Superfield Equations in the Berezin-Kostant-Leites Category

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The same kind of results holds for the irreducible representations of the super Poincaré group in the context of superfields, cf. [WB], [BK], [DM]. In order to realize these representations in a space of superfields, one can consider a priori a suitable action functional for superfields on Minkowski superspacetime, and then obtain differential equations se- lecting the representation as the Euler-Lagrange equations corresponding to that action functional. This Lagrangian field-theoretic approach involves the differential geometry of the underlying supermanifold (here Minkowski superspacetime), in order to carry out the calculus of variations, and is most conveniently dealt with by applying the functor of points. This is what is implicitly done in the physics literature, and it leads naturally to spinor fields with anticommuting Grassmannian-valued components. For instance, one can obtain in this way the Wess-Zumino equations for massive chiral superfields (cf. [WB]).

Starting with the category sMan of supermanifolds in the sense of Berezin, Kostant, Leites (cf. [DM]), the extension corresponding to the functor of points is given by the category sF of contravariant functors from sMan to the category Set of ordinary sets. There is an embedding of sMan in sF, given by associating to every supermanifold M the representable functor B 7→ HomsMan(B, M). The necessity of this category for working with superﬁelds and superfunctionals is apparent in the fact that functional spaces exist in sF but are not representable in sMan; for instance the simplest scalar superﬁelds Φ from M to R give rise to the functor which associates to B the set HomsMan(B × M, R) (“inner Hom functor”), but no supermanifold corresponds to this functor.

Thus, it is a priori a necessity to view the solutions of the equations associated to a representation of the super-Poincarégroup as a functor. However, the first result of this paper is that this functor is representable by the solutions of supersymmetric equations that in- volve only ordinary fields, with real or complex valued components. More precisely, with definitions and notations that will be given in Section 3, we have the following theorem:

Theorem 1. Let E be the solution functor of the Wess-Zumino equations for massive chiral superfields, that is, the contravariant functor from the category of supermanifolds to the category of sets defined by 2 E(B) := {Φ : Mcs(B) −→ OB(|B|)0¯ | Da˙ Φ = 0 and (D) Φ − 2m Φ = 0} Then E is representable by the super-vector space 4 2 E := {f ∈ OMcs (R ) | Da˙ f = 0 and (D) f − 2m f = 0} in the sense that there is a natural isomorphism of functors: E≃LE, where LE(B) := (E0⊗Oˆ B(|B|)0¯) ⊕ (E1⊗Oˆ B(|B|)1¯) for every supermanifold B. Moreover, this isomorphism is super Poincaré-equivariant.

The above theorem is a result about the nature of the fields. It is expected that the super-vector space E should correspond, in a natural way, to an irreducible unitary representation of the super Poincarégroup. Our second result will prove that this is indeed the case, i.e. the space E of solutions of the above supersymmetric equations realizes the irreducible representation of super Poincaréwith mass m and superspin 0. More precisely, and again with definitions and notations that will be given in Section 3, we have the fol-

2 lowing theorem:

Theorem 2. Let Mcs be the linear supermanifold associated to the super vector space V ⊕ SC (where V is a four-dimensional Lorentzian vector space, and SC the corresponding four-dimensional complex space of Dirac spinors). The irreducible unitary representation of the super-Poincar´ealgebra of mass m and superspin 0 can be realized as the sub-super 4 ∞ 4 • ∗ vector space of OMcs (R ) = C (R , SC) made of the superfunctions satisfying the differential equations: V

2 ¯ D1˙ f = 0 , D2˙ f = 0 and (D) f = 2m f.

In components, this representation space corresponds to: 2 ∞ 4 ∞ 4 ∗ ( + m )ϕ = 0 {(ϕ, ψ) ∈C (R , C) ×C (R ,S ) | µ } . + εacΓ ∂ ψ − mψ¯ = 0 ab˙ µ c b˙

The category sF, containing functional spaces of superfields, provides an elegant formalism allowing to rewrite the functional super-calculus in physics, which is particularly useful in quantum field theory, for instance in the computation of Batalin-Vilkoviski cohomology for quantization of gauge theories; thus, it is legitimate to question the interest of an interpretation of supersymmetric particles in an ordinary geometrical manner. A first reason comes from the interpretation of comparisons of the theory with possible experiments: when the operator methods of QFT are applied, renormalized quantum amplitudes have to be expressed with ordinary numbers, even if the operators belong to the super category. Then, the first interest of the geometric correspondence is to make this fact more understandable. A second reason, from a theoretical point of view, has to do with the interpretation of local interactions. First, is it possible to get ordinary nonlinear PDEs that are equivalent, in the sense of Theorems 1 and 2, to super-solutions of a supersymmetric super-Lagrangian? And, developing further the quantum theory of this supersymmetric super-Lagrangian, is it possible to get again a correspondence between the obtained quantum corrections and quantum corrections of an ordinary field theory? A negative answer would be particularly interesting. Last but not least, the equations that appear, the linear ones and their nonlinear natural deformations, give new systems of partial differential equations which could be useful in differential geometry.

The first idea in our approach to proving Theorem 2 is a supersymmetric generalization of a construction that produces the Poincaré-invariant differential operators which cut out a given irreducible representation H in the space of spin-tensor fields. Roughly speaking, this construction can be described in the ordinary case as follows. Considering for instance + ∗ 2 2 the massive case, denote by Om ⊂ V the orbit for Spin(V ) given by |p| = m and lying + in the forward timelike cone. Then choose a preferred point in Om, and let K be the stabilizer of that point. The representation W of Spin(V ) that defines the spin-tensor fields can be restricted to K, which allows the construction of a Spin(V )-equivariant vec- + tor bundle over Om. We show in Section 3 that this bundle has a natural equivariant trivialization, which can be used to associate to every K-equivariant linear map from W + ∗ to another Spin(V )-module E, a Spin(V )-equivariant symbol ζ : Om −→ W ⊗ E. These symbols, in turn, give rise to Poincaré-equivariant differential operators on spacetime, by inverse Fourier transform.

3 The second idea is a suitable supersymmetric generalization of the Fourier transform, which we introduce and study in Section 6. One observation there is that the Hodge star can be interpreted as a purely odd super Fourier transform.

The article is organized as follows. In Section 2, we recall briefly the main result about the classification of irreducible unitary representations of the Poincarégroup, which also serves as an opportunity to introduce our notations and terminology for the rest of the paper. In Section 3, we discuss the necessary notions to formulate Theorems 1 and 2. In Section 4, we prove Theorem 1. In Section 5, we explain in more detail our general construction of the Poincaré-invariant differential operators corresponding to a given irreducible representation. In Section 6, we introduce the relevant notion of super Fourier transform and discuss its main properties. In Section 7, we prove Theorem 2. Finally, Section 8 is an appendix in which we give a non-supersymmetric example (massive particle of arbitrary spin) that illustrates the construction in Section 5.

2 Wigner representations

In all this paper, V denotes a real vector space of dimension d equipped with an inner product h , i of signature (1, d − 1). We choose one of the two connected components of the timelike cone C = {v ∈ V | hv, vi > 0}, and denote it by C+, the other ∨ ∗ component being denoted by C−. Dually, we set C := {p ∈ V | hp,pi > 0} and ∨ ∨ C± := {p ∈ C | p(v) ≥ 0 ∀v ∈ C±}. Also, we denote by Spin(V ) the double cover of the connected Lorentz group of V (preserving space and time orientation), and by Π(V ) the Poincar´egroup of V , deﬁned as the semidirect product V ⋊ Spin(V ).

It is well-known that irreducible unitary representations of the Poincarégroup are in one- to-one correspondence with pairs (O,σ), where O is an orbit for the contragredient action of Spin(V ) on V ∗, and σ : K −→ U(F ) a finite-dimensional irreducible unitary representation of the stabilizer K of a preferred momentum q ∈ O. This result goes back to Wigner for the Poincarégroup (cf. [Wig]), and can be seen as a special case of a theorem of Mackey concerning the irreducible unitary representations of more general semidirect products of groups (cf. [Mac]).

± ∗ 2 ∨ For each m > 0, the sheet of hyperboloid Om := {p ∈ V | hp,pi = m } ∩ C± is an orbit; the parameter m here is called the mass. Also, the one-sided light-cone ± ∗ ∨ O0 := {p ∈ V | hp,pi = 0} ∩ C± is an orbit: it corresponds to the massless case. + + Irreducible unitary representations of the Poincar´egroup corresponding to Om or O0 are called of positive energy.

+ 0 When the orbit is Om for some m> 0, one can choose q = me as preferred momentum. + In this case, the stabilizer K is isomorphic to Spin(d − 1). When the orbit is O0 , one can choose q = e0 + ed−1 as preferred momentum. In this case, the stabilizer is isomorphic to Rd−2 ⋊ Spin(d − 2), but only the (ﬁnite-dimensional) irreducible unitary representations of the compact part Spin(d − 2) of the stabilizer will lead to physically meaningful representations of the Poincar´egroup Π(V ). Thus, we will call little group and denote by K either the stabilizer Spin(d − 1) in the massive case, or the compact quotient Spin(d − 2) of the stabilizer in the massless case.

4 The irreducible unitary representations of positive energy of the Poincar´egroup that come from a (ﬁnite-dimensional) irreducible unitary representation of the little group will be called Wigner representations.

3 Superﬁeld equations and super Poincar´erepresentations

In this section, we prepare the ground for and then formulate each of our results. In what follows, we restrict ourselves to d = 4 (so V is a vector space with an inner product of signature (1, 3)). We will deal with representations in various dimensions in a subsequent extended article.

Denote by SC the irreducible complex representation of the Clifford algebra Cℓ(V ). Then ∗ ∗ ∗ a ∗ ¯b˙ ∗ ∗ C2 SC = S+ ⊕ S−, and we fix once for all a basis {θ } in S+ and {θ } in S−, so that S+ ≃ ∗ C2 R4 and S− ≃ . Complex superspacetime is defined as the supermanifold Mcs = ( , OMcs ), ∞ ∗ 4 where OMcs (U) := C (U, C) ⊗ SC for every open set U ⊂ R . A superfunction on Mcs can beV written: f = ϕ + ψ θa + η θ¯b˙ + Fθ1θ2 + Gθ¯1˙ θ¯2˙ +Γµ A θaθ¯b˙ + λ θ1θ2θ¯b˙ + µ θ¯1˙ θ¯2˙ θa + Hθ1θ2θ¯1˙ θ¯2˙ a b˙ ab˙ µ b˙ a ∞ C • ∗ where ϕ , ψa, ηb˙ , F, G, Aµ, λb˙ , µa,H ∈ C (U, ) (we have dim SC = 16 complex- valued functions). V

µ a b˙ ∂ ∂ ∂ The global coordinates (x ,θ , θ¯ ) induce a global frame on Mcs: ( , , ), ∂xµ ∂θa ∂θ¯b˙ but this is not the natural frame to consider: in fact, Mcs has a natural super Lie group structure (with super Lie algebra V ⊕ SC), though not an abelian one (supertranslations do not commute, as [SC,SC] 6= 0). This leads to consider a global moving frame made of left-invariant super vector ﬁelds:

∂ ∂ µ b˙ ∂ ∂ µ b ∂ Pµ = , Da = − Γ θ¯ and Da˙ = − Γ θ ∂xµ ∂θa ab˙ ∂xµ ∂θ¯a˙ ba˙ ∂xµ and we have: [P , D ] = [P , D ] = 0, [D , D ] = [D , D ]=0 and [D , D ]=2Γµ P . µ a µ a˙ a b a˙ b˙ a b˙ ab˙ µ

The generalized supermanifold of superfields is F := Hom(Mcs, C). It is by definition the contravariant functor from the category of (finite-dimensional) supermanifolds sMan to Set given by F(B) = Hom(B × Mcs, C) for all supermanifolds B. One would like to think of F as some kind of infinite-dimensional supermanifold (|F|, OF ).

♯ ♯ ∞ 4 IfΦgeom = (Φˇ, Φ ) : B×Mcs −→ C is a morphism, then Φ : C (C, C) −→ OB (|B|) ⊗ˆ OMcs (R ) ♯ R4 is entirely determined by Φ (IdC) ∈ (OB(|B|) ⊗ˆ OMcs ( ))0¯. Thus, the information about ♯ • ∗ ∞ R4 C Φgeom is fully contained in Φ (IdC) ∈ ( SC ⊗C ( , ) ⊗ˆ OB(|B|))0¯. µ a b˙ V ∞ Using x ,θ , θ¯ as coordinates on Mcs, we may write, for any f ∈C (C, C), ˙ ˙ ˙ ˙ Φ♯(f)= ϕ♯(f)+ θa(ψ )♯(f)+ θ¯b(η )♯(f)+ θ1θ2F ♯(f)+ θ¯1θ¯2G♯(f)+Γµ θaθ¯b(A )♯(f) a b˙ ab˙ µ 1 2 ¯b˙ ♯ ¯1˙ ¯2˙ a ♯ 1 2 ¯1˙ ¯2˙ ♯ +θ θ θ (λb˙ ) (f)+ θ θ θ (µa) (f)+ θ θ θ θ H (f) ♯ ∞ R4 C ♯ ∞ R4 C a ♯ where ϕ (f) ∈C ( , ) ⊗ˆ OB(|B|)0¯, (ψa) (f) ∈C ( , ) ⊗ˆ OB(|B|)1¯ (while θ (ψa) (f) ∈ ∗ ∞ R4 C S+ ⊗C ( , ) ⊗ˆ OB(|B|)1¯), etc...

5 To the superﬁeld Φgeom, we can associate a map Φ : Mcs(B) −→ OB(|B|)0¯, so that

˙ ˙ ˙ ˙ Φ(y,ξ, ξ¯)= ϕ(y)+ ξaψ (y)+ ξ¯bη (y)+ ξ1ξ2F (y)+ ξ¯1ξ¯2G(y)+Γµ ξaξ¯bA (y) a b˙ ab˙ µ

1 2 ¯b˙ ¯1˙ ¯2˙ a 1 2 ¯1˙ ¯2˙ +ξ ξ ξ λb˙ (y)+ ξ ξ ξ µa(y)+ ξ ξ ξ ξ H(y) a a where ϕ(y) ∈ OB(|B|)0¯, ξ ∈ OB(|B|)1¯ and ψa(y) ∈ OB(|B|)1¯ (so ξ ψa(y) ∈ OB(|B|)0¯), etc...

Let Φ : Mcs(B) −→ OB(|B|)0¯ be a scalar superfield. We say that Φ is chiral (resp. antichiral) if D1˙ Φ= D2˙ Φ = 0 (resp. D1Φ= D2Φ = 0). For any chiral superfield, let 1 A(Φ) = ΦΦ¯ d2ξ d2ξ¯ d4y + mΦ2 d2ξ d4y + ZMcs(B) ZMcs(B) 2 The superfield equation corresponding to the above action functional is

(D)2Φ − 2m Φ = 0

2 Let E(B) := {Φ : Mcs(B) −→ OB(|B|)0¯ | Da˙ Φ=0 and (D) Φ − 2m Φ = 0} 2 4 4 2 ( + m )ϕ = 0 ≃ {(ϕ, ψ) ∈ Map(R (B), O (|B|) )×Map(R (B), (O (|B|) ) ) | µ } B 0¯ B 1¯ εacΓ ∂ ψ − mψ¯ = 0 ab˙ µ c b˙ E is the solution functor of the superﬁeld equation. It is a generalized supermanifold on which the super Poincar´egroup SΠ(V ) acts.

We are ready to state the ﬁrst result of this paper, whose proof will be given in the next section.

This theorem shows that it is possible to obtain supersymmetric field equations as partial differential equations in terms of ordinary real or complex valued fields, without hidden anticommuting variables.

In fact, the above supersymmetric equations correspond in a natural way to an irreducible representation of the super Poincar´egroup. This is not very surprising, but a careful state- ment and proof are not available in the literature, up to our knowledge. This motivates our second result, stated below after introducing the necessary notations and terminology.

Let S be an irreducible real spinorial representation of Spin(V ). We know (cf. Deligne’s “Notes on Spinors” in [DM] or Chapter 6 in [Var1]) that there exists a Spin(V )-equivariant

6 symmetric bilinear map Γ : S ×S −→ V . With this, one can define a super Lie algebra ex- tending the PoincaréLie algebra, by setting sπ(V ) := spin(V )⊕V ⊕S, with the nontrivial brackets defined as follows: [spin(V ),S] is given by the action of spin(V ) on S, whereas [V,S] =0 and [s1,s2] = −2 Γ(s1,s2) for all s1,s2 ∈ S. The super Lie algebra sπ(V ) is called super Poincaréalgebra.

Let ρ : sπ(V ) −→ gl(H) be an irreducible unitary representation of the super Poincaréal- gebra on a super Hilbert space H = H0¯ ⊕H1¯. One can show that similarly to the non-super case, this representation is associated to a pair (O,σ), where: O is an orbit of Spin(V ) on ∗ ∨ V (except that now, this orbit will necessarily lie in the closure of C+: supersymmetry forces the positivity of the energy), and σ : k ⊕ S −→ gl(F ) is a finite-dimensional irreducible unitary representation of k ⊕ S, where k is the Lie algebra of the little group K. Given a preferred momentum q ∈ O, there is a natural inner product on S, defined by 1 hs1,s2iq := 2 q([s1,s2]) = −q(Γ(s1,s2)). Now since ρ is a morphism of super Lie alge- bras, we have [ρ(s1), ρ(s2)] = ρ([s1,s2]), and on F , we have ρ([s1,s2]) = iq([s1,s2])IdF . It follows that ρ(s1) ◦ ρ(s2)+ ρ(s2) ◦ ρ(s1) = 2ihs1,s2iq , which shows that F carries a representation of the Clifford algebra Cℓ(S, h , iq).

• C2 If F is irreducible (as a Cliﬀord module), then F ≃ . Thus, F = F0¯ ⊕ F1¯, where 0 C2 2 C2 C C 1 C2 C2 F0¯ ≃ ⊕ = ⊕ and F1¯ ≃ V = (as K-modules). We will considerV the irreducibleV unitary representation H ofV the super Poincar´ealgebra sπ(V ) that + • C2 corresponds to Om for some m> 0 (“massive case”) and F ≃ (“superspin 0”). V

2 ¯ D1˙ f = 0 , D2˙ f = 0 and (D) f = 2m f.

In components, this representation space corresponds to: 2 ∞ 4 ∞ 4 ∗ ( + m )ϕ = 0 {(ϕ, ψ) ∈C (R , C) ×C (R ,S ) | µ } . + εacΓ ∂ ψ − mψ¯ = 0 ab˙ µ c b˙

The proof of this theorem will be given in Section 7. It relies on two components. One of them is a supersymmetric version of a general construction that produces Poincaré- invariant differential operators which cut out a given Wigner representation in the space of spin-tensor fields. We will explain this construction in Section 5. The second component is a suitable supersymmetric generalization of the Fourier transform, which we will define and study in Section 6.

4 Proof of Theorem 1

Given a ﬁnite-dimensional super-vector space V = V0¯ ⊕ V1¯, we denote by LV the supermanifold naturally associated to V (“linear supermanifold”), so that LV = (V0¯, OLV ),

7 ∞ • ∗ where OLV (U) := C (U) ⊗ V1 for every open set U ⊂ V0¯. V

On the other hand, given a super-vector space V = V0¯ ⊕ V1¯ (not necessarily finite- dimensional), we can associate to V a functor LV : sManop −→ Set (“generalized supermanifold”), defined by LV (B) := (OB (|B|)0¯ ⊗ˆ V0¯) ⊕ (OB (|B|)1¯ ⊗ˆ V1¯) for every supermanifold B. Note that if V is finite-dimensional, then this functor is representable in the category sMan, precisely by the linear supermanifold LV : we have Hom(·, LV ) ≃LV (natural isomorphism of functors).

Now let M be a supermanifold and W = W0¯ ⊕ W1¯ a finite-dimensional super-vector space. To define supersymmetric linear sigma-models, one needs a notion of “space of maps” from M to the linear supermanifold LW . Such a space of maps should constitute some kind of infinite-dimensional supermanifold; in this article, we treat such spaces as functors (generalized supermanifolds). Thus, we consider the inner hom functor Hom(M, LW ) : sManop −→ Set, defined by Hom(M, LW )(B) := Hom(B × M, LW ).

One naturally expects that in a suitable category of infinite-dimensional supermanifolds, Hom(M, LW ) should be representable by an infinite-dimensional linear supermanifold. Since we are working with functors, this leads us to expect that there should be a natural isomorphism of functors Hom(M, LW ) ≃ LV for some infinite-dimensional vector space V = V0¯ ⊕ V1¯. The following lemma shows that this indeed the case.

Lemma 4.1. Let M be a supermanifold and W = W0¯ ⊕ W1¯ a ﬁnite-dimensional super- vector space. We have the following natural isomorphism of functors

Hom(M, LW ) ≃ L(OM (|M|) ⊗ W )

Proof. For every supermanifold B,

Hom(M, LW )(B) = Hom(B × M, LW ) ≃ HomsAlg(OLW (W0¯) , OB(|B|) ⊗ˆ OM (|M|)) ∗ ≃ HomsVect(W , OB(|B|) ⊗ˆ OM (|M|)) ≃ (OB(|B|) ⊗ˆ OM (|M|) ⊗ W )0¯

≃ (OB(|B|)0¯ ⊗ˆ (OM (|M|) ⊗ W )0¯) ⊕ (OB(|B|)1¯ ⊗ˆ (OM (|M|) ⊗ W )1¯)

≃ L(OM (|M|) ⊗ W )(B)

Now recall that Minkowski superspacetime is the linear supermanifold Mcs = L(V ⊕ SC), 4 4 thus Mcs = (R , OMcs ) where we think of R as the linear manifold corresponding to V ∞ • ∗ 4 and OMcs (U)= C (U) ⊗ SC for every open set U ⊂ R . Applying the previous lemma with M = Mcs and W = RV gives

4 Hom(Mcs, R) ≃ L(OMcs (R ))

R4 Now if E = E0¯ ⊕ E1¯ is the sub-super-vector space of OMcs ( ) made of the solutions 2 of the equations Da˙ f = 0 and (D) f − 2m f = 0, the functor LE is a subfunctor of 4 L(OMcs (R )). Also, the solution functor E of the Wess-Zumino equations for massive chiral superfields is a subfunctor of Hom(Mcs, R). Now for every supermanifold B, the 4 bijection L(OMcs (R ))(B) −→ Hom(Mcs, R)(B) coming from the above natural isomorphism clearly sends LE(B) to E(B). Therefore, the above natural isomorphism induces a natural isomorphism between the subfunctors LE and E, and it is not difficult to see that the latter preserves the action of the super Poincarégroup.

8 5 Field equations corresponding to Wigner representations

We will describe in general terms how, for a given Wigner representation H of the Poincaré group, one can find a suitable space of classical fields, and a set of linear partial differential equations on those fields whose space of solutions corresponds to H in a natural way.

More precisely, consider a finite-dimensional representation of the group Spin(V ) on some complex vector space W (i.e., a “spin-tensor” representation). We may define a classical field of type W on Minkowski spacetime to be a compactly supported smooth section of the equivariant vector bundle W := Π(V ) ×Spin(V ) W , associated by this representation to d d the principal Spin(V )-bundle Π(V ) −→ R . The space Γc(R , W) of classical fields of type W carries a (highly reducible) representation of the Poincarégroup Π(V ). Ultimately, one would be interested in finding some decomposition of this space of fields into irreducible representations. One step in this direction is to start with a Wigner representation H, find a suitable spin-tensor representation W , and construct a natural inner product on d Γc(R , W) in such a way that after separating (i.e. dividing out the kernel of that inner product) and completing, the resulting Hilbert space contains a subspace unitarily isomorphic to the Wigner representation H we started with.

The first question is the choice of the spin-tensor representation W . As will be clear from the following construction, one must choose the representation W of Spin(V ) so that when restricted to the little group K, it contains a subrepresentation isomorphic to the irreducible unitary representation σ : K −→ U(F ) that defines H. Once such a choice of d W has been made, if one’s goal is to pick H out of Γc(R , W) (as representations of Π(V )), it is reasonable to try first to perform the “algebraic analog”, that is, pick F out of W (as representations of K). One way to achieve this is to have a K-equivariant linear map u from W to some other representation E of Spin(V ), such that Ker u ≃ F . As for the inner product, one starts also by “working at the algebraic level”, by choosing an appropriate K-invariant Hermitian form h , i0 on W , whose restriction to F is positive-definite.

Next, one extends the previous algebraic considerations, which can be thought of as taking place in the category of finite-dimensional representations of K, to an equivalent category of equivariant vector bundles. Although the inner product we have in mind can be ulti- d mately defined on the space Γc(R , W) of all fields (or on their Fourier transforms, which is easier), we will content ourselves here with an on-shell description in momentum space.

Thus, for m ≥ 0, we consider the equivariant vector bundle W(m) := Spin(V ) ×K W on O+ , associated to the principal K-bundle Spin(V ) −→ O+ by the restricted representa- m m c tion of K on W . The K-equivariant linear map u : W −→ E defines then a morphism of equivariant vector bundles W(m) −→ E(m) (where E(m) := Spin(V ) ×K E), which, in + W + E turn, defines a linear map betweenc sectionsb Γ(Om, (bm)) −→ Γ(Om, (m)). The Wigner representation H corresponds then to the kernel of this linear map, and the inner product + Wc b on H corresponds to the inner product on Γ(Om, (m)) defined by the K-invariant Her- mitian form on W . Finally, taking the inverse Fourier transform of the condition defining + W c H in Γ(Om, (m)) gives the desired partial differential equations. c In fact, one does not need to work with sections of vector bundles: there is a natural, W E equivariant trivialization of the the vector bundles (m) and (m). The reason is that these equivariant vector bundles are in fact “tractor bundles”, that is, they are associated c b to representations of K which are in fact restrictions of representations of Spin(V ). In W + such a situation, there is always an equivariant trivialization: (m) −→ Om × W , given c 9 E + by [h, w] 7→ (hq, hw) (and, similarly, (m) −→ Om × E, given by [h, e] 7→ (hq, he)). One W E can check that in these trivializations,b the bundle morphism (m) −→ (m) becomes the equivariant map ζ : O+ −→ W ∗ ⊗ E given by ζ (p) = h · u · h−1, where h ∈ Spin(V ) u m u p c p b p ˆ + is such that hp q = p. Then, H corresponds to the subspace of maps Φ : Om −→ W satisfying the condition ζu(p)(Φ(ˆ p)) = 0.

Typically, the spin-tensor representation W is of real type, i.e. it admits a Spin(V )- invariant conjugation. Then H would also correspond to the space of maps Φˆ : Om −→ W satisfying the reality condition Φ(ˆ −p) = Φ(ˆ p). By inverse Fourier transform, one gets ﬁelds valued in a real vector space, satisfying partial diﬀerential equations corresponding to the condition ζu(p)(Φ(ˆ p)) = 0.

+ As for the inner product, one can check that it is given on the space of maps from Om −1 −1 + + to W by hΦˆ, Ψˆ i = hh Φ(ˆ p), h Ψ(ˆ p)i0 dβ (p), where β is the Spin(V )-invariant + p p m m ZOm + volume form on Om (and hp ∈ Spin(V ) is such that hp q = p).

This construction can be used to produce in a natural way the field equations corresponding to the usual irreducible representations of the Poincarégroup, and so one can obtain in this way the Klein-Gordon equation, the Dirac equation and the Maxwell equation. More generally, we illustrate this construction by producing the field equations in dimension 4 that correspond to a massive particle of arbitrary spin (cf. the appendix, Section 8).

6 Super Fourier transform

Notions of Fourier transform in superspace have appeared previously in the literature, namely in [DeW], [Rog] and [DeB]. These authors work in a different category of supermanifolds, and [DeW], [Rog] use a kernel that transforms under the orthogonal group. On the other hand, [DeB] uses instead a kernel that transforms under the group Sp(2n, R) (here, n is the odd dimension). Inspired by this, we use the standard supermetric on Mcs to define a natural version of the Fourier transform for Minkowski superspacetime, taking ∞ R4 a ¯a˙ ∞ ∗ a a˙ superfunctions in Cc ( )[θ , θ ] to superfunctions in C (V )[τ , τ¯ ]. Once an expression for the super Fourier transform of a superfunction is obtained, we see that the purely odd part of the transform coincides with the Hodge isomorphism defined by the invariant symplectic structure on the spinors. From this, it is easy to check that the super Fourier ∂ transform has natural properties such as exchanging the odd derivative with exterior ∂θ1 ∂ multiplication by iτ 2, and multiplication by θ1 with the contraction −i . The 1 ↔ 2 ∂τ 2 exchange is not surprising since the super Fourier transform is defined via a symplectic structure. Finally, we define supersymmetric symbols ζ ¯ and ζdτa that correspond, via dτ¯a˙ the super Fourier transform, to the supertranslation-invariant odd vector fields Da and Da˙ canonically defined on Mcs.

Deﬁnition 6.1. The super Fourier transform of a (compactly supported) superfunction ∞ R4 • ∗ ∞ R4 a ¯a˙ ∞ ∗ a a˙ f ∈Cc ( , SC) ≃Cc ( )[θ , θ ] is the element ⋆f ∈C (V )[τ , τ¯ ] deﬁned by: V b¯ ⋆f := e−i(hp,xi+ε+(τ,θ)+ε−(¯τ,θ)) f dx dθ dθ¯ ZMcs b

10 Note that if we deﬁne the bosonic Fourier transform of f by f(p) := e−ihp,xi f(x) dx , ZR4 − ¯ b then ⋆f(p)= e−i(ε+(τ,θ)+ε (¯τ,θ)) f(p) dθ dθ¯. Z b b a ¯b˙ 1 2 ¯1˙ ¯2˙ Proposition 6.2. Let f(x) = ϕ(x)+ ψa(x) θ + ηb˙ (x) θ + F (x) θ ∧θ + G(x) θ ∧θ +Γµ A (x) θa ⊗θ¯b˙ + λ (x) (θ1 ∧θ2)⊗θ¯b˙ + µ (x) θa ⊗(θ¯1˙ ∧θ¯2˙ ) + H(x) (θ1 ∧θ2)⊗(θ¯1˙ ∧θ¯2˙ ) ab˙ µ b˙ a 4 • ∗ be the expansion of a generic superfunction f : R −→ SC. Then the super Fourier transform of f is given by V

a b˙ 1 2 1˙ 2˙ ⋆f(p) = H(p) + iµa(p) τ + iλb˙ (p)τ ¯ + G(p) τ ∧ τ + F (p)τ ¯ ∧ τ¯ −bΓµ A (p)bτ a ⊗τ¯b˙ + iη (p) (τ 1 ∧τb2)⊗τ¯b˙ + iψb(p) τ a ⊗(¯τ 1˙ ∧τ¯2˙ )b + ϕ(p) (τ 1 ∧τ 2)⊗(¯τ 1˙ ∧τ¯2˙ ) ab˙ µ b b˙ a Proof.b b b b a ¯b˙ 1 2 ¯1˙ ¯2˙ We have f(p) = ϕ(p) + ψa(p) θ + ηb˙ (p) θ + F (p) θ ∧ θ + G(p) θ ∧ θ +Γµ A (pb) θa ⊗θ¯b˙ + λ (p) (θb1 ∧θ2)⊗θ¯b˙ + µ (p) θa ⊗b(θ¯1˙ ∧θ¯2˙ ) + H(pb) (θ1 ∧θ2)⊗(θ¯1˙ ∧θ¯2˙ ) ab˙ µ b b˙ b a To deriveb the stated expressionb for the superb Fourier transform ⋆fb(p), start by expanding 1 2 2 1 1˙ 2˙ 2˙ 1˙ −i(ε+(τ,θ)+ε−(¯τ,θ¯)) −iε+(τ,θ) −iε−(¯τ,θ¯) −i(τ θ −τ θ ) −i(¯τ θ¯ −τ¯ θ¯ ) the exponential e = e e = e b e = (1 − iτ 1θ2 + iτ 2θ1 + τ 1τ 2θ1θ2) (1 − iτ¯1˙ θ¯2˙ + iτ¯2˙ θ¯1˙ +τ ¯1˙ τ¯2˙ θ¯1˙ θ¯2˙ ) Then, multiply the result by the expansion of f(p), retaining only the coeﬃcients of 1 2 ¯1˙ ¯2˙ (θ ∧ θ ) ⊗ (θ ∧ θ ), and ﬁnally perform a Berezinb integration.

Corollary 6.3. The purely odd super Fourier transform coincides with the Hodge dual (with respect to the symplectic form ε on SC).

Proof. Just compare the expansions of f(p) and ⋆f(p). b b Proposition 6.4.

∂f ∂f ˙ 1. ⋆( )= iε τ b (⋆f) , ⋆( )= iε τ¯b (⋆f) a ab ¯a˙ a˙ b˙ ∂θd ∂dθ ∂ b ˙ b∂ 2. ⋆(θaf)= −iεab (⋆f) , ⋆(θ¯a˙ f)= −iεa˙ b (⋆f) ∂τ b ∂τ¯b˙ d b d b Proof. Follows from Proposition 6.2.

4 • ∗ Recall that for every superfunction f : R −→ SC,

∂f µ b˙ ∂f V ∂f µ b ∂f we have Daf = − Γ θ¯ and Daf = − Γ θ . ∂θa ab˙ ∂xµ ˙ ∂θ¯a˙ ba˙ ∂xµ 1 1 ˙ Also, we set (D)2f := εabD D f and (D)2f := εa˙ bD D f. 2 a b 2 a˙ b˙

We deﬁne the following symbols, acting on elements of C∞(V ∗)[τ a, τ¯a˙ ]: c˙d˙ µ b ˙ ζda (p) := (iεab eτ ⊗ Id) − (Id ⊗ ε Γac˙ pµ ιτ¯d ) cd µ b˙ d ζd¯a˙ (p) := (Id ⊗ iεa˙ b˙ eτ¯ ) − (ε Γca˙ pµ ιτ ⊗ Id) 1 ab 1 a˙ b˙ ζ 2 := ε ζda ◦ ζdb and ζ ¯ 2 := ε ζ ¯ ◦ ζ ¯ . (d) 2 (d) 2 da˙ db˙

11 Proposition 6.5.

1. ⋆Daf(p)= ζda (p)(⋆f(p)) , ⋆Da˙ f(p)= ζd¯a˙ (p)(⋆f(p)) d \d2 b \2 b 2. ⋆(D) f(p)= ζ(d)2 (p)(⋆f(p)) , ⋆(D) f(p)= ζ(d¯)2 (p)(⋆f(p)) Proof. Follows directly fromb the preceding proposition. b

7 Proof of Theorem 2

4 The superfields we are considering are elements of the super-vector space OMcs (R ) = ∞ 4 • ∗ ∞ 4 • ∗ C (R ) ⊗ SC = C (R , SC), so adopting the notations of Section 3, we have here • ∗ • ∗ • ∗ W = SVC = S+ ⊗ VS− , since SC = S+ ⊕ S−. V V V On the other hand, we have seen in Section 4 that the irreducible unitary representation H of the super-Poincaréalgebra sπ(V ) that is of mass m> 0 and superspin 0 corresponds • 2 to the irreducible Cℓ(S, h , iq)-module F = C . V As explained in Section 3, we need to a way to “pick F out of W ” as representations of the stabilizer, and so we need some equivariant linear map on W whose kernel would be F . Now W is a Clifford module, but it is certainly reducible and a multiple of the irreducible Clifford module F . In fact, upon restriction to K ≃ Spin(3), we get S+ ≃ S−, and with a convenient choice of bases, one can write W = • C2 ⊗ • C2 = • C2 ⊗ (C ⊕ C2 ⊕ 2 C2) = • C2 ⊕ ( • C2 ⊗ C2) ⊕ • C2 This directV sumV of three termsV shows that weV have at leastV two waysV of realizing FV inside W , one by choosing the • C2 occurring as the first term (“chiral choice”), and the other • by choosing the C2 occurringV as the third term (“antichiral choice”). V A canonical way to perform these choices is via the following linear maps. Recall that 0 0 hs1,s2iq = −q(Γ(s1,s2)), so with q = me , we have hs1,s2iq = −me (Γ(s1,s2)). We set Γ0 := e0 ◦ Γ, and define the following endomorphisms of W : c˙d˙ 0 b ˙ da := (iεab eτ ⊗ Id) − (Id ⊗ ε Γac˙ ιτ¯d ) cd 0 ¯ ˙ d da˙ := (Id ⊗ iεa˙ b˙ eτ¯b ) − (ε Γca˙ ιτ ⊗ Id) ¯ ¯ This allows to consider F as the “chiral subspace” Ker d1˙ ∩ Ker d2˙ , which is in fact K- invariant.

Thus, we see that chirality of superﬁelds in dimension 4|4 appears already at the purely algebraic level. The next step is to proceed in the spirit of Section 3, associating actual symbols to the “algebraic symbols” da and d¯a˙ . Not surprisingly, this gives exactly the symbols ζda (p) and ζd¯a˙ (p) deﬁned in Section 5.

It remains to calculate the action of these symbols on the expansions of the superﬁelds in momentum space, and impose the right mass-shell equation, involving the second-order symbol ζ(d)2 (p).

Lemma 7.1. A superfunction ⋆f ∈C∞(V ∗)[τ a, τ¯a˙ ] satisﬁes the condition

b ζd¯a˙ (p)(⋆f(p)) = 0 b 12 µν if and only if µa = G = ηb˙ = 0 , H(p)= η ipµ Aν(p) , Aµ(p)= −ipµ ϕ(p) and λ (p)= εac Γµ ip ψ (p) , so that b˙ b ab˙ b µ bc b b b b b b ⋆f(p) = |p|2 ϕ(p) − εac Γµ p ψ (p)τ ¯b˙ + F (p)τ ¯1˙ ∧ τ¯2˙ + Γµ ip ϕ(p) τ a ⊗ τ¯b˙ ab˙ µ c ab˙ µ a 1˙ 2˙ 1 2 1˙ 2˙ +biψa(p) τ ⊗ (¯τb ∧ τ¯ ) + ϕ(p) (τb ∧ τ ) ⊗ (¯τb ∧ τ¯ ) b b b b˙ cd µ ∂ Proof. Applying ζ ¯ (p) = iε τ − ε Γ p to the expansion of ⋆f(p) given in da˙ a˙ b˙ ca˙ µ ∂τ d Proposition 6.2, the result follows after a calculation, taking into account thatb we have Γµ Γν − Γµ Γν = ηµν . 11˙ 22˙ 12˙ 21˙

Lemma 7.2. Let ⋆f ∈ C∞(V ∗)[τ a, τ¯a˙ ] be a superfunction satisfying the condition of the preceding lemma. Then b

2 c˙d˙ dc µ ν 2 a ζ 2 (p)(⋆f(p)) = −|p| F (p) + (ε ε Γ Γ p p iψ (p) − |p| iψ (p)) τ (d) ac˙ dd˙ µ ν c a − 4 |p|2 ϕ(bp) τ 1 ∧ τ 2 + Γµbip F (p) τ a ⊗ τ¯b˙ + 2Γµ p ψb (p) (τ a ∧ τ cb) ⊗ τ¯b˙ ab˙ µ ab˙ µ c ˙ ˙ − F (p) (bτ 1 ∧ τ 2) ⊗ (¯τ 1 ∧ τ¯2) b b b b c˙d˙ µ ∂ Proof. Applying ζda (p) = iεab τ − ε Γac pµ to the expansion of ⋆f(p) given in ˙ ∂τ¯d˙ Lemma 7.1 gives b ˙ ζ (p)(⋆f(p)) = ε |p|2 iϕ(p) τ b − ε εdc Γµ ip ψ (p) τ b ⊗ τ¯b da ab ab db˙ µ c − ε Γµbp φ(p) (τ b ∧ τ c) ⊗ τ¯b˙ + ε iF (p) τ b ⊗ (¯τ 1˙ ∧b τ¯2˙ ) − ε ψ (p) (τ b ∧ τ c) ⊗ (¯τ 1˙ ∧ τ¯2˙ ) ab cb˙ µ b ab ab c + εc˙d˙ εdc Γµ bΓν p p ψ (p) + εc˙d˙ Γµ Γbν p p iϕ(p) τ c + Γµ p bF (p)τ ¯c˙ ac˙ dd˙ µ ν c ac˙ cd˙ µ ν ac˙ µ µ c c˙ µ 1 2 c˙ − Γac˙ ipµ ψc(p) τ ⊗ τ¯ b+ Γac˙ pµ ϕ(p) (τ ∧ τ ) ⊗bτ¯ b The next stepb is to apply to the aboveb another instance of ζda (p), then contract with the symplectic form ε on S+ and divide by 2. After calculation and simpliﬁcation, one obtains the stated expression for ζ(d)2 (p)(⋆f(p)). b To conclude the proof of Theorem 3, we impose the equation

ζ(d)2 (p)(⋆f(p)) = 2m ⋆f(p)

Comparing the expansions of both sides,b we see that b −F (p) = 2m ϕ(p) and −4 |p|2 ϕ(p) = 2m F (p) whichb gives b b b (|p|2 − m2) ϕ(p) = 0 Since 2Γµ p ψ (p) (τ a ∧ τ c) ⊗ τ¯b˙ = 2 εac Γµ p ψ (p) (τ 1 ∧ τ 2) ⊗ τ¯b˙ , we also obtain ab˙ µ c ab˙b µ c b b εac Γµ p ψ (p) = −im ψ (p) ab˙ µ c b˙ b b Finally, taking the inverse super Fourier transform gives the equations in Theorem 2.

13 8 Appendix: an example in dimension 4 with arbitrary spin

In this appendix, we illustrate the content of Section 5 with an example in four-dimensional Minkowski spacetime: the case of a particle of mass m> 0 and of spin s ≥ 1.

We know that the corresponding Wigner representation H(m,s) is obtained by induction from the irreducible unitary representation of K ≃ Spin(3) ≃ SU(2) of spin s. The space 2s 2 of this representation is Fs ≃ Sym C . ∞ R4 We want to realize H(m,s) in a space of spin-tensor ﬁelds C ( ,W ). We can take 2α 2β W = Sym S+ ⊗ Sym S− with α + β ≥ s. We consider the minimal choice α + β = s. 1 3 Of course, α, β ∈ {0, 2 , 1, 2 , 2, ...}, and we consider in what follows the interesting case α> 0 and β > 0. We need an SU(2)-equivariant map u on the Spin(V )-module W whose 2α ∗ 2β ∗ kernel is Fs. At this point, we need to know how to decompose Sym S+ ⊗ Sym S− into irreducible representations of SU(2).

1 3 Recall that the irreducible complex representations of SU(2) are classiﬁed by {0, 2 , 1, 2 , 2, ...}. 1 3 2s ∗ More precisely, for each s ∈ {0, 2 , 1, 2 , 2, ...}, the vector space Sym S+ carries the irre- 2s ∗ ducible representation of SU(2) of highest weight s. We say that Sym S+ is the irreducible representation of spin s of SU(2); its dimension is 2s + 1, and its internal structure can be described as follows. iθ 0 A canonical choice of Cartan subalgebra of su(2) is t := { ; θ ∈ R}. Under 0 −iθ 2s ∗ t, the representation Sym S+ decomposes into one-dimensional weight spaces: 2s ∗ Sym S+ = L−s ⊕ L−s+1 ⊕ L−s+2 ⊕···⊕ Ls−2 ⊕ Ls−1 ⊕ Ls iθ 0 where acts on Lj by λ 7→ 2j(iθ)λ. 0 −iθ

C 1 For instance, the spin 0 representation is the trivial representation on , the spin 2 rep- ∗ 2 ∗ resentation is S = L 1 ⊕ L 1 , and the spin 1 representation is Sym S = L−1 ⊕ L0 ⊕ L1. + − 2 2 +

Lemma 8.1. Assume in addition that α ≥ β. Then, as representation of SU(2),

2α ∗ 2β ∗ 2(α+β) ∗ 2(α+β−1) ∗ 2(α−β) ∗ Sym S+ ⊗ Sym S− ≃ Sym S+ ⊕ Sym S+ ⊕ . . . ⊕ Sym S+

∗ ∗ Proof. As representations of SU(2), S+ and S− become equivalent. Thus, we need to 2α ∗ 2β ∗ decompose Sym S+ ⊗ Sym S+. We start by writing the weight-space decomposition of each factor: we have

2α ∗ Sym S+ = L−α ⊕ L−α+1 ⊕ L−α+2 ⊕···⊕ Lα−2 ⊕ Lα−1 ⊕ Lα and 2β ∗ Sym S+ = L−β ⊕ L−β+1 ⊕ L−β+2 ⊕···⊕ Lβ−2 ⊕ Lβ−1 ⊕ Lβ

Taking the tensor product, and using the fact that Lj ⊗ Lk = Lj+k, we obtain

2α ∗ 2β ∗ Sym S+ ⊗Sym S+ = L−α−β ⊕2L−α−β+1 ⊕3L−α−β+2 ⊕···⊕3Lα+β−2 ⊕2Lα+β−1 ⊕Lα+β which implies easily the result.

Notice that by the above lemma,

2α−1 ∗ 2β−1 ∗ 2(α+β−1) ∗ 2(α+β−2) ∗ 2(α−β) ∗ Sym S+ ⊗ Sym S− ≃ Sym S+ ⊕ Sym S+ ⊕ . . . ⊕ Sym S+

14 and therefore we have (also by the above lemma):

2α ∗ 2β ∗ 2s ∗ 2α−1 ∗ 2β−1 ∗ Sym S+ ⊗ Sym S− ≃ Sym S+ ⊕ (Sym S+ ⊗ Sym S−)

Consequently, we deﬁne

2α ∗ 2β ∗ 2α−1 ∗ 2β−1 ∗ u : Sym S+ ⊗ Sym S− −→ Sym S+ ⊗ Sym S− 0 ∗ 0 ∗ proceeding as follows. First, extend e ∈ V by C-linearity to obtain an element e ∈ VC . ∗ ∗ 0 ∗ ∗ C Then compose with ΓC : S+ ⊗ S− −→ VC. This gives a map e ◦ ΓC : S+ ⊗ S− −→ . 2α ∗ 2β ∗ ∗ ∗ 2α−1 ∗ 2β−1 ∗ Now let ι : Sym S+ ⊗ Sym S− ֒→ S+ ⊗ S− ⊗ Sym S+ ⊗ Sym S− be the canonical inclusion. Finally, set u := ((e0 ◦ ΓC) ⊗ Id) ◦ ι. Then u is SU(2)-equivariant (since e0 is SU(2)-equivariant).

In fact, we have the following exact sequence of SU(2)-modules:

2s ∗ 2α ∗ 2β ∗ 2α−1 ∗ 2β−1 ∗ 0 −→ Sym S+ −→ Sym S+ ⊗ Sym S− −→ Sym S+ ⊗ Sym S− −→ 0 the second nontrivial map being u, and the ﬁrst nontrivial map being (Id ⊗ c˜) ◦ ια,β , where 2s ∗ 2α ∗ 2β ∗ ∗ ∗ ια,β : Sym S+ ֒→ Sym S+ ⊗ Sym S+ is the canonical inclusion and c : S+ −→ S− is an SU(2)-equivariant isomorphism.

∗ It is easy to check that the corresponding symbol ζu : Om −→ W ⊗ E (where E := 2α−1 ∗ 2β−1 ∗ Sym S+ ⊗ Sym S−) is given by

ζu(p) = ((p ◦ ΓC) ⊗ IdE) ◦ ι Let Ξ : W ⊕ (V ∗ ⊗ W ) −→ E be deﬁned by Ξ(w, p ⊗ w)=Ξ(1)(p ⊗ w) , where Ξ(1) : V ∗ ⊗ W −→ E is given by

(1) ∗ Ξ = (tr ⊗ IdE) ◦ (IdVC ⊗ ΓC ⊗ IdE) ◦ (j ⊗ ι) ∗ ∗ .where j : V ֒→ VC is the canonical inclusion

(1) ∗ Then Ξ(w, p ⊗ w)=Ξ (p ⊗ w) = ((tr ⊗ IdE) ◦ (IdVC ⊗ ΓC ⊗ IdE)) (p ⊗ ι(w)) = (tr ⊗ IdE)(p ⊗ (ΓC ⊗ IdE)(ι(w))) = ((p ◦ ΓC) ⊗ IdE)(ι(w)) = ζu(p)(w) for every p ∈ Om and w ∈ W . Thus, ζu is the symbol of a ﬁrst-order diﬀerential operator ∞ 4 ∞ 4 ∞ 4 Du : C (R ,W ) −→ C (R , E). For φ ∈C (R ,W ), (1) Duφ(x)=Ξ(φ(x) , −idφ(x))=Ξ (−idφ(x))

We denote this “divergence-type” diﬀerential operator Du by δα,β. In conclusion, we have the following proposition:

Proposition 8.2. Let H(m,s) be the Wigner representation of mass m > 0 and spin s (obtained by induction from an irreducible unitary representation F of the stabilizer K). Also, let W be the vector bundle on Minkowski spacetime R4 associated to the represen- 2α ∗ 2β ∗ tation W = Sym S+ ⊗ Sym S− of the group Spin(V ). Assume that F appears in the decomposition of W under K (which is equivalent to α + β ≥ s). Then H(m,s) is selected by the condition ζu(p)(φ(p)) = 0 in momentum space, and by the following equations in spacetime: b ( + m2)φ = 0 δα,βφ = 0 ∞ R4 2α ∗ 2β ∗ ∞ R4 2α−1 ∗ 2β−1 ∗ where δα,β : C ( , Sym S+ ⊗ Sym S−) −→ C ( , Sym S+ ⊗ Sym S−) .

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