op (M , M )(pt n ) are in bijective correspondence with R × (Λ n ) 1 if M = R and with (Λ n ) 1 if M ⊂ R is a proper interval. These pt n -relative automorphisms describe ordinary and supertranslations. For a generic which reproduces the usual supersymmetry transformations. The diagram (6.30) commutes, since using (8.22) one can easily compute that , for any object pt n in SPt op , which can be parametrized by the odd elements ζ ∈ (Λ n ) 1 . Explicitly, on the generators where Q := ∂ θ − θ ∂ t is the generator of supersymmetry transformations and ζ ∈ (Λ n ) 1 . On the superalgebra of observables A(M ) itself, these Λ C n -S * Alg-automorphisms can be understood as an odd superderivation Q : and it satisfies the superderivation property for all homogeneous F 1 , F 2 ∈ O c (M ), as a consequence of (eA M ,M ) pt n (χ) being a Λ C n -S * Algmorphism. We may decompose Φ M into its component quantum fields (careful: neither ψ M nor φ M are natural in the enriched setting) via for all F = f + θ h ∈ O c (M ), from which we recover the usual supersymmetry transformations for all f, h ∈ C ∞ (t 0 , t 1 ).

3|2-dimensions
Our second example is the free Wess-Zumino model in 3|2-dimensions. Once again we provide all the data listed in Definition 4.2.
Representation theoretic data: We take W = R 3 together with the standard Lorentz metric g = diag(1, −1, −1). The corresponding spin group is Spin(W, g) ≃ SL(2, R) and we take ρ S : Spin(W, g) × S → S the defining representation of SL(2, R) on S = R 2 . Notice that SL(2, R) is the two-fold cover of the identity component SO 0 (1, 2) of the special pseudoorthogonal group and that ρ W : Spin(W, g) × W → W is given by composing the covering map with the defining representation of SO 0 (1, 2).
The standard basis {p α } of W = R 3 , with α = 0, 1, 2, is an orthonormal basis of (W, g). We denote the metric coefficients in this basis by g αβ := g(p α , p β ) and the coefficients of the inverse metric in the dual basis {p α } by g αβ . Elements w ∈ W will be indicated by w = w α p α , with summation over repeated indices understood. For the coefficients of the dual vector g(w, · ) = w α g αβ p β ∈ W * we shall also write w β = w α g αβ . Notice that w α = w β g βα . The choice of basis {p α } fixes an orientation on W .
The representation ρ S induces up to SL(2, R)-equivalence a unique representation of the Clifford algebra Cl(W, g) in terms of purely imaginary matrices γ α , with α = 0, 1, 2, on the complexification S C of S. These matrices thus satisfy the Clifford relations γ α γ β + γ β γ α = 2 g αβ id S C . (8.28) Furthermore, there exists a charge conjugation matrix C which satisfies where T denotes matrix transposition. A possible representation is given in terms of the Pauli matrices by γ 0 = σ 2 , γ 1 = i σ 1 , γ 2 = i σ 3 and C = γ 0 . We define the antisymmetrized product and note the identities where ǫ αβδ is totally antisymmetric with ǫ 012 = 1 and L is an arbitrary endomorphism of S C .
As the pairing Γ we shall take Γ : S ⊗ S −→ W , s 1 ⊗ s 2 −→ (s 1 , γ α C −1 s 2 ) p α , (8.32) where ( · , · ) is the standard inner product on S = R 2 . (Notice that γ α C −1 is a real matrix.) The pairing Γ is symmetric, Spin(W, g)-equivariant and positive with respect to the forward light cone C = {w ∈ W : g(w, w) > 0 and w 0 > 0}. For ǫ we take which is Spin(W, g)-invariant, antisymmetric and non-degenerate. We choose a symplectic basis {q a } of S = R 2 , with a = 1, 2, and use the index notation ǫ ab := ǫ(q a , q b ) with ǫ 12 = −ǫ 21 = 1. We further set ǫ ab := ǫ ab for the coefficients of the symplectic structure on the dual vector space S * in the dual basis {q a } and notice that ǫ ab ǫ bc = −δ a c . Elements s ∈ S will be indicated by s = s a q a , with summation over repeated indices understood. For the coefficients of the dual spinor ǫ(s, · ) = s a ǫ ab q b ∈ S * we shall also write s b = s a ǫ ab . Notice that s a = s b ǫ ab . The choice of basis {q a } fixes an orientation on S. The two pairings Γ and ǫ read in our bases as Γ(s 1 , s 2 ) = −s a 1 s b 2 i γ α ab p α , ǫ(s 1 , s 2 ) = s a 1 s b 2 ǫ ab . (8.34) In order to state explicitly the super-Lie bracket relations in the super-Poincaré super-Lie algebra sp = (spin ⊕ W ) ⊕ S corresponding to this choice of data, we recall that the Lie algebra spin may be spanned by generators L αβ = −L βα , with α, β = 0, 1, 2, and that the Lie algebra actions induced by ρ W and ρ S read as ρ W * : spin ⊗ W −→ W , L αβ ⊗ p γ −→ g αγ p β − g βγ p α , The objects in ghSCart: We characterize explicitly the objects M = (M, Ω, E) in the category ghSCart for the case of 3|2-dimensions under the following simplifying assumptions: As in the 1|1-dimensional case, we assume that the underlying topological spaces M are connected and furthermore that the structure sheaf O M is globally isomorphic to C ∞ The category SLoc: We define a full subcategory SLoc of ghSCart by the conditions that 1.) the underlying topological spaces are connected, 2.) the structure sheaves are globally isomorphic to C ∞ M ⊗ • R 2 and 3.) the supergravity supertorsion constraints given in [WZ77, Eqns. (10) and (11)] hold true. In order to discuss the latter constraints, we consider an arbitrary (local) coordinate system x µ on M , with µ = 0, 1, 2, and use the notation X M for the combined super-coordinate system {x µ , θ m }, where θ m are global geometric odd coordinates. (With abuse of notation we will denote the indices on X M by the same symbol as the supermanifold.) We set |M | := |X M | for the Z 2 -parity of X M . Analogously, we use the notation P A for the combined generators {p α , q a } of the supertranslation super-Lie algebra t = W ⊕ S and set |A| := |P A |. Consequently, we can expand the supervielbein as E = E A ⊗ P A = dX M E A M ⊗ P A and its inverse as E A = E M A ∂ M . Using this notation, we may expand and compute the supertorsion (3.7) Here, Ω β α and Ω b a are defined, for arbitrary w = w α p α ∈ W and s = s a q a ∈ S, as ρ W * (Ω ⊗ w) =: w α Ω β α ⊗ p β , ρ S * (Ω ⊗ s) =: s a Ω b a ⊗ q b . (8.43) Note that the signs in (8.42b) are only correct if E is even.
For any object M in SLoc and any constant m ≥ 0, we define the super-differential operator then the retarded/advanced super-Green's operators G ± M for P M can be written as where G ± +m 2 and G ± i ∇+m are the retarded/advanced Green's operators for the component differential operators + m 2 and i ∇ + m, respectively. Finally, the super-differential operators (8.52) are the components of a natural transformation since they are constructed geometrically in terms of the supervielbein E and a constant m ≥ 0. Hence, we have constructed an example of a super-field theory according to Definition 4.2. Applying Theorem 5.11 we further obtain a super-QFT, which in the present case describes the quantized free Wess-Zumino model in 3|2-dimensions.
Enriched super-field theory: We shall now discuss the super-field theory defined above in the enriched setting. We consider two objects M and M ′ in eSLoc op and any object pt n in SPt op . Before discussing the set eSLoc op (M ′ , M )(pt n ) in more detail, we remark that, since any χ ∈ eSLoc op (M ′ , M )(pt n ) preserves by definition the pt n -relative supervielbeins, and P M is constructed geometrically, the super-field theory discussed in this example automatically satisfies the axioms of an enriched super-field theory given in Definition 6.15.
Instead of fully characterizing the set eSLoc op (M ′ , M )(pt n ), we aim for analyzing a presumably large subset which contains supersymmetry transformations by considering a wellmotivated ansatz. For a generic F ′ ∈ O c (M ′ ) expanded as in (8.46), we consider SMan/pt nmorphisms χ : M/pt n → M ′ /pt n of the form Thus, we find that a non-zero Q is only possible if the Levi-Civita connection ω is vanishing, and hence also the super-spin connection Ω is vanishing on M . In this case χ defined as above is an element of eSLoc op (M ′ , M )(pt n ) if and only if the super-spin connection Ω ′ on M ′ is also vanishing.
This rather restrictive condition for the existence of interesting enriched morphisms χ ∈ eSLoc op (M ′ , M )(pt n ) originates from our requirement that the supervielbein E is even and that χ must preserve the pt n -relative supervielbein. In the treatment of supergravity one usually deals with supervielbeins which are not purely even and considers, in the terminology of this paper, enriched morphisms which have to preserve the pt n -relative supervielbein and pt n -relative super-spin connection only up to local Lorentz transformations. This class of enriched morphisms contains the so-called supergravity transformations [WB92].
Supersymmetry transformations in the enriched super-QFT: We close the discussion of this example in analogy to the 1|1-dimensional case by illustrating the structure of supersymmetry transformations. As discussed above, these transformations appear only if we consider an object M = (M, Ω, E) in eSLoc with Ω = 0, such as for example the 3|2-dimensional super-Minkowski space. Given any such object, we can use the SSet-functor eA : eSLoc → eS * Alg constructed in Theorem 7.11 to obtain a superalgebra of observables A(M ) and a SSet-morphism