Wavefront sets and polarizations on supermanifolds

In this paper we develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable notion of super wavefront set for superdistributions which generalizes Dencker's polarization sets for vector-valued distributions to supergeometry. In particular, our super wavefront sets detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criteria when two superdistributions may be multiplied. As an application of our framework, we study the singularities of distributional solutions of a supersymmetric field theory.


I. INTRODUCTION AND SUMMARY
Supergeometry has its origins in theoretical physics, where it is used as a refined model of spacetime that treats Bosonic and Fermionic degrees of freedom on an equal footing. The basic concept is that of a supermanifold, which loosely speaking is a manifold with even (Bosonic) and odd (Fermionic) local coordinates. Quantum field theories on supermanifolds unify Bosonic and Fermionic quantum fields in a single entity called a super quantum field.
They are very interesting from the perspective of a quantum field theorist because of their improved renormalization behavior. Such special features of supergeometric quantum field theories are collectively called non-renormalization theorems 7,14 .
During the last decade, our mathematical understanding of perturbative quantum field theory on Lorentzian manifolds has steadily improved, mainly due to the development of perturbative algebraic quantum field theory (pAQFT), see e.g. Ref. 5 for a recent review. In this framework, a key role is played by the class of Hadamard states, which are distinguished from a physical viewpoint since they share the same ultraviolet behavior of the Minkowski vacuum and they yield finite quantum fluctuations of all observables. From a mathematical perspective, they are defined in terms of a prescribed singular structure of the truncated two-point function associated to the state 12 . Hence, in this respect, microlocal analysis serves as one of the main techniques used in pAQFT since its role is to analyze carefully the singularities of distributions like propagators and n-point functions. This proves essential not only for identifying Hadamard states but also for performing the perturbative construction and its renormalization.
The goal of this paper is to develop the foundations of microlocal analysis on supermanifolds. Our work is based on and extends earlier investigations of Rempel and Schmitt 13 on pseudodifferential operators on supermanifolds. As a new development, we introduce a supergeometric generalization of the wavefront set, which is a suitable concept to encode polarization information about the singularities of distributions on supermanifolds. See also Ref. 4 for a first work in this direction, which however discards the polarized character of superdistributions. Our super wavefront sets are motivated by the polarization sets of Dencker 3 for vector-valued distributions. However, they are constructed in such a way that they transform in a natural way under supermanifold morphisms and not only vector bun-dle morphisms. The techniques which we develop in this paper will be the basis to identify and to construct Hadamard states in the context of quantum field theories on supermanifolds. As mentioned before, these are characterized by a prescribed singular behavior of the associated, truncated two-point function and they are the building block for a covariant construction of Wick-polynomials. The latter are then used to introduce interaction terms within the perturbative framework. Hence, the results of this paper are expected to play a major role in extending pAQFT to supergeometric quantum field theories 9 , a longer term research goal that we hope to achieve in future works. This would provide a rigorous framework to prove (and extend to curved supermanifolds) the non-renormalization theorems in Refs. 7 and 14.
The outline of the remainder of this paper is as follows: In Section II we fix our notations and give a brief review of some basic aspects of the theory of supermanifolds. In Section III we assign to each supermanifold X = ( X, O X ) a polarization bundle π : P * X → T * X over the cotangent bundle of the underlying smooth manifold X; this is a super vector bundle that encodes the local polarization information of superfunctions and superdistributions on X. Our polarization bundle is a special case of the general construction by Rempel and Schmitt in Section 8 of Ref. 13: It corresponds to a particular choice of what they call "admissible tuple", which is strongly motivated by the fact that it enables us to detect ellipticity and hyperbolicity of the operators appearing in supergeometric field theories, see Examples IV.7 and IV.8. In Section IV we introduce super pseudodifferential operators on supermanifolds, define their super principal symbols as bundle mappings between the polarization bundles, and develop their calculus. The main definitions in this section are taken from Ref. 13 (see in particular Sections 7 and 8), which we however can present in a simplified form because of our particular choice of "admissible tuple" for the polarization bundles. We also present examples of super pseudodifferential operators which are relevant for physics, in particular the equation of motion operators (and their associated propagators) of the supergeometric field theories studied in Ref. 9. Crucially, as we have already indicated above, our concept of super principal symbols is able to detect ellipticity (or hyperbolicity) of these operators. As our first genuinely new result, we introduce in Section V polarization sets for supermanifolds, motivated by Ref. 3, and thereby define the super wavefront set of a superdistribution. We analyze the transformation property of the super wavefront set under supermanifold morphisms and their compatibility with the action of super pseudodifferential operators. In Section VI we generalize to supermanifolds the ordinary pullback theorem for distributions on manifolds, see Theorem 8.2.4 in Ref. 10. By including the polarization information of superdistributions (and their singularities), this leads to a refinement of the ordinary pullback theorem. An important example is given by the super diagonal mapping, which provides criteria when two superdistributions may be multiplied. As an application, we analyze in Section VII the singularities of distributional solutions to the equation of motion of the 3|2-dimensional Wess-Zumino model.

II. PRELIMINARIES
We briefly recall some basic aspects of the theory of supermanifolds which are frequently used in our work. For a detailed introduction to this subject see, for example, Refs. 1 and 2 and also Section 2 in Ref. 9 for a short summary.
A superspace is a pair X = ( X, O X ) consisting of a topological space X (second-countable and Hausdorff) and a sheaf of supercommutative superalgebras O X on X, called the struc- where U αβ := U α ∩ U β is the intersection, there exists a unique section f ∈ O X (U ) such that f α = res U,Uα (f ). Loosely speaking, this means that a family of local sections of O X which match in all overlaps can be glued to a unique global section and that any global section arises in that way.
The standard example of a superspace is R m|n := (R m , C ∞ R m ⊗ ∧ • R n ), where ∧ • R n denotes the Grassmann algebra with n generators. The sections over any open U ⊆ R m are given where Z n 2 := {0, 1} n , {θ a ∈ R n : a = 1, . . . , n} is the standard basis of R n and f I ∈ C ∞ (U ).
is a pair ( χ, χ * ) consisting of a continuous map χ : X → Y and a sheaf homomorphism A supermanifold (of dimension m|n) is a superspace X = ( X, O X ) which is locally isomorphic to R m|n . More explicitly, this means that for any point x ∈ X there exists an open neighborhood U ⊆ X of x such that X| U := (U, O X | U ) is isomorphic as a superspace to , for some open subset W ⊆ R m . We say that χ : X → Y is a morphism between two supermanifolds X = ( X, O X ) and Y = ( Y , O Y ) if it is a superspace morphism.
Every supermanifold X = ( X, O X ) comes together with a filtration for any open U ⊆ X, where is the superideal of nilpotents and J k X (U ) is its k-th power, k ≥ 2. Locally, i.e. for sufficiently small U ⊆ X, by definition there exists an isomorphism O X (U ) superalgebras for some open W ⊆ R m . Applying this isomorphism to the filtration (II.5) we obtain which implies that locally J k X (U ) = 0 for all k > n. Indeed, in this case C ∞ (W )⊗∧ ≥k R n = 0. Due to the sheaf condition the same statement holds globally, i.e. J k X (U ) = 0 for all k > n and U ⊆ X open.
Let us also recall that to any m|n-dimensional supermanifold X = ( X, O X ) there is canonically assigned an m-dimensional manifold; it is specified by the topological space X together with the structure sheaf O X /J X . The underlying continuous map χ : X → Y of any supermanifold morphism χ : X → Y is smooth with respect to this manifold structure.
The supermanifold morphism ι X,X : ( X, O X /J X ) → ( X, O X ), given by ι X,X = id X and the quotient mapping ι * X,X : O X → O X /J X , embeds the underlying smooth manifold into the supermanifold.

III. POLARIZATION BUNDLES
The space of superdistributions on a supermanifold X is locally given by where U ⊆ R m is an open subset and D (U ) denotes the space of distributions on U . Hence, superdistributions locally carry polarization information in the Grassmann algebra ∧ • R n .
We now construct a bundle over the cotangent bundle T * X of the underlying manifold X, which describes the polarization information of superdistributions and their singularities.
Our construction in this section is a special case of the general construction by Rempel and Schmitt in Section 8 of Ref. 13.
Let us start with the case where the supermanifold is a superdomain, i.e. U m|n := (U, C ∞ U ⊗ ∧ • R n ) ⊆ R m|n for some open U ⊆ R m . In this case the polarization bundle is defined as the trivial bundle where the fibers are the complexified Grassmann algebras and T * U = U × R m is the cotangent bundle over U .
Now consider a supermanifold morphism χ : U m|n → V m |n between two superdomains.
The underlying smooth map χ : U → V induces a fiber-wise pullback map T * χ : T * χ(x) V → T * x U of cotangent vectors, for any point x ∈ U . Our goal is to construct a suitable fiber-wise map between the polarization bundles such that the diagram commutes, for any point x ∈ U .
To approach this problem, we have to analyze in more detail the superalgebra homomor- is the pullback of functions along the underlying smooth map χ : U → V . We now show that the other components (χ * V ) j i are relative differential operators along χ * . Recall, e.g. from Theorem 4.1.11 in Ref. 1, that the superalgebra homomorphism χ * V is uniquely specified by its action on the supercoordinates (y µ , ζ a ) of V m |n . We have that where J k U m|n (U ) is the filtration explained in (II.5), see also (II.7). For a generic f ∈ C ∞ (V ) ⊗ ∧ • R n , we use the component expansion f = I∈Z n 2 f I ζ I and obtain Using the first property in (III.5) and Taylor expansion in the odd coordinates, we observe that where Q l is a differential operator of order l and λ 2l ∈ ∧ 2l R n . Using also the second property in (III.5) and the fact that the odd coordinates θ a on U m|n are nilpotent, we obtain Here (D χ ) j i are matrices of differential operators of order j−i 2 . In summary, we have shown that, for any supermanifold morphism χ : U m|n → V m |n between two superdomains, the corresponding superalgebra homomorphism χ * V can be factorized uniquely as where D χ is a matrix of differential operators.
We now define the mapping P * χ in (III.2) component-wise by where σ l denotes the principal symbol of a differential operator of order l.
Given now two supermanifold morphisms χ : and hence for the non-vanishing components of ((χ • χ) * W ) j i . Combining this with (III.10) and the multiplicativity of principal symbols, it is easy to check that the polarization mapping in Because of this result, the concept of polarization bundle globalizes from superdomains to supermanifolds: Let X = ( X, O X ) be any m|n-dimensional supermanifold and choose an open cover {U α ⊆ X} and isomorphisms to superdomains, i.e. a superatlas. In all overlaps U αβ := U α ∩ U β this gives rise to transition supermanifold morphisms which satisfy χ αα = id Wα m|n for all α as well as the cocycle condition χ βγ • χ αβ = χ αγ on all triple overlaps U αβγ := U α ∩ U β ∩ U γ . In any superchart W α m|n we take the trivial polarization bundle P * W α m|n from (III.1). The global polarization bundle P * X on the supermanifold X is then given by gluing these local bundles via the transition functions g αβ := P * χ βα ; the cocycle condition for the g αβ follows from (III.13). It is important to stress that, even though the local polarization bundles (III.1) look like Grassmann algebra bundles, the transition functions g αβ in general do not preserve the product structure and the Z-grading on the fibers -note the outer-diagonal terms in (III.10), which depend on k. However, the coarser Z 2 -grading on the fibers of the local bundles is preserved by the transition functions. Hence the polarization bundle π : P * X → T * X is a complex super vector bundle for any supermanifold X = ( X, O X ).

IV. SUPER PSEUDODIFFERENTIAL OPERATORS
We introduce super pseudodifferential operators on supermanifolds and define their super principal symbols. As in the case of a manifold, the definition is local, and we first consider the case where the supermanifold is a superdomain U m|n ⊆ R m|n . The main definitions in this section are taken from Ref. 13 (see in particular Sections 7 and 8). However, we will study the properties of super pseudodifferential operators in more detail and also provide interesting examples from supergeometric field theory.

A linear map
In the following all pseudodifferential operators are implicitly assumed to be properly supported and classical, see e.g. Ref. 15 for the relevant definitions. Recall, in particular, that properly supported pseudodifferential operators map compactly supported functions to compactly supported functions, hence they can be composed. The composition is again a properly supported pseudodifferential operator. Given any supermanifold isomorphism χ : U m|n → V m|n and a super pseudodifferential operator A on U m|n , consider the linear map It defines a super pseudodifferential operator on V m|n because the components of χ * V and its inverse are both (matrices of) relative differential operators, cf. (III.8).
Definition IV.1. We say that a super pseudodifferential operator A on U m|n is of order l if its components A j i are (matrices of) pseudodifferential operators on U of order j−i 2 + l, i.e., The super principal symbol of A ∈ sΨDO l (U m|n ) is the super vector bundle map with components given by Example IV.2. Let χ : U m|n → V m|n be a supermanifold isomorphism between two superdomains, and consider the unique factorization χ * V = χ * • D χ given in (III.8). Then D χ is a super pseudodifferential operator of order 0, i.e. D χ ∈ sΨDO 0 (V m|n ). In the case where U = V and χ = id U , the super principal symbol of D χ is the polarization mapping (III.10), We collect some useful properties of super pseudodifferential operators and their super principal symbols. The proofs of these statements follow easily from our definitions and are omitted.
Lemma IV.3. Let A ∈ sΨDO l (U m|n ) and B ∈ sΨDO l (U m|n ). Then the following statements hold true: Lemma IV.4. Let A ∈ sΨDO l (U m|n ) and B ∈ sΨDO l (U m|n ). Then the following statements hold true: Super pseudodifferential operators and their super principal symbols are easily globalized to supermanifolds by slightly adapting the globalization procedure for the pseudodifferential operators on manifolds, see e.g. Chapter I, Section 5 in Ref. 16. Let X = ( X, O X ) be an m|n-dimensional supermanifold and O X,c ( X) the space of compactly supported global sections of the structure sheaf. Consider a maximal superatlas ρ α : X| Uα → W α m|n . A super pseudodifferential operator A ∈ sΨDO l (X) of order l on X is a continuous linear map is an element in sΨDO l (W α m|n ). Here ext denotes the extension (by zero) maps for compactly supported sections. To each A ∈ sΨDO l (X) we associate a super principal symbol, which is a super vector bundle morphism To study the singularities of distributions, the notion of ellipticity is crucial.
Definition IV.5. We say that a super pseudodifferential operator E ∈ sΨDO l (X) is elliptic if the super principal symbol σ l (E) is invertible on T * X \ 0.
Many properties of elliptic pseudodifferential operators on ordinary manifolds are still valid in our framework. In particular, we obtain Lemma IV. 6. Let E ∈ sΨDO l (X) be an elliptic super pseudodifferential operator. Then there exists a super pseudodifferential operator F ∈ sΨDO −l (X) such that where sΨDO −∞ (X) := l∈R sΨDO l (X). F is called a parametrix for E.
Proof. The proof is as in the case of ordinary manifolds, see e.g. Theorem 5.1 in Ref. 15.
We shall now give examples of super differential and super pseudodifferential operators A ∈ sΨDO l (X) which have their origin in supersymmetric field theory.
Example IV.7. Let X = R 1|1 be the superline. The dynamics of a superparticle on X is governed by a super differential operator, which in global supercoordinates (t, θ) on R 1|1 reads as cf. Section 8.1 in Ref. 9. In our component notation, the operator P is given by Notice that P ∈ sΨDO 3 2 (R 1|1 ). Its super principal symbol is invertible for all (t, k) ∈ T * R \ 0, hence P is elliptic. Specifically, the inverse is In this case a parametrix F of P from Lemma IV.6 is explicitly given by the integral kernel operator on M and m ≥ 0 is a mass term. Notice further that the component notation in (IV.15) is in block-matrix form, because ∧ 1 R 2 R 2 is two-dimensional; in particular, the Dirac operator is a 2 × 2-matrix of differential operators. The operator P ∈ sΨDO 1 (X) is of order 1, and in local coordinates x µ and k µ on T * M its super principal symbol is given by Using the Clifford algebra relations {γ µ , γ ν } = 2 g µν for the gamma-matrices, it is easy to check that σ 1 (P )(x, k) is invertible for all (x, k) ∈ T * M \ 0 which are not light-like (i.e. k µ k ν g µν (x) = 0). More explicitly, we have

V. SUPER WAVEFRONT SETS
We start with the case where the supermanifold is a superdomain U m|n ⊆ R m|n . Then the space of superdistributions D (U ) ⊗ ∧ • R n is the dual of C ∞ c (U ) ⊗ ∧ • R n , and both C ∞ c (U ) ⊗ ∧ • R n and C ∞ (U )⊗∧ • R n are dense sub-spaces. We say that a superdistribution u ∈ D (U )⊗ We define the super wavefront set of a superdistribution on X motivated by the approach of Dencker 3 for vector-valued distributions. The starting point is the polarization bundle π : P * X → T * X introduced in Section III. We denote by π : P * X := π −1 T * X \ 0 −→ T * X \ 0 (V.1) the restriction of the polarization bundle to the cotangent bundle with the zero-section removed.
Definition V.1. The super wavefront set (of order l) of a superdistribution u ∈ D (U ) ⊗ ∧ • R n is defined as the intersection We collect some important properties of the super wavefront sets defined above.
Proposition V.2. For any u ∈ D (U ) ⊗ ∧ • R n , the following properties hold true: where π : P * U m|n → T * U \ 0 is the projection (III.1) and WF(u I ) ⊆ T * U \ 0 denotes the ordinary wavefront set of u I ∈ D (U ).
Proof. To show item a), take any (x, k, λ) ∈ sWF l (u). By assumption there exists A ∈ sΨDO l (U m|n ) such that Au smooth and σ l (A)(x, k) λ = 0. Composing this A with any elliptic super pseudodifferential operator E ∈ sΨDO l −l (U m|n ) of order l − l, Hence, (x, k, λ) ∈ sWF l (u), which completes the proof.
Item b): We prove the inclusion "⊆" by contradiction. Suppose that there exists WF(u I ). The latter condition implies that, for each I ∈ Z n 2 , there exists A I ∈ ΨDO l (U ) such that A I u I is smooth and σ l (A I )(x, k) = 0. We define A ∈ sΨDO l (U m|n ) by placing the A I in their corresponding diagonal entry of the matrix and setting all other entries to zero. By construction, we have that Au is smooth and that the super principal symbol σ l (A)(x, k) is invertible. This implies that λ = 0 and leads to a contradiction.
Then there exists A ∈ sΨDO l (U m|n ) such that Au is smooth and σ l (A)(x, k) is invertible at (x, k). Thus, by a straightforward refinement of Lemma IV.6, as in Proposition 6.9 in Ref. 16 we construct a microlocal parametrix F ∈ sΨDO −l (U m|n ). From the existence of this microlocal parametrix F we conclude that all components u I of u are smooth at (x, k).
Hence (x, k) / ∈ I∈Z n 2 WF(u I ), which is a contradiction. Proof. The statement is a special instance of (V.3).
Example V.5. Let us consider the superdomain U m|2 and the superdistribution where v ∈ D (R m ) is an ordinary distribution and 0 denotes the zero vector in ∧ 1 R 2 R 2 according to our block-matrix component notation. Then the super pseudodifferential Loosely speaking, this shows that our notion of super wavefront sets both picks out the leading singularities to determine the polarization and assigns a higher weight to the components of a superdistribution with a lower number of θ-powers. Notice that this is a direct consequence of our definition of orders and super principal symbols for super pseudodifferential operators in Definition IV.1. Hence this feature generalizes to superdomains in higher odd-dimensions U m|n .
The super wavefront set of a superdistribution behaves well with respect to the action of super pseudodifferential operators.
Proposition V.6. Let u ∈ D (U ) ⊗ ∧ • R n and A ∈ sΨDO l (U m|n ). Then where the equality holds true whenever A is elliptic.
If A is elliptic, we use Lemma IV.6 to obtain an elliptic F ∈ sΨDO −l (U m|n ), such that . Equality in (V.8) is then shown by replacing the role of u with Au and that of A with F .
Remark V.7. More generally, equality in (V.8) holds true microlocally above any point Given any supermanifold isomorphism χ : U m|n → V m|n , the fibre-wise polarization mapping from (III.10) defines a super vector bundle isomorphism We now show that the super wavefront sets transform well under supermanifold isomorphisms.

(V.10)
Proof. This is a direct consequence of Lemma IV.4 b).
This transformation property of the super wavefront set under the action of all supermanifold isomorphisms allows us to globalize super wavefront sets from superdomains to supermanifolds: Let u be a superdistribution on a supermanifold X = ( X, O X ). We use a superatlas ρ α : X| Uα → W α m|n and describe u in terms of a family of local superdistributions u α ∈ D (W α ) ⊗ ∧ • R n , which satisfy the gluing conditions on all overlaps U αβ . Here χ βα are the transition supermanifold morphisms. The super wavefront set of u is then obtained by gluing all subsets sWF(u α ) ⊆ P * W α m|n via the transition functions g αβ = P * χ βα of the polarization bundle. Proposition V.8 guarantees that this construction defines a global super wavefront set sWF(u) ⊆ P * X.

VI. PULLBACK AND MULTIPLICATION THEOREMS
Given a generic supermanifold morphism χ : X → Y , we cannot pull back a generic superdistribution u on Y to a superdistribution on X. However, depending on the explicit form of χ, certain superdistributions u on Y may admit a (unique) pullback to X. It is the goal of this section to develop a suitable criterion to select a class of superdistributions which admit a pullback. Let us now consider a supermanifold morphism χ : U m|n → V m |n between two superdomains. The case of a generic supermanifold morphism χ : X → Y between two supermanifolds follows from this by localizing χ in suitable superatlases of X and Y . Recalling that χ * V admits a unique factorization (III.8) into a matrix of differential operators D χ and the component-wise pullback χ * along the underlying smooth map, we analyze the pullback of superdistributions in two steps: Given any superdistribution u ∈ D (V ) ⊗ ∧ • R n on V m |n , the first step is to act with the differential operator D χ on u, which is always well-defined and results in an auxiliary superdistribution where the components are now in the Grassmann algebra ∧ • R n with n generators. In the second step, we would like to pull back D χ u along χ * . However, this operation is not always well-defined. If we assume the condition has a unique continuous extension to those superdistributions u ∈ D (V ) ⊗ ∧ • R n which satisfy the condition (VI.3).
Remark VI.2. Another condition which would guarantee the existence of χ * V u is given by In fact, using Proposition V.2, the condition (VI.6) is equivalent to the strong condition WF(u J ) ∩ N χ = ∅ for all components u J . Because differential operators preserve wavefront sets, it follows that for any I, which implies (VI. 3). Notice that the condition (VI.6) is much coarser than our condition (VI.3). Loosely speaking, it does not take into account those components of u which "vanish algebraically under pullback" due to the differential operator D χ . Let us illustrate this important point by an example: Consider the supermanifold morphism χ : { * } → U m|n which maps a point into the superdomain U m|n . Then is the mapping which "forgets" all higher components in the Grassmann algebra and evaluates the lowest component at the point χ( * ) ∈ U . We can clearly extend χ * U to all superdistributions D (U ) ⊗ ∧ • R n with smooth lowest component u (0,...,0) ∈ C ∞ (U ) by setting u = I∈Z n 2 u I θ I −→ u (0,...,0) ( χ( * )) . (VI.9) Because N χ = T * χ( * ) U is the cotangent space at χ( * ), the condition (VI.6) is violated as soon as any u I has a singularity at this point. In contrast, our condition (VI.3) just involves the lowest component u (0,...,0) of the superdistribution, because the matrix of differential operators reads as D χ = 1 0 · · · 0 and hence D χ u = u (0,...,0) .
In the remaining part of this section we specialize the result of Theorem VI.1 to the important case where χ is the super diagonal mapping (VI.10) The underlying smooth map ∆ : U → U × U , x → (x, x) is the diagonal map and ∆ * U ×U : where µ : ∧ • R n ⊗ ∧ • R n → ∧ • R n denotes the product in the Grassmann algebra ∧ • R n . The normal set of ∆ can be characterized explicitly and it is given by Given two superdistributions u, v ∈ D (U ) ⊗ ∧ • R n , their product (if it exists) is given by Expanding into components u = I∈Z n 2 u I θ I and v ∈ J∈Z n 2 u J θ J , we Due to the factorization (VI.11), the product of u and v (if it exists) is computed by first multiplying in the Grassmann algebra and then pulling back the result component-wise via ∆ * , i.e.
As a consequence of Theorem VI.1, we have 16) or equivalently, whenever all components u I , v J ∈ D (U ), for which θ I θ J = 0, can be multiplied in the sense of ordinary distributions, cf. Theorem 8.2.4 in Ref. 10.
Remark VI.4. It is important to stress that the condition (VI.16) in the corollary above does not impose conditions on the components u I and v J which multiply trivially on account of the Grassmann algebra structure, i.e. for which θ I θ J = 0. This is a clear advantage compared to the alternative (and much coarser) condition (VI.6).

VII. SINGULARITIES IN SUPERGEOMETRIC FIELD THEORY
In this section we apply the techniques developed in this paper to analyze the singularities of the supergeometric field theory introduced in Example IV.8. For simplifying our explicit computations, we consider only the case where M = R 3 is the Minkowski spacetime, i.e.
We next observe that the composition P •P of (VII.1) with the super (pseudo-)differential operator (of order 1) gives the component-wise Klein-Gordon equation In particular, each component u I of any u satisfying P u = 0 satisfies the Klein-Gordon equation Qu I = 0, which entails the following inclusion i.e. any integral curve c : R → Ω Q of H Q which satisfies c(0) ∈ WF(u I ) remains in WF(u I ).
Following the ideas of Dencker 3 , we now shall study the propagation of polarizations in our example. Given any integral curve c : R → Ω Q of H Q as in (VII.9), we consider the restriction of N P given in (VII.3) to c, which gives rise to a vector bundle N P | c −→ R . (VII.10) Using (VII.4), we can compute its total space N P | c = s, φ + ψ θ : γ µ k µ ψ = 0 . (VII.11) As the solution space of the Dirac-constraint γ µ k µ ψ is one-dimensional (in 3 dimensions), the vector bundle N P | c → R is of rank two. Following Definition 4.1 in Ref. 3, a Hamiltonian orbit for our operator P is a sub-line bundle L ⊆ N P | c , where c is an integral curve as above and L is spanned by a section w ∈ Γ ∞ (N P | c ) that satisfies D P w = 0. Here D P := H Q + 1 2 {σ 1 ( P ), σ 1 (P )} + i σ 1 ( P ) σ s 0 (P ) is a partial connection (cf. Equation (4.6) in Ref.
3), where σ s 0 (P ) denotes the subprincipal symbol of P . Clearly, the vector bundle N P | c can be spanned by the sections w ∈ Γ ∞ (N P | c ) satisfying D P w = 0. In our example, we find that Notice that the connection coefficients (i.e. the second term in the expression above) act trivially on the fibers of N P | c (this follows from (VII.11)), hence the expression for D P simplifies to Any Hamiltonian orbit in our example is therefore of the form R × span C φ + ψ θ ⊆ N P | c , (VII.14) for some 0 = φ + ψ θ ∈ ∧ • R 2 satisfying γ µ k µ ψ = 0.
Combining Theorem VI.1 with this knowledge about propagation of singularities, it follows that any distributional solution to our supersymmetric field equation can be restricted to a Cauchy surface. The initial conditions for a well-posed supergeometric Cauchy problem, however, need to account for compatibility conditions between the Cauchy data in different degrees, see Refs. 6 and 8. A detailed discussion will be given in ongoing work.

ACKNOWLEDGMENTS
We would like to thank Helmut Abels, Federico Bambozzi, Ulrich Bunke and Felix Finster