{'fraction_non_alphanumeric': 0.06607659574468085, 'fraction_numerical': 0.01126808510638298, 'mean_word_length': 3.7144920558497834, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 17, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Killing Vector Fields and Superharmonic Field Theories

The harmonic action functional allows a natural generalisation to semi-Riemannian supergeometry, referred to as superharmonic action, which resembles the supersymmetric sigma models studied in high energy physics. We show that Killing vector fields are infinitesimal supersymmetries of the superharmonic action and prove three different Noether theorems in this context. En passant, we provide a homogeneous treatment of five characterisations of Killing vector fields on semi-Riemannian supermanifolds, thus filling a gap in the literature.


Introduction
Symmetries belong to the main ingredients of modern physical theories. In high energy physics, supersymmetry is a conjectured transition between the two types of elementary particles in nature: bosons and fermions, which differ in their statistics. The observed properties of these particles are, to a good agreement, described by quantum field theories, see e.g. [PS95] for a standard treatment. The quantum theories are, usually, based on classical field theories. In this context, bosons and fermions are described by even and odd fields, respectively. Free bosonic string theory is modelled on the well-known harmonic action [Jos01], while a class of supersymmetric extensions is given by so called supersymmetric sigma models [DF99].
In this article, we study a similar such extension which, from a mathematical point of view, is more natural and also more general in that it can be formulated for every pair of semi-Riemannian supermanifolds. We shall, therefore, refer to the resulting field theories as superharmonic. In contrast to the aforementioned sigma models, the superharmonic action always allows a "superspace formulation", meaning that all fields occuring can be included in a single superfield, a mathematical model of which is a map with flesh [Hél09].
Killing vector fields are known to be infinitesimal symmetries of the harmonic action. According to the Noether principle, every symmetry of a classical field theory should induce a conserved quantity. In the harmonic theory, this is indeed the case ( [BE81], [Hél02]), while it is a priori not clear for the superharmonic action. Indeed, whereas certain elements of supergeometry, such as the divergence of a vector field, differ from the classical theory of manifolds merely by a number of signs, others are non-trivial extensions. Examples for the latter include integration theory and maps with flesh.
In the present article, we show that Killing vector fields on the domain as well as on the target space supermanifold are infinitesimal symmetries of the superharmonic action. As our main results, we formulate and prove three different Noether theorems in this context. This is the subject matter of Sec. 4. There are several equivalent definitions for a Killing vector field on a semi-Riemannian manifold, most of which have been generalised to supergeometry ( [MSV96], [Goe08], [ACDS97]), while a homogeneous treatment of the subject is yet missing. Another aim of this article is to fill this gap. In Sec. 3, we prove equivalence of five characterisations of Killing vector fields. It turns out that the definition used in [ACDS97] is a non-equivalent variation. To start with, Sec. 2 reviews elements of the theory of semi-Riemannian supermanifolds needed later.

Semi-Riemannian Supermanifolds
Superharmonic field theories are based on semi-Riemannian supermanifolds. For later use, we briefly recall the relevant background here.
Throughout the article, we adopt the Berezin-Kostant-Leites definition of supermanifolds and their morphisms in terms of sheaves as in [Lei80]. In particular, a supermanifold is a ringed space (M, O M ), and a morphism Φ : (M, O M ) → (N, O N ) of supermanifolds consists of two parts Φ = (ϕ, φ). We shall occasionally abuse notation and write M instead of (M, O M ). Modern monographs on the general theory of supermanifolds include [Var04] and [CCF11] while aspects of Riemannian supergeometry are studied in [Goe08]. References for more specialised topics will be given at suitable positions throughout the text.
Following the conventions of [Gro11], Note the use of the shorthand notation X ∈ SM , meaning X ∈ SM (U ) for p ∈ U ⊆ M . Similarly, we yield a canonical (classical) vector fieldX ∈ Γ(T M ) on M by setting It is well-known that superfunctions f are not detemined by their valuesf (p) and, likewise, vector fields X are not determined by their values X(p). With respect to local coordinates (ξ 1 , . . . , ξ n , ξ n+1 , . . . , ξ n+m ) of M , the tuple ( ∂ ∂ξ 1 , . . . , ∂ ∂ξ n+m ) is a local O M -basis of SM which is adapted in the sense that the first n vector fields are even and the remaining m ones are odd. Likewise, the tuple ( ∂ ∂ξ 1 (p), . . . , ∂ ∂ξ n+m (p)) is an adapted basis of the super vector space S p M .
Tensor calculus on supermanifolds is based on superlinear algebra. For U ⊆ M sufficiently small, SM (U ) is a free O M (U )-supermodule of rank n|m = dim M . In general, consider a supercommutative superalgebra A and free A-supermodules M and N of rank m|n and r|s, respectively. With respect to adapted right bases (f 1 , . . . , f n+m ) and (g 1 , . . . , g r+s ) of M and N , respectively, any superlinear map L : M → N can be identified with a matrix L ∈ Mat A (m|n, r|s). Similarly we denote, for an even (super-)bilinear form B ∈ Hom A (M ⊗ A M, A), the corresponding matrix with entries B jk := B(f j , f k ) by the same symbol B ∈ Mat(n|m, A).
Let GL n|m (A) denote the group of even and invertible n|m-matrices with entries in A. The orthosymplectic group of dimension (t, s)|2m is defined as follows.
where g 0 denotes the standard supermetric which, with respect to the standard basis for A t+s|2m , is given by the matrix The corresponding super Lie algebra is where gl n|m (A) := gl n|m ⊗ A is the super Lie algebra of all matrices with entries in A.
For A = R, we define osp (t,s)|2m := osp (t,s)|2m (R). By means of choosing a basis {T j } of osp (t,s)|2m , it is easy to see that osp (t,s)|2m (A) = osp (t,s)|2m ⊗ A. Now let g be an even (i.e. parity-preserving), nondegenerate and supersymmetric bilinear form (a supermetric for short). By an extension of the Gram-Schmidt procedure as detailed e.g. in Sec. 2.8 of [DeW84], there is an adapted basis (e 1 , . . . , e t+s+2m ) of M , such that g = g 0 on the level of matrices, which we shall call an OSp (t,s)|2m -basis. In particular, the values of t, s, m ∈ N are uniquely determined by g. As in Sec. 3.5 of [Han12] with slightly different conventions, we introduce the even map J, which is defined with respect to an OSp (t,s)|2m -basis {e i } as follows.
This is such that g (e k , Je j ) = (−1) |e k | δ kj and Je k = (−1) |e k | h km e m and, moreover, g identifies any other bilinear form K ∈ Hom A (M ⊗ A M, A) with a superlinear map K : M → M of the same parity as K via The supertrace of K with respect to g is defined as str g K := strK = (−1) |f j |(|K|+1)K j j where the second equation holds for any adapted (right) basis {f j } of M upon identifying K with a supermatrix. An explicit calculation then shows that, regardless of the parity of K, the following formula holds for every OSp (t,s)|2m -basis f j = e j .
We shall use the preceding superlinear algebra for the tensor calculus on a super- which can be taken as a definition [DM99]. Sections of any sheaf of multilinear forms will be commonly denoted as tensors or tensor fields. The differential df of a superfunction f ∈ O M is defined by the formula df [X] := (−1) |f ||X| X(f ) for X ∈ SM . In particular, we may consider the differential of coordinate functions ξ i . Our sign conventions are such that any even bilinear form An even bilinear form g which is non-degenerate and (super-)symmetric is called a semi-Riemannian supermetric. Locally, the above treatment on superlinear algebra applies here. In particular, there exists an OSp (t,s)|2m -basis {e j } with t, s and m being intrinsic invariants of g.
for Y, Z ∈ SN , together with super(bi)linear extensions for general sections of SΦ, A connection on N is an even R-linear sheaf morphism ∇ : SN → S * N ⊗ O N SN satisfying the (graded) Leibniz rule. If (N, g) is a semi-Riemannian supermanifold, there is a unique connection which is (graded) metric and torsion-free, called the Levi-Civita connection [Goe08]. In general, for a connection ∇ on N , there is a canonical pullback connection ∇ Φ : SΦ → ϕ * S * M ⊗ ϕ * O M SΦ (see [GW12]), that, in the following, we shall denote simply by ∇. With respect to local coordinates {η i } on N , it reads Lemma 2.1. Let (N, g) be a semi-Riemannian supermanifold and ∇ a superconnection on N which is metric. Then the pullback ∇ Φ is metric in the following sense.
holds true for every X ∈ ϕ * SM as well as Y, Z ∈ SΦ.
Proof. This follows from a straightforward calculation in local coordinates using (5).
There is a second type of pullback for tensors and functions f ∈ O N on N , which yields corresponding objects on M . We define and analogous for other tensors. The canonical identification with an (ordinary) tensor on M stated follows from the general theory of ringed spaces, e.g. from Thm. 4.4.14 of [Ten75] applied to the induced morphism (ϕ, id) : Moreover, one verifies that, in case Φ is a diffeomorphism, it holds which is the definition of the pullback in [Goe08]. The next lemma will be needed in calculations below.
Lemma 2.2. Let f ∈ O N be a superfunction and Y ∈ SM be a super vector field. Then Proof. The assertion holds by the following calculation in coordinates {η i } on N .
The matrix groups GL n|m (A) and OSp (t,s)|2m (A) give rise to super Lie groups GL n|m and OSp (t,s)|2m , respectively. They are examples of matrix super Lie groups and, for the treatment of G-structures below, are most conveniently described in terms of S-points. GL n|m is the functor which sends any supermanifold S to the multiplicative group GL n|m (O S (S)) of even and invertible n|m-matrices with values in O S (S) and any mor- . This functor is representable such that, on the level of S-points, multiplication is ordinary matrix multiplication. The same construction can be applied to natural subgroups of GL n|m (A). For example, the generalised super Lie group OSp (t,s)|2m is the functor defined through S → OSp (t,s)|2m (O S (S)) and the corresponding morphism map. It is representable by the stabliliser condition [BCF09]. For our applications, it will be sufficient to consider all super Lie groups occuring as generalised. This is also the point of view of [ACDS97].
We shall next describe frame fields and G-structures on a supermanifold, where G is a super Lie group. In particular, we will see that supermetrics are equivalent to OSp (t,s)|2m -structures.
A tuple (X 1 , . . . , X n , X n+1 , . . . , X n+m ) of n even and m odd vector fields on an open set U ⊆ M is called a frame field, if the tuple (X 1 (p), . . . , X n+m (p)) of tangent vectors is a basis of the supervector space T p M for every p ∈ U . If U is sufficiently small (such that it lies in a coordinate chart), this condition is equivalent to (X 1 , . . . , X n+m ) being an adapted O M (U )-basis. The frame fields form a sheaf on M , denoted U → F(U ).
The super Lie group GL n|m induces a sheaf of groups over M via GL n|m (U ) := GL n|m (O M (U )), where the latter corresponds to a morphism U → GL n|m of supermanifolds. GL n|m (U ) acts naturally from the right on the frame fields F(U ). Explicitly, where A jk denotes the jk-th entry of the matrix of A ∈ GL n|m (U ). If U is sufficiently small, we see that this action is simply transitive.

Definition 2.3 ([ACDS97]
). Let G ⊆ GL n|m be a Lie subgroup. A G-structure on a supermanifold M is a sheaf F G of subsets F G ⊆ F such that G(U ) acts on F G (U ) and for all points, there is a neighbourhood for which the action is simply transitive.
Thus, in particular, every supermetric g defines an Osp (t,s)|2m -structure F g . Conversely, assume that F OSp is an Osp (t,s)|2m -structure. We construct a supermetric g as follows. Let (X 1 , . . .) ∈ F OSp and define g by g(X i , X j ) := g 0 (e i , e j ) and superbilinear extension. This definition does not depend on the chosen frame field in F OSp and is such that F g = F OSp .
An automorphism of M is an isomorphism Φ : M → M of supermanifolds. Its differential (4) can be identified with a sheaf morphism dΦ : Definition 2.5. Let Φ : M → M be an automorphism of a supermanifold M . Let g be a semi-Riemannian supermetric and F G be a G-structure on M .
thus proving one direction. The converse follows from the same characterisation together with superbilinearity.

Killing Vector Fields
In this section, we shall give five equivalent characterisations of Killing vector fields on supermanifolds. While the building blocks of the theory are mostly available in the literature, a homogeneous treatment such as below is still missing, and there is sometimes some confusion about the concept. In particular, we will see how the notion of a Killing vector field as treated in [ACDS97] is different from the canonical one. We start with the Lie derivative of tensors.

The Lie Derivative
The Lie derivative of a tensor on a manifold M with respect to a vector field X ∈ Γ(T M ) is defined by means of the flow ϕ of X which has the defining properties ∂ ∂t ϕ x (t) = X • ϕ x (t) and ϕ x (0) = x. By a standard theorem [War83], every vector field X possesses a unique smooth flow ϕ : Now let X be a vector field on a supermanifold andX be its canonical projection.
We recall results concerning the flow of super vector fields from [MSV93]. To integrate also odd vector fields, the derivation ∂ ∂t needs to be endowed by an odd part to a (super-)derivation D. As argued in [MSV93], one is naturally let to one of three integration models to the three different super Lie algebra structures on R 1|1 . Due to the ev-morphism, all of the following does, however, not depend on the choice of the integration model and, for brevity, we shall denote either derivation by D ∈ S(R 1|1 ). To simplify notation, we shall also denote its lift to D(X) by the same symbol D.
Definition 3.1. Let X be a super vector field. Its flow is a morphism Φ = (ϕ, φ) : D(X) → M such that the following equations hold which are referred to as the flow condition and initial condition, respectively.
In calculations, only the homogeneous part of D with the same parity as X occurs since the flow equation splits into two equations (according to the Z 2 -decomposition of X and D), of which one vanishes if X is homogeneous. By a slight abuse of notation, we may thus write |D| = |X|. For the strong flow condition D • φ = φ • X without the ev morphism to hold, X must satisfy certain conditions as shown in [MSV93]. where, on the right hand side, the pullback Φ * B is implicitly understood to be restricted to vector fields in SM ⊆ S(D(X)).
Lemma 3.5. Let f ∈ O M be a superfunction and Y ∈ SM be a super vector field. Then, for X, Y and f of homogeneous parity, Proof. Using the defining properties of the flow, we calculate The second assertion holds by the following calculation.
Here, we used Lem. 2.2 and t, τ -independence of Y .
Lemma 3.6. Let Φ : D(X) → M be the flow of a vector field X. Then the initial condition implies the generalisation for multilinear forms B.
Proof. We prove the statement for a one-form F , the general case is analogous. Let Y ∈ SM be a vector field. We let η 1 , . . . , η n+m denote local coordinates on M and ξ 1 , . . . , ξ n+m+2 the corresponding coordinates on D(X) such that ξ i = η i for i ∈ {1, . . . , n + m} and ξ n+m+1 = t and ξ n+m+2 = τ . Then By assumption, Y t = Y τ = 0 such that we may replace ∂ ∂ξ j by ∂ ∂η j , which commutes with ev, such that Proof. The first assertion follows from a straightforward calculation, using (7), the derivation property of D as well as ev| t=0 (f · g) = (ev| t=0 f ) · (ev t=0 g) and Lem. 3.5 and Lem. 3.6. The second assertion is shown analogous.
Any multilinear form can be (locally) written as the tensor product of one-forms of the form df for a superfunction, multiplied with a superfunction. By Lem. 3.5 the Lie derivative of these building blocks is independent of the integration model chosen (i.e. independent of D). By Lem. 3.7, the Lie derivative of a general multilinear form is uniquely determined by the building blocks and thus also independent of the integration model. In particular, for an even bilinear form, one starts with as in (3). An explicit calculation yields the following result.
The right hand side of Lem. 3.8 is taken as the definition of the Lie derivative in [Kli05] (up to a global sign). For semi-Riemannian supermetrics, we have the following characterisation, which naturally generalises the analogous formula in the classical case. The proof is based on ∇ being metric and torsion-free and (up to signs) the same as in the classical case, thus omitted.
Lemma 3.9. Let g be a semi-Riemannian supermetric and ∇ the Levi-Civita superconnection. Then the Lie derivative of g can be written For the proof of the following characterisation theorem, we shall need the flow equation for superbilinear forms as stated in the next lemma, which is easily generalised to general multilinear forms. A similar result is stated in [MSV93] as Prp. 4.3.

Killing Vector Fields
for a superfunction f as well as df . Writing B in local form as in (3), a further calculation using these building blocks as well as Lem. 3.7 and (7) yields the assertion. It is clear that all steps can be done without ev provided that the strong flow condition holds.
Let g be a semi-Riemannian supermetric on M with associated OSp (t,s)|2m -structure F g = F OSp (t,s)|2m as in Lem. 2.4. Let U ⊆ M be open. We write E ∈ F g (U ) as E = (X 1 , . . . , X t+s+2m ) such that g(X i , X j ) = g 0 (e i , e j ). Any supermatrix with entries in O M (U ) acts on E via (8). Following [ACDS97], we call U "small" if it is such that the action of OSp (t,s)|2m (U ) on F g (U ) is simply transitive. Moreover, we set L X E := ([X, X 1 ] , . . . , [X, X t+s+2m ]) for a vector field X ∈ SM . With this notation, our characterisation theorem can be stated as follows.
Theorem 3.12. Let X ∈ SM be a vector field. Then the following conditions are equivalent.
(iii) The metric Φ * g = g is preserved by the flow Φ of X.
Proof. (i) ⇐⇒ (ii) is immediate by Lem. 3.9. (iii) =⇒ (i): Assume that Φ * g = g. Then, by t,τ -indepence of g, we yield : Let X be a Killing vector field, we want to show that Φ * g = g follows. Let Y, Z be vector fields of pure parity on M and consider the superfunction Since g is purely even and Φ, being a morphism, is even, f has a fixed parity and thus f 0 and f 1 are of opposite parity (because of τ ). By assumption, L X g = 0 vanishes. Therefore, by the flow equation of Lem. 3.11, we have The first and second terms are of opposite parity (as stated above). Therefore, each summand vanishes individually such that f 1 = 0 and ∂ t f 0 = 0. Consider the local representation f 0 = J f 0 J θ J with odd coordinates θ i on M and multiindices J, for which we yield ∂ t f 0 J = 0 (an equation for ordinary functions). Now, by Lem. 3.6, we have the initial condition where f 0 (t = 0) = J f θ J (t = 0)θ J (the right hand side can be expanded into an analogous sum). Therefore, we conclude for all vector fields Y, Z of fixed parity. Therefore, Φ * g = g which was to be shown.
(i) ⇐⇒ (v): Let E = (X 1 , . . .) ∈ F g (U ) and X ∈ SM . There is a supermatrix L ∈ gl t+s|2m (O M (U )) such that [X, X i ] = X m · L mi . By Lem. 3.8, we have where e i is the standard basis of O M (U ) t+s|2m and g 0 is the standard supermetric as in (1). It follows immediately that L X g = 0 is equivalent to L ∈ osp (t,s)|2m ⊗ O M (U ).
Finally, we consider Killing vector fields on spinor supermanifolds, the example of supermanifolds as considered in [ACDS97]. Let (M, g) be a spin manifold and consider a parallel non-degenerate suitable bulinear form g 1 on the spinor bundle S. We also assume that g 1 is skew-symmetric (consult [Har90] for a classification of such forms) such that g + g 1 induces a Riemannian supermetric on the split supermanifold (M, Γ( S)). There is a canonical monomorphism ι : Γ(T M ) ⊕ Γ(S * ) → SM of sheaves which induces an isomorphism ι : T p M ⊕ S * p → S p M for every p ∈ M . It follows that {ι(X i ), ι(s * j )} is a (local) basis for SM if {X i } is a basis of Γ(T M ) and {s * j } is a basis of S * . We identify sections s * ∈ Γ(S * ) with s ∈ Γ(S) via g 1 .
Lemma 3.13. Let s * ∈ Γ(S * ). Then the super vector field ι(s * ) is Killing if and only if s is a parallel spinor.
In particular, we obtain existence results for Killing vector fields on spinor supermanifolds from the classification of spin manifolds admitting parallel spinors [MS00]. The definition of a Killing vector field in [ACDS97] is as in Thm. 3.12(v) but with OSp (t,s)|2m replaced by some super Poincaré group. For the case of ι(s * ), this is proved to be equivalent to s being a twistor spinor, which is different from our characterisation.
Proof. It suffices to consider L ι(s * ) g with vector fields from the image of ι inserted.
Next, we consider Y, Z ∈ Γ(T M ) such that ι(Y ), ι(Z) ∈ SM 0 . The type of supermetric considered here vanishes upon insertion of an even and an odd vector field. We are thus led to L ι(s * ) (ι(Y ), ι(Z)) = 0 which is again no condition.
Next, we find L ι(s * ) (ι(t * ), ι(Y )) = −g 1 (t, ∇ Y s). A necessary and sufficient condition for the last term to vanish for all t * and Y is ∇s ≡ 0.

Superharmonic Field Theories
The classical harmonic action functional for maps ϕ : M → N between semi-Riemannian manifolds (M, h) and (N, g) reads Critical points of this functional are called harmonic maps. For an exhaustive treatment of the Riemannian case, consult [Xin96]. In this section, we study a natural generalisation of (9) for semi-Riemannian supermanifolds (M, h) and (N, g) and prove three Noether theorems in this context. Throughout, M is assumed to be compact and superoriented as explained below after introducing some terminology. For a superfunction f ∈ O M , we write f > 0 if f (p) > 0 for all p. Moreover, we set |f | := m · f , where m : M → {±1} is such that |f | > 0. It can be shown that, in case f is even and f > 0, there is a unique square root √ f which is constructed using Taylor-like expansion in odd coordinates. Now, M is called superoriented if it has an atlas of coordinate charts such that, for every coordinate transformation Φ = (ϕ, φ) : R n|m → R n|m , both det(dϕ) > 0 and sdet(dΦ) > 0 hold [Sha88]. Here, the first condition (classical orientedness) is needed to make the integral over densities (sections of the superdeterminant sheaf, see Chp. 3 of [DM99]) welldefined. In case of the second condition, the metric h induces a canonical volume form (density) dsvol h on M as follows. Let ξ = (x, θ) be local coordinates on M and denote by [d n xd m θ] the induced local density. Moreover, let sdet(h) denote the super-determinant of the matrix h ∂ ξ l , ∂ ξ k . With this notation, dsvol h can be defined by the local expression which is independent of the coordinates used. A treatment of Riemannian volume forms in a slightly different style can be found in Sec. 3.5 of [Han12].
Instead of plain morphisms Φ = (ϕ, φ) : M → N of supermanifolds, we consider morphisms Φ : M × R 0|L → N in order to obtain odd "component fields" (for simplicity, think of terms of φ) as models for fermions. Following [Hél09], we call such morphisms maps with flesh, while the same concept occurs with several names in the literature, see [DF99] and [Khe07]. For the following treatment, it suffices to consider a fixed value of L ∈ N which is large enough to make the calculations consistent, cf. the discussion in [Hél09]. For a functorial point of view (considering all values of L simultaneously), we refer to [Han12]. In the following, we shall simply write Φ : M → N for maps with flesh, leaving the superpoint R 0|L implicit. The differential dΦ is then implicitly rescricted to the tangent sheaf of M tensored with the algebra R L of superfunctions of R 0|L . Tensors on SM are similarly endowed to that sheaf by R L -multilinear extension. For details, consult [Gro11].
With these preparations, we define the superharmonic action functional as with str h as in (2), and where {e j } is a local OSp (t,s)|2m -frame on (M, h).
is called 2nd fundamental form.
To show supersymmetry, it thus suffices to consider coordinate vector fields. One verifies that the expression of  The corresponding theorem in [Han12] is formulated for a special case (in particular, h is Riemannian there), but the proof provided there applies to the general case.
We also need the divergence of a super vector field. Classically, it can be defined as tr(∇X). In the super case, an additional sign occurs since the map Y → ∇ Y X (which has the parity of X) is not a superlinear map in case X is odd.
For the characterisation in a local OSp (t,s)|2m -frame, beware that the supertrace is defined with respect to right coordinates. We define next the analogon for a vector field along Φ. Consider the super-bilinear form (X, Y ) → (−1) |X||ξ| g Φ (∇ X ξ, dΦ[Y ]). Again, the sign is necessary to make it a super-bilinear form.
Definition 4.6. Let ξ ∈ SΦ be a vector field along Φ. We define its divergence to be Lemma 4.7. Let ξ ∈ SΦ and set W ξ := g Φ (ξ, dΦ[e j ]) Je j which has the parity of ξ.
Proof. The assertion is shown by the following calculation, using |W ξ | = |ξ| as well as the metric property of both h and g Φ .

Target Space Symmetries
Killing vector fields ξ ∈ Γ(T N ) on the target space are infinitesimal symmetries of the harmonic action (9). This can be seen as follows. Consider the flow F t of ξ. We alter ϕ by moving along the flow lines via ϕ t (x) := F t (ϕ(x)). In physicists' notation, this means that, infinitesimally, ϕ(x) → ϕ(x) + ξ(ϕ(x))dt. The infinitesimal change of A(ϕ) by ξ thus becomes Consider now the context of the superharmonic action (10). We let ξ ∈ SN be a vector field on N and denote by F : D(ξ) × N → N its flow as in Def. 3.1. Note that F is a plain morphism of supermanifolds while Φ is a map with flesh. The analogon of Definition and Lemma 4.8. The infinitesimal change of A by ξ ∈ SN is Proof. By compactness of M , we may interchange integration and differentiation, such that In this calculation, the third equation holds since only F * g depends on the flow coordinates on D(ξ). It is proved by a straightforward calculation in local coordinates.
It follows that, again, Killing vector fields on N are infinitesimal symmetries! According to the Noether principle, there should be an induced conserved quantity. We will show next that this is indeed the case. The need the following analogon of Lem. 3.9.
Lemma 4.9. Let ∇ = ∇ Φ denote the pullback connection as in (5). Then Proof. Choosing local coordinates {i = η i } on N , the assertion is reduced to Lem. 3.9 as follows.

Domain Space Symmetries
We have seen that the infinitesimal change of the superharmonic action (10) by a Killing vector field on the target space vanishes. Let us now consider the corresponding infinitesimal change by a vector field ξ ∈ SM on the domain space with flow F : D(ξ)×M → M .
Definition and Lemma 4.11. The infinitesimal change of A by ξ ∈ SM is Proof. Analogous to the proof of Lem. 4.8, we calculate As usual, ξ is called an infinitesimal symmetry if this expression vanishes for all morphisms Φ. Opposed to the target space situation, the vanishing of the Lie derivative L ξ (Φ * g) (for all Φ) is harder to achieve. In case symmetry is present, it is usually only such that L ξ (Φ * g) is some exact term depending on Φ (but integrated over to zero).
Theorem 4.12 (Noether). Let ξ ∈ SM be a Φ-Killing vector field, i.e. such that Proof. Here, Φ * g is a supermetric. However, Lem. 3.9 is not directly applicable since it would lead to the Levi-Civita connection ∇ Φ * g of this supermetric rather than to the Levi-Civita connection ∇ = ∇ h of h. We thus step back and use Lem. 3.8 as well as torsion-freeness of ∇ to obtain 0 = L ξ (Φ * g) (e j , Je j ) = ξΦ * g (e j , Je j ) − Φ * g (∇ ξ e j , Je j ) + (−1) |ξ||e j | Φ * g ∇ e j ξ, Je j − (−1) |ξ||e j | Φ * g (e j , ∇ ξ (Je j )) + Φ * g e j , ∇ Je j ξ Now, by Lem. 2.1, g Φ is metric, and we thus obtain We combine the first and third and the second and fifth terms, respectively, such that We combine the first and fourth and second and sixth terms, respectively, and use Lem. 4.2 such that The second statement now follows immediately from Lem. 4.7.

Domain Space Symmetries II
In the previous subsection, we have considered Φ-Killing vector fields ξ ∈ SM . We will next prove a Noether theorem for the more common vector fields ξ ∈ SM which are Killing with respect to the metric h, thus generalising a classical result due to Baird and Eells [BE81]. We denote the super energy of Φ by e(Φ) := 1 2 str h Φ * g = 1 2 Φ * g (e j , Je j ) and define the stress-energy tensor by where ξ ∈ SM is a vector field. As for the signs, cf. the discussion in the context of Def. 4.5. Here, the second term cancels with the fourth and fifth such that  Theorem 4.15 (Noether). Let ξ ∈ SM be a Killing vector field and Φ be superharmonic. Then divY ξ = 0 vanishes.