Symmetries and conservation laws of a nonlinear sigma model with gravitino
Jürgen Jost, Enno Ke?ler, Jürgen Tolksdorf, Ruijun Wu, Miaomiao Zhu
aa r X i v : . [ m a t h . DG ] M a r SYMMETRIES AND CONSERVATION LAWS OFA NONLINEAR SIGMA MODEL WITH GRAVITINO
JÜRGEN JOST, ENNO KEßLER, JÜRGEN TOLKSDORF, RUIJUN WU, MIAOMIAO ZHU
Abstract.
We show that the action functional of the nonlinear sigma model with gravitinoconsidered in [18] is invariant under rescaled conformal transformations, super Weyl transfor-mations and diffeomorphisms. We give a careful geometric explanation how a variation of themetric leads to the corresponding variation of the spinors. In particular cases and despite usingonly commutative variables, the functional possesses a degenerate super symmetry. The corre-sponding conservation laws lead to a geometric interpretation of the energy-momentum tensorand supercurrent as holomorphic sections of appropriate bundles. Introduction
The main motivation for the introduction of the two-dimensional nonlinear supersymmetricsigma model in quantum field theory, or more specifically super gravity and super string theory,are its symmetries, see for instance [3, 9]. Furthermore, as argued in [20], the functional isdetermined by its symmetries together with suitable bounds on the order of its Euler–Lagrangeequations. While super symmetric models are usually formulated using anti-commutative vari-ables, in [18] an analogue of the two-dimensional nonlinear supersymmetric sigma model usingonly commutative variables was introduced. Here we would like to give a detailed geometricaccount of the symmetries of the purely commutative model considered in [18].In physics as well as in geometry, symmetries of a system always have important implications.In particular, differentiable symmetries of a system give rise to conservation laws—this is ba-sically what the famous Noether’s theorem states, see e.g. [14, Chapter 1], [15, Section 2.3.2]and [11]. To see how Noether’s principle helps, let us take a closer look at the harmonic maptheory as an example; this will also be the prototype for our work.The
Dirichlet energy for a smooth map φ : ( M, g ) → ( N, h ) from a Riemannian surface to aRiemannian manifold is given by E ( φ ) = 12 Z M | d φ | g ∨ ⊗ φ ∗ h d vol g = 12 Z M g αβ ∂φ i ∂x α ∂φ j ∂x β h ij ( φ ( x )) p det g ( x ) d x where g ∨ denotes the dual metric on the cotangent bundle and φ ∗ h the induced metric on thepullback bundle φ ∗ T N . The critical points of the energy functionals are named harmonic maps in [10], the study of which has been among the central topics in mathematics for a long time.Essential in the study of harmonic maps are the diffeomorphism invariance of the Dirichletenergy E ( φ ◦ f ; g f ) = E ( φ ; g ) , ∀ f ∈ Diff( M ) , Date : October 16, 2018.
Key words and phrases. nonlinear sigma model, gravitino, Noether’s theorem, symmetry, energy-momentumtensor, supercurrent, Dirac-harmonic map.Miaomiao Zhu was supported in part by the National Science Foundation of China (No. 11601325). We denote by g f the non-negative definite symmetric 2-form defined by g f ( X, Y )( x ) := g f ( x ) ( T f ( X ) , T f ( Y )) and as a peculiarity in the case that the domain is two dimensional, the conformal invariance E ( φ ; e u g ) = E ( φ ; g ) , ∀ u ∈ C ∞ ( M ) . Those two symmetries link the theory of harmonic maps on a Riemann surface to the Teichmüllertheory, see [16].The conservation laws corresponding to diffeomorphism invariance and conformal invarianceare expressed in terms of the energy-momentum tensor . The energy-momentum tensor is definedas the variation of the Dirichlet energy with respect to the metric, that is, for a smooth family ( g t ) of Riemannian metrics on M , dd t E ( φ, g t ) = − Z M (cid:28) ∂g t ∂t , T ( φ ; g t ) (cid:29) d vol g t . Explicitly the energy-momentum tensor is a symmetric 2-tensor T ( φ ; g t ) = T αβ d x α ⊗ d x β givenby T αβ = h ij ( φ ) ∂φ i ∂x α ∂φ j ∂x β − | d φ | g ∨ t ⊗ φ ∗ h g tαβ . The energy-momentum tensor is symmetric and traceless—though this is clear from the expres-sion, it roots in the fact that we are taking variations in the symmetric -tensors and that theenergy functional is conformally invariant in dimension two. Actually, consider the conformalfamily ( g t = e tu g ) , the conformal invariance says that E ( φ ; e tu g ) = E ( φ ; g ) , and consequently t (cid:12)(cid:12)(cid:12) t =0 E ( φ, e tu g ) = − Z M h ug, T ( φ ; g ) i d vol g = − Z M u · Tr g ( T ( φ ; g )) d vol g , which is equivalent to say that Tr g ( T ( φ ; g )) = 0 since the function u can be arbitrary.The conservation law corresponding to the diffeomorphism invariance of the Dirichlet energyis that the energy-momentum tensor is divergence-free on shell . Indeed, for a differentiablefamily ( f t ) of diffeomorphisms of M with f = Id M and a harmonic map φ , we have t (cid:12)(cid:12)(cid:12) t =0 E ( φ ◦ f t ; g f t ) = − Z M (cid:28) ∂∂t g f t (cid:12)(cid:12)(cid:12) t =0 , T ( φ ; g ) (cid:29) d vol g , When we denote the generator of f t by X ∈ Γ( T M ) the differential of the metric is given bythe following Lie-derivative dd t (cid:12)(cid:12)(cid:12) t =0 g f t = L X g. Consequently, when the map φ is harmonic, we have Z M hL X g, T ( φ ; g ) i d vol g = − Z M h X, div g ( T ( φ ; g )) i d vol g , where the second equality is recalled in Section 4. As the vector field X could be arbitrary, itfollows that the energy-momentum tensor is divergence-free. for any x ∈ M and any X, Y ∈ Γ( T M ) , which is commonly referred to as “pullback metric” when f is animmersion in differential geometry. We reserve the symbol f ∗ g for the induced metric on the pullback bundle f ∗ T M . Here we use the phrase “on shell”, as it is common in physics, to say that the Euler–Lagrange equations aresatisfied.
YMMETRIES AND CONSERVATION LAWS 3
The fact that the energy-momentum tensor is symmetric, traceless and divergence-free hasimportant geometric and analytic consequences. For instance, the space of symmetric, tracelessand divergence-free 2-tensors can be identified with the space of holomorphic quadratic differ-entials. Using existence results of harmonic maps and the Theorem of Riemann–Roch, one canthen show that the Teichmüller space is a ball of dimension p − , where p is the genus of thesurface.The action functional in [18] extends the Dirichlet energy to include spinor fields: a Dirac-term in the field ψ and additional terms inolving the gravitino field χ . While the Euler–Lagrangeequations for ψ yield a Dirac-type equation, the gravitino is more considered as a parameterin the theory, similar to the metric g . By analogy to the models studied in physics we expectconformal invariance, super Weyl invariance (an analogue to conformal invariance affectingthe gravitino) and diffeomorphism invariance. We will show that the functional introducedin [18] possesses indeed the aforementioned symmetries. The major difficulty in the study ofthose symmetries is that spinors depend on the metric. In order to express the variation ofthe spinors in terms of the variation of the metric we use the methods developped in [2]. Incontrast to the original physical model, the functional studied here possesses only a “degeneratesupersymmetry” in special cases.The conservation laws corresponding to those four symmetries are expressed in terms of theenergy-momentum tensor and the supercurrent, which is the variation of the action with re-spect to the gravitino. We show that rescaled conformal and super Weyl invariance lead toalgebraic constraints on the energy-momentum tensor T and the supercurrent J . Diffeomor-phism invariance and degenerate supersymmetry yield equations for the divergences of T and J . All four symmetries together allow for an interpretation of T and J as holomorphic sectionsof appropriate bundles. This is in full analogy to the super geometric setting, where T and J constitute tangent vectors to the moduli space of so-called super Riemann surfaces, see [17, 19].The article is organized as follows: First we consider a Dirac action as a toy model, as wellas to set up the notation for later use. The Weyl symmetry and the diffeomorphism invarianceare checked and the corresponding conservation laws are obtained. Then we put the main effortto study the nonlinear sigma model with gravitino in [18], and the four kinds of symmetriesmentioned above are discussed. We give explicit computation of the energy-momentum tensorand supercurrent in that case. The corresponding conservation laws are derived and explained.Finally we make a supplement about the divergence operators, both on symmetric 2-tensorsand on spinors, which are frequently used in the context.2. Dirac action and its conservation laws
In this section we consider a Dirac action. While various Dirac actions play a prominent rolein physics, we will use the Dirac action as an elaborate toy model. At the example of the Diracaction we will set up the notation and theory necessary to study symmetries and conservationlaws of a sigma-model involving spinors.The geometric setup is as in [18]. Let ( M, g ) be a closed oriented Riemannian surface andlet ξ : P Spin ( M, g ) → P SO ( M, g ) be a spin structure on ( M, g ) . The irreducible representation ofthe Clifford bundle Cl , induces the real spin representation µ : Spin(2) → GL ( V ) where V isa representation space and is isomorphic to R . From this one can form the associated spinorbundle S g := P Spin ( M, g ) × µ V. JÜRGEN JOST, ENNO KEßLER, JÜRGEN TOLKSDORF, RUIJUN WU, MIAOMIAO ZHU
Note that here by the lower index in S g we emphasize the dependence on the Riemannianmetric g . There are a canonical fiberwise real inner product denoted by g s , and a canonicalmetric connection ∇ s . The connections on Γ( S g ) and on the algebra bundle Cl( M, − g ) satisfya Leibniz rule, see e.g. [21]. Choosing an isomorphism between V and Cl , we get a Cliffordmap denoted by γ : T M → End( S g ) which satisfies the Clifford relation γ ( X ) γ ( Y ) + γ ( Y ) γ ( X ) = − g ( X, Y ) for all X, Y ∈ Γ( T M ) . Sections of S g will be referred to as (pure) spinors, which is used to describe matter fields withnon-integer spins in physics.On the spinor bundle S g there is a spin Dirac operator /∂ g : Γ( S g ) → Γ( S g ) which is definedby /∂ g σ := γ ( e α ) ∇ se α σ for σ ∈ Γ( S g ) , where ( e α ) is a local oriented g -orthonormal frame. It is easily to check that /∂ g is well-defined,that is, independent of the g -orthonormal frame. This Dirac operator is an elliptic self-adjointoperator, and has a real spectrum which is unbounded in either side in R . A spinor q ∈ Γ( S ) is called harmonic if it is in the kernel of /∂ g , i.e. /∂ g q = 0 . One can refer to [13, 1] for moreknowledge about harmonic spinors.Here we consider a Dirac action as a functional defined on spinors and Riemannian metricsby DA ( σ ; g ) := Z M g s ( σ, /∂ g σ ) d vol g for g ∈ Met ( M ) and for σ ∈ Γ( S g ) where Met ( M ) denotes the space of all Riemannian metrics on M . From the facts above on thespectrum about /∂ g one knows that this action functional cannot be bounded from either side.Recall that the spin Dirac operator /∂ g is self-adjoint with respect to the L -inner product. Aneasy calculation shows that for a smooth family ( σ t ) t of spinors, dd t (cid:12)(cid:12)(cid:12) t =0 DA ( σ t ; g ) = 2 Z M (cid:28)(cid:18) dd t (cid:12)(cid:12)(cid:12) t =0 σ t (cid:19) , /∂ g σ (cid:29) d vol g . Thus the Euler–Lagrange equation for the Dirac action is /∂ g σ = 0 , that is, the critical spinorsare the harmonic spinors.2.1. Rescaled conformal invariance of the Dirac action.
First observe that mathemati-cally rigorously, one can not keep the spinor σ fixed while varying the metric, because the spinstructure depends on the metric g and so do the spinor bundle S g and the Dirac operator /∂ g .To overcome this difficulty we take the approach developed in [2].Let g and g ′ be two Riemannian metrics on M . There is a unique self-adjoint endomorphism H ∈ End(
T M ) such that g ′ ( · , · ) = g ( H · , · ) . Thus if we set b ≡ b gg ′ := H − / ∈ Aut(
T M ) , then b : ( T M, g ) → ( T M, g ′ ) is an isometry of Riemannian vector bundles. More explicitly, g ( · , · ) = g ′ ( b ( · ) , b ( · )) .Next we lift the isomorphism b to the spin level. Since b : ( T M, g ) → ( T M, g ′ ) is SO(2) -equivariant, it defines a principal bundle isomorphism b : P SO ( M, g ) → P SO ( M, g ′ ) Note that in most applications in physics a mass term is added to the Lagrangian.
YMMETRIES AND CONSERVATION LAWS 5 which lifts to an isomorphism ˜ b : P Spin ( M, g ) → P Spin ( M, g ′ ) on the spin level which covers b . Here it is important to assume that P Spin ( M, g ′ ) corresponds to the same topological spinstructure as P Spin ( M, g ) . Let now S g and S g ′ be spinor bundles associated to P Spin ( M, g ) and P Spin ( M, g ′ ) via the same representation µ : Spin → SO ( V ) . Since ˜ b is Spin(2) -equivariant, itinduces an isometry between the spinor bundles as Riemannian vector bundles, denoted by β ≡ β gg ′ : S g → S g ′ , which is to say, g s ( · , · ) = g ′ s ( β ( · ) , β ( · )) . Moreover, note that for a vector v ∈ T M and a spinor σ ∈ S g , β ( γ ( v ) σ ) = γ ′ ( b ( v )) β ( σ ) . where γ ′ denotes the Clifford map with respect to the metric g ′ .Next we consider the conformal behavior of the Dirac action. First recall the followingProposition: Proposition 1 (see [12, Prop. 1.3.10]) . Let u ∈ C ∞ ( M ) , then g ′ = e u g is a Riemannianmetric conformal to g . The spin Dirac operators /∂ g and /∂ g ′ satisfy /∂ g ′ β ( σ ) = e − u β (cid:18) /∂ g σ + m − γ (grad( u )) σ (cid:19) , where m = dim M . Moreover, after a rescaling by e − m − u , the Dirac operator transformshomogeneously: /∂ g ′ β ( e − m − u σ ) = e − m +12 u β ( /∂ g σ ) . This implies that the dimension of the space of harmonic spinors is a conformal invariant.Let us now specify to the surface case, m = 2 . The map β allows us to transform the spinoraccording to the change of the metric. However, by Proposition 1 an additional rescaling isnecessary to make the Dirac action invariant. Indeed, DA ( βσ ; e u g ) = Z M e u g s ( σ, /∂ g σ ) d vol g , which is in general not equal to DA ( σ ; g ) . In contrast, if the rescaling by e − m − u = e − u istaken into account the Dirac action is invariant:(1) DA ( e − u βσ ; e u g ) = Z M g ′ s (cid:16) e − u βσ, e − u β ( /∂ g σ ) (cid:17) e u d vol g = DA ( σ ; g ) . Hence, strictly speaking the Dirac action is not conformally invariant. We will rather use theterm “rescaled conformal invariance” for the invariance given in Equation (1).2.2.
The energy-momentum tensor of the Dirac action.
We need to calculate the vari-ation of the Dirac action with respect to the Riemannian metric. To this end we need aparametrized version of the theory presented in the last subsection. Let ( g t ) t be a smooth familyof Riemannian metrics parametrized by t in a neighborhood of zero in R such that g = g . The t -derivative at t = 0 is a smooth symmetric -form, say k ∈ Γ(Sym( T ∗ M ⊗ T ∗ M )) . Conversely,any k ∈ Γ(Sym( T ∗ M ⊗ T ∗ M )) can be represented in this way, for instance, ( g t := g + tk ) t issuch a family for | t | small enough. JÜRGEN JOST, ENNO KEßLER, JÜRGEN TOLKSDORF, RUIJUN WU, MIAOMIAO ZHU
As before there is a unique family of self-adjoint endomorphisms ( H t ) t ⊂ End(
T M ) such that g t ( · , · ) = g ( H t · , · ) . We will denote by b t ≡ b gg t and β t ≡ β gg t : S g → S g t . Notice that dd t (cid:12)(cid:12)(cid:12) t =0 b t = − K ∈ End(
T M ) , where K is the endomorphism associated to k by metric dualization, which is also the t -derivative of H t at t = 0 . Let { e α } be a local oriented g -orthonormal frame, then { E α ( t ) = b t ( e α ) } is a local oriented g t -orthonormal frame, and dd t (cid:12)(cid:12)(cid:12) t =0 E α ( t ) = dd t (cid:12)(cid:12)(cid:12) t =0 b t ( e α ) = − Ke α = − K βα e β . We also need to consider the volume forms of different metrics: d vol g t = √ det H t d vol g . The t -derivative at t = 0 is dd t (cid:12)(cid:12)(cid:12) t =0 d vol g t = (cid:18) dd t (cid:12)(cid:12)(cid:12) t =0 p det H t (cid:19) d vol g = 12 Tr g ( K ) d vol g . Transport the Dirac operator /∂ g t : Γ( S g t ) → Γ( S g t ) via the isometry β t to obtain a differentialoperator on Γ( S g ) : /∂ g t := β − t ◦ /∂ g t ◦ β t : Γ( S g ) → Γ( S g ) . From [2] and [22], we know that dd t (cid:12)(cid:12)(cid:12) t =0 /∂ g t = − γ ( e α ) ∇ sK ( e α ) + 14 γ (cid:16) grad(Tr g k ) − div g ( k ) ♯ (cid:17) , where the g -divergence div g ( k ) is recalled in Section 4.Now we are ready to derive the variation of the Dirac action for the metric g while keepingthe spinors essentially “unchanged”. The variation formula for metrics can be obtained in thefollowing way: dd t (cid:12)(cid:12)(cid:12) t =0 DA ( β t σ ; g t ) = dd t (cid:12)(cid:12)(cid:12) t =0 Z M g s ( t ) (cid:0) β t σ, /∂ g t β t σ (cid:1) d vol g t = dd t (cid:12)(cid:12)(cid:12) t =0 Z M g s ( σ, β − t /∂ g t β t σ ) d vol g t = dd t (cid:12)(cid:12)(cid:12) t =0 Z M g s ( σ, /∂ g t σ ) d vol g t = − Z M h k, T ( σ ; g ) i d vol g , where T ( σ ; g ) = T αβ e α ⊗ e β is the energy-momentum tensor, with coefficients T αβ = 12 D σ, γ ( e α ) ∇ se β σ + γ ( e β ) ∇ se α σ E g s − h σ, /∂ g σ i g s g αβ . By construction the energy-momentum tensor is symmetric, but not necessarily traceless; infact, Tr g ( T ) = − (cid:10) σ, /∂ g σ (cid:11) g s . YMMETRIES AND CONSERVATION LAWS 7
This can be explained by the conservation law associated to the rescaled conformal invarianceas follows. t (cid:12)(cid:12)(cid:12)(cid:12) t =0 DA ( e − tu βσ, e tu g ) = Z M (cid:28) − uσ, /∂σ (cid:29) − h ug, T i d vol g = − Z M u (cid:0)(cid:10) σ, /∂σ (cid:11) + Tr g ( T ) (cid:1) d vol g The conservation law to the rescaled conformal invariance prescribes the trace of the energy-momentum tensor. As the Dirac action is in general not conformally invariant, the trace isnon-zero. Notice that on shell the energy-momentum tensor is traceless.2.3.
Diffeomorphism invariance of the Dirac action.
While diffeomorphism invariance ofthe Dirac action may seem obvious at first glance, we need a precise formulation to obtain thecorresponding conservation law.Let f be a smooth diffeomorphisms of M . Pull the metric g back via f to get a Riemannianmetric g f on T M . The differential
T f of f is an isometry of Riemannian vector bundles whichcovers the map f ; that is, the following diagram commutes: ( T M, g f ) ( T M, g ) M M
T ff
This induces a map on the orthonormal frame bundles, which is also denoted by
T f . As
T f is SO(2) -equivariant, there exists a unique spin structure P Spin ( M, g f ) → P SO ( M, g f ) such that T f lifts to the corresponding principal
Spin(2) -bundles. Indeed, as explained in [21, TheoremII.1.7], the spin structures on M are in one-to-one correspondence to elements of H ( M, Z ) and P Spin ( M, g f ) is given by the pullback of the cohomology class corresponding to P Spin ( M, g ) .Hence, in general, the topological spin structures corresponding to P Spin ( M, g ) and P Spin ( M, g f ) do not need to coincide. The situation is summarized the following commutative diagram inthe left:(2) P Spin ( M, g f ) P Spin ( M, g ) P SO ( M, g f ) P SO ( M, g ) M M f T fT ff ( S g f , g fs ) ( S g , g s ) M M Ff An irreducible spin representation µ : Spin(2) → SO ( V ) will give rise to spinor bundles associ-ated to the above principal Spin(2) -bundles, and the isomorphism f T f induces an isometry F of the corresponding spinor bundles, as shown in the commutative diagram above in the right.In particular note that F being an isometry means that for any y ∈ M with f ( y ) ≡ x ∈ M ,and for any σ , σ ∈ ( S g ) x , g s | x ( σ , σ ) = g fs | y (cid:16) F − | y σ , F − | y σ (cid:17) . JÜRGEN JOST, ENNO KEßLER, JÜRGEN TOLKSDORF, RUIJUN WU, MIAOMIAO ZHU
As a result, the Dirac operators /∂ g on S g and /∂ g f on S g f are the “same” up to the isometry F in the sense that ( /∂ g ) x = F | y ◦ ( /∂ g f ) y ◦ F − | y . Remark.
One should note that this formula holds because F is induced by an isometry f : ( M, g f ) → ( M, g ) . Then T f preserves the Levi-Civita connections on the tangent bundles, and hence F preserves the spin connection. As a comparison, although the morphism β constructed in theprevious subsection is also an isometry, it will not preserve the Dirac operator. Indeed, themap b preserves the metrics but not the Lie brackets (this can be seen already in the case of aconformal perturbation of the Riemannian metric), hence b does not preserve the Levi-Civitaconnection and consequently β does not necessarily preserve the spin connections. This is thereason why a change of the Riemannian metric will give rise to a change of the Dirac operator,which we have used before.Now we explain the diffeomorphism invariance. The claim is that(3) DA ( σ ; g ) = DA ( F − ◦ σ ◦ f ; g f ) . Notice that F − ◦ σ ◦ f is a section of the bundle S g f → ( M, g f ) , which is clear from thediagram (2). Then we have DA ( F − ◦ σ ◦ f ; g f ) = Z M g fs | y (cid:16) F − | y ( σ ◦ f ) y , ( /∂ g f ) y (cid:16) F − | y ( σ ◦ f ) y (cid:17)(cid:17) d vol g f ( y )= Z M g s | f ( y ) (cid:16) σ ( f ( y )) , F | y ( /∂ g f ) y F − | y ( σ ( f ( y ))) (cid:17) d vol g f ( y )= Z M g s | x (cid:0) σ ( x ) , ( /∂ g ) x σ ( x ) (cid:1) d vol g ( x )= DA ( σ ; g ) . Thus the claim (3) is confirmed.To obtain the corresponding conservation law, we take a (local) one-parameter group ofdiffeomorphisms ( f t ) of M with f = Id M and dd t (cid:12)(cid:12)(cid:12) t =0 f t = X ∈ Γ( T M ) . For example, the flow generated by X is such a family; the flow is global since M is assumed tobe compact. Write M t = f − t ( M ) = f − t ( M ) and denote the pullback metrics on M by g t ≡ g f t .The differential T f t is again an isometry and hence can be viewed as a map of the principal SO(2) -bundles. Note that M t = M and hence g is also a Riemannian metric on T M t . Thesetwo metrics can be related by an isometry b t ≡ b gg t : ( T M t , g ) → ( T M t , g t ) as before and can belift to the corresponding principal Spin(2) -bundles. In order to construct the lift, we notice thata diffeomorphism homotopic to the identity preserves the topological spin structure. Therefore
YMMETRIES AND CONSERVATION LAWS 9 we have the following commutative diagram: P Spin ( M, g ) P Spin ( M t , g t ) P Spin ( M, g ) P SO ( M, g ) P SO ( M t , g t ) P SO ( M, g ) M M t My Id( y ) = y x = f t ( y ) ˜ b t f T f t b t T f t Id f t The bottom line exhibits the pointwise behavior of the maps on base manifolds. Note thatin this diagram all the horizontal maps are diffeomorphisms/isomorphisms. The associatedcommutative diagram of spinor bundles is given by ( S g , g s ) ( S g t , g s ( t )) ( S g , g s ) M M t M β t Id F t f t The diffeomorphism invariance of the Dirac action says that for any t , DA ( σ ; g ) = DA ( F − t ◦ σ ◦ f t ; g t ) . Taking linearizations of both sides, we get t (cid:12)(cid:12)(cid:12) t =0 DA ( F − t ◦ σ ◦ f t ; g f t )= dd t (cid:12)(cid:12)(cid:12) t =0 Z M t g s ( t ) | y (cid:16) F t − | y ( σ ◦ f t ) y , ( /∂ g t ) y (cid:16) F t − | y ( σ ◦ f t ) y (cid:17)(cid:17) d vol g t ( y )= dd t (cid:12)(cid:12)(cid:12) t =0 Z M t g s ( t ) | y (cid:16) β t ◦ β − t ◦ F t − | y ( σ ◦ f t ) y , ( /∂ g t ) y (cid:16) β t ◦ β − t ◦ F t − | y ( σ ◦ f t ) y (cid:17)(cid:17) d vol g t ( y ) . Write σ t ≡ β − t ◦ F t − | y ( σ ◦ f t ) y . Then using the spinor Lie-derivative as introduced in [2], wehave dd t (cid:12)(cid:12)(cid:12) t =0 σ t ( y ) = dd t (cid:12)(cid:12)(cid:12) t =0 β − t ◦ F t − | y ( σ ◦ f t ) y = L SX σ ( y ) . Then, from the invariance (3) it follows that t (cid:12)(cid:12)(cid:12) t =0 Z M t g s ( t ) (cid:0) β t σ t , /∂ g t β t σ t (cid:1) d vol g t = 2 Z M hL SX σ, /∂ g σ i d vol g − Z M hL X g, T i d vol g = 2 Z M h X, div σ ( /∂ g σ ) i d vol g + Z M h X, div g ( T ) i d vol g . Here the expression div σ is a formal analogue of the divergence for spinors, explained in Section 4.As X can be arbitrary, it follows that for any σ , the conservation law reads σ ( /∂ g σ ) + div g ( T ) = 0 In particular, along critical spinors which are the harmonic spinors, the energy-momentumtensor is divergence-free. This confirms our expectation: the energy-momentum tensor isdivergence-free on shell.2.4.
Conclusions and remarks.
We summarize our discussion about the Dirac action in thefollowing theorem.
Theorem 1.
Let S g be a spinor bundle over ( M, g ) , and consider the following Dirac actiondefined on pure spinors σ ∈ Γ( S g ) and Riemannian metrics g by DA ( σ ; g ) = Z M g s ( σ, /∂ g σ ) d vol g . (1) The total variation formula is δDA = Z M h δσ, EL ( σ ) i − h δg, T ( σ ; g ) i d vol g . where the Euler–Lagrange equation for the spinor is EL ( σ ) = /∂ g σ = 0 , and the energy-momentum tensor in a local oriented orthonormal frame ( e α ) is T ( σ ; g ) = (cid:26) D σ, γ ( e α ) ∇ se β σ + γ ( e β ) ∇ se α σ E g s − h σ, /∂ g σ i g s g αβ (cid:27) e α ⊗ e β . (2) This Dirac action is invariant under rescaled conformal transformations. That is, any u ∈ C ∞ ( M ) induces a conformal metric g ′ = e u g and there is a map β : S g → S g ′ suchthat DA ( e − u βσ ; g ′ ) = DA ( σ ; g ) .(3) The Dirac action is invariant under diffeomorphisms. That is, for any f ∈ Diff( M ) ,there is an induced map F : S g f → S g such that DA ( F − ◦ σ ◦ f ; g f ) = DA ( σ ; g ) .(4) The energy-momentum is a symmetric 2-tensor, with trace Tr g ( T ) = −h σ, /∂ g σ i g s . When σ is harmonic, T is traceless and divergence-free, hence corresponds to a holomorphic qua-dratic differential on M . The methods we presented here do generalize to higher dimensions and all results exceptthe rescaled conformal invariance do hold there. It is also straightforward to adapt the theorypresented here to models with mass-term or other terms which do not involve derivatives ofspinor fields.3.
The case of the nonlinear sigma model with gravitinos
We now turn to the full nonlinear sigma model with gravitino as considered in [18]. Let usfirst briefly recall its construction. As before φ : ( M, g ) → ( N, h ) is a map from a Riemannsurface to a Riemannian manifold, and a fixed spin structure P Spin ( M, g ) ξ −→ P SO ( M, g ) hasbeen chosen. ( S g , g s ) is a rank-four real a spinor bundle associated to this spin structure.On the twisted vector bundle S g ⊗ φ ∗ T N there is the induced metric g s ⊗ φ ∗ h and inducedmetric connection, denoted by e ∇ . A section ψ of this bundle, called a “vector spinor”, will serveas a super partner of the map φ . Note that a spin Dirac operator /D g can be defined in thesame manner as above, i.e., /D g ψ = γ ( e α ) e ∇ e α ψ. YMMETRIES AND CONSERVATION LAWS 11
Let { y i } be a local coordinate of N , then ψ can be locally written as ψ = ψ j ⊗ φ ∗ (cid:16) ∂∂y j (cid:17) , whereeach ψ j is a (local) pure spinor. The Dirac operator then acts as /D g ψ = γ ( e α ) (cid:18) ∇ se α ψ j ⊗ φ ∗ (cid:18) ∂∂y j (cid:19) + ψ j ⊗ ∇ φ ∗ T Ne α ∂∂y j (cid:19) = /∂ g ψ j ⊗ φ ∗ (cid:18) ∂∂y j (cid:19) + γ ( e α ) ψ j ⊗ φ ∗ (cid:18) ∇ T NT φ ( e α ) ∂∂y j (cid:19) . (4)A super partner of the Riemannian metric g is given by a gravitino χ , which is a section ofthe bundle S g ⊗ T M . This bundle splits into two orthogonal summands, with the projectionsgiven by
P χ = − γ ( e β ) γ ( e α ) χ α ⊗ e β , Qχ = − γ ( e α ) γ ( e β ) χ α ⊗ e β . For further geometric properties of gravitinos in super geometry we refer to [17].Then the action functional is given by A ( φ, ψ ; g, χ ) := Z M | d φ | g ∨ ⊗ φ ∗ h + h ψ, /Dψ i g s ⊗ φ ∗ h − h ( ⊗ φ ∗ )( Qχ ) , ψ i g s ⊗ φ ∗ h − | Qχ | g s ⊗ g | ψ | g s ⊗ φ ∗ h −
16 R N ( ψ ) d vol g , (5)where R N stands for the pullback of the curvature of N under φ , and the curvature term in theaction is defined by −
16 R N ( ψ ) = −
16 R
Nijkl ( φ ) (cid:10) ψ i , ψ k (cid:11) g s (cid:10) ψ j , ψ l (cid:11) g s . The Euler–Lagrange equation of A are computed in [18]: EL ( φ ) = τ ( φ ) −
12 R φ ∗ T N ( ψ, e α · ψ ) φ ∗ e α + 112 S ∇ R ( ψ )+ h∇ Se β ( e α · e β · χ α ) , ψ i S + h e α · e β · χ α , ∇ S ⊗ φ ∗ T Ne β ψ i S , EL ( ψ ) = /Dψ − | Qχ | ψ − SR ( ψ ) − ⊗ φ ∗ ) Qχ (6)Here τ ( φ ) is the tension field of the map φ and SR ( ψ ) is a term involving the curvature of thetarget and is of third order in ψ . The term S ∇ R ( ψ ) involves derivatives of R N and is of fourthorder in ψ . For the precise form of the curvature terms we refer to [18].From the case of the Dirac action treated before and the analogous two-dimensional nonlinearsuper symmetric sigma model considered in super string theory (see for example [3, 9]) that themodel is invariant under (rescaled) conformal transformations, super Weyl transformations anddiffeomorphisms. We will see that this expectation is met. The corresponding conservationslaws will be formulated with the help of the energy-momentum tensor and the supercurrentwhich is given by the variation of the action with respect to the gravitino. The rescaled con-formal and super Weyl symmetry lead to algebraic properties of energy-momentum tensor andsupercurrent respectively. The diffeomorphism invariance leads to a coupled differential equa-tion for the energy-momentum tensor and the supercurrent which holds on shell. At first sightit might be surprising that the conservation law for the diffeomorphism invariance couples theenergy-momentum tensor and the supercurrent. Notice, however, that by “on shell” we onlyassume that the equations of motion for the map φ and the vector spinor ψ hold. An important feature of the model considered in physics is super symmetry. As the modelintroduced in [18] was built only with commuting variables, full supersymmetry cannot beexpected. However, we show that in special cases there is a remainder of super symmetry,which we call “degenerate supersymmetry”. The conservation law associated to degenerate supersymmetry leads to a differential equation for the supercurrent. Together with the conservationlaw from diffeomorphism invariance we can conclude in this special case that energy-momentumtensor and supercurrent are holomorphic quantities.One sees immediately that the Dirac-harmonic functional introduced in [4, 5] and some ofits variants [6, 7] are special cases of this action functional. Hence the computations presentedhere contain the several variants of the Dirac-harmonic action functional. This leads to aninteresting application of degenerate super symmetry to Dirac-harmonic maps, showing thattwo known simple solutions are actually related by degenerate super symmetry.3.1.
Rescaled conformal symmetry and super Weyl symmetry.
As explained beforethe spin structure, spinor bundle and Dirac operator depend on the chosen metric g . If we nowchoose a metric g ′ = e u g that is conformal to g we also need the isometric bundle isomorphisms b : ( T M, g ) → ( T M, g ′ ) and β : ( S g , g s ) → ( S g ′ , g ′ s ) defined in Section 2.1. Furthermore, as inthe case of the Dirac action an additional rescaling of the spinors is needed. Hence we obtainthe following rescaled conformal invariance:(7) A (cid:16) φ, e − u ( β ⊗ ) ψ ; e u g, e − u ( β ⊗ b ) χ (cid:17) = A ( φ, ψ ; g, χ ) . Indeed, under the conformal transformations, the volume form rescales as d vol g ′ = e u d vol g .The harmonic term behaves as in the classical theory and the Dirac term is very similar to thecase considered in Section 2.1. The three remaining terms can be checked easily using the factthat b and β are isometries. For the third term it holds: − h ( ⊗ φ ∗ ) Q ′ e − u ( β ⊗ b ) χ, e − u ( β ⊗ ) ψ i g ′ s ⊗ φ ∗ h = 2 e − u (cid:28) γ ′ ( b ( e β )) γ ′ ( b ( e α )) β ( χ β ) ⊗ b ( e α ) , β ( ψ j ) ⊗ φ ∗ (cid:18) ∂∂y j (cid:19)(cid:29) g ′ s ⊗ φ ∗ h = 2 e − u (cid:28) β ( γ ( e β ) γ ( e α ) χ β ) ⊗ ( e − u e α ) , β ( ψ j ) ⊗ φ ∗ (cid:18) ∂∂y j (cid:19)(cid:29) g ′ s ⊗ φ ∗ h = 2 e − u (cid:10) γ ( e β ) γ ( e α ) χ β ⊗ e α , ψ (cid:11) g s ⊗ φ ∗ h = − e − u h ( ⊗ φ ∗ ) Qχ, ψ i g s ⊗ φ ∗ h . For the fourth term notice | Q ′ ( β ⊗ b ) e − u χ | g ′ s ⊗ g ′ = e − u | ( β ⊗ b ) Qχ | g ′ s ⊗ g ′ = e − u | Qχ | g s ⊗ g , and | ( β ⊗ ) e − u ψ | g ′ s ⊗ φ ∗ h = e − u g ′ s ( βψ i , βψ j ) h ij ( φ ) = e − u g s ( ψ i , ψ j ) h ij ( φ ) = e − u | ψ | g s ⊗ φ ∗ h , which implies(8) | Q ′ ( β ⊗ b ) e − u χ | g ′ s ⊗ g ′ | ( β ⊗ ) e − u ψ | g ′ s ⊗ φ ∗ h = e − u | Qχ | g s ⊗ g | ψ | g s ⊗ φ ∗ h . The curvature term again uses that β is an isometry: R(( β ⊗ ) e − u ψ ) = e − u R Nijkl g ′ s (cid:0) βψ i , βψ k (cid:1) g ′ s (cid:0) βψ j , βψ l (cid:1) = e − u R Nijkl g s (cid:0) ψ i , ψ k (cid:1) g s (cid:0) ψ j , ψ l (cid:1) = e − u R( ψ ) . (9) YMMETRIES AND CONSERVATION LAWS 13
This completes the verification of the rescaled conformal invariance.In contrast to the rescaled conformal invariance, super Weyl invariance affects only the grav-itino. As only Qχ enters the action functional, we have(10) A ( φ, ψ ; g, χ + ζ ) = A ( φ, ψ ; g, χ ) . for Qζ = 0 . The property (10) is called super Weyl invariance.We now take the opportunity to recall some properties of the bundle S g ⊗ T M and of theprojections P and Q , which have been presented in detail in [18]. The Riemann surface M possesses an almost complex structure J M such that g ( J M X, Y ) = d vol g ( X, Y ) for all vectorfields X and Y . Left multiplication of spinors by the negative volume form − ω = − e · e ∈ Cl ( M, − g ) yields an almost complex structure on S g that is compatible with J M , that is γ ( J M X ) s = − γ ( X ) ωs for all spinors s . The decomposition of S g in even and odd part yields S g = Σ ⊕ Σ , where both summands are isomorphic as associated vector bundles to P Spin ( M, g ) .The almost complex structure ω restricts to Σ and the restriction will be denoted by J Σ . Forthe complex line bundle W = (Σ , J Σ ) it holds that W ⊗ C W = T ∗ M . With this preparation wecan now identify the summands in S g ⊗ T M = ker Q ⊕ Im Q to be ker Q = S g = W ⊕ W Im Q = S g ⊗ C T M = ( W ⊗ C W ∨ ⊗ C W ∨ ) ⊕ ( W ⊗ C W ∨ ⊗ C W ∨ ) Hence both ker Q and Im Q are naturally holomorphic vector bundles.3.2. Supercurrent.
The supercurrent is the variation of the action with respect to the grav-itino. As the gravitino enters the action only algebraically, computation of the supercurrent isstraightforward. Fix ( φ, ψ ) as well as the metric g and vary the gravitino via X ( t ) = X α ( t ) ⊗ e α ∈ Γ( S g ⊗ T M ) with X (0) = χ and dd t (cid:12)(cid:12)(cid:12) t =0 X α ( t ) = ζ α . Then dd t (cid:12)(cid:12)(cid:12) t =0 A ( φ, ψ ; g, X ( t )) = dd t (cid:12)(cid:12)(cid:12) t =0 Z M − h ( ⊗ φ ∗ )( QX ( t )) , ψ i g s ⊗ φ ∗ h − | QX ( t ) | g s ⊗ g | ψ | g s ⊗ φ ∗ h d vol g . This can be computed as follows. dd t (cid:12)(cid:12)(cid:12) t =0 Z M h γ ( e α ) γ ( e β ) X α ⊗ φ ∗ e β , ψ i g s ⊗ φ ∗ h d vol g = Z M (cid:28) γ ( e α ) γ ( e β ) (cid:18) dd t X α (cid:19) ⊗ φ ∗ e β , ψ (cid:29) g s ⊗ φ ∗ h d vol g (cid:12)(cid:12)(cid:12) t =0 = Z M (cid:28)(cid:18) dd t X α (cid:19) , h φ ∗ e β , γ ( e β ) γ ( e α ) ψ i φ ∗ h (cid:29) g s d vol g (cid:12)(cid:12)(cid:12) t =0 = Z M h ζ α , h φ ∗ e β , γ ( e β ) γ ( e α ) ψ i φ ∗ h i g s d vol g , dd t (cid:12)(cid:12)(cid:12) t =0 Z M h X β , γ ( e α ) γ ( e β ) X α i g s | ψ | g s ⊗ φ ∗ h d vol g = Z M (cid:28)(cid:18) dd t X α (cid:19) , γ ( e α ) γ ( e β ) X α (cid:29) g s | ψ | g s ⊗ φ ∗ h d vol g (cid:12)(cid:12)(cid:12) t =0 = Z M h ζ β , γ ( e α ) γ ( e β ) χ α i g s | ψ | g s ⊗ φ ∗ h d vol g . Hence setting the supercurrent to be J = J α ⊗ e α ∈ Γ( S g ⊗ T M ) with(11) J α = 2 h φ ∗ e β , γ ( e β ) γ ( e α ) ψ i φ ∗ h + | ψ | γ ( e β ) γ ( e α ) χ β , we obtain dd t (cid:12)(cid:12)(cid:12) t =0 A ( φ, ψ ; g, X ( t )) = Z M h ζ , J i g s ⊗ g d vol g . The conservation law associated to the super Weyl symmetry is obtained as follows. For any ζ ∈ Γ( S g ⊗ T M ) , we have QP ζ = 0 and hence t (cid:12)(cid:12)(cid:12) t =0 A ( φ, ψ ; g, χ + tP ζ ) = Z M h P ζ , J i = Z M h ζ , P J i g s ⊗ g d vol g Since ζ can be arbitrary, we conclude that J satisfies P J = 0 , and hence J ∈ Γ(Im Q ) . Remark.
Note that for a section ζ = ζ α ⊗ e α ∈ Γ( S g ⊗ T M ) , the followings are equivalent: P ζ = 0 ⇔ γ ( e α ) ζ α = 0 ⇔ ζ = γ ( e ) γ ( e ) ζ ⇔ ζ = − γ ( e ) γ ( e ) ζ , which are all equivalent to say that ζ = ζ α ⊗ e α lies in Im Q = ( S g , J Σ ⊕ J Σ ) ⊗ C T M , where J Σ is the complex structure on Σ . Note that the bundle Im Q is a holomorphic vector bundleover the Riemann surface M . Later we will show that in some particular cases J is actually anholomorphic section.3.3. Energy-momentum tensor.
Let ( g t ) t be a family of Riemannian metrics on M with dd t (cid:12)(cid:12)(cid:12) t =0 g t = k ∈ Γ(Sym( T ∗ M ⊗ T ∗ M )) , and let K ∈ End(
T M ) be the associated endomorphism. As in the discussions of Dirac actions,the spinor bundles change theoretically with the metrics, so the vector spinors and gravitinosneed to be carried using isometries along with the variation of metrics. Thus, one needs tocalculate the linearization dd t (cid:12)(cid:12)(cid:12) t =0 A ( φ, ( β t ⊗ ) ψ ; g t , ( β t ⊗ b t ) χ ) = dd t (cid:12)(cid:12)(cid:12) t =0 Z M I + II + III + IV + V d vol g t , where the roman numerals I , . . . , V denote the summands under integral of the action functional.We calculate them as follows.(i) The energy of the map can be analyzed as usual: dd t (cid:12)(cid:12)(cid:12) t =0 I = dd t (cid:12)(cid:12)(cid:12) t =0 E α ( φ i ) E α ( φ j ) h ij ( φ ) = − K βα · (2 h φ ∗ e α , φ ∗ e β i φ ∗ h ) . (ii) Locally write the vector spinor as ψ = ψ j ⊗ φ ∗ (cid:16) ∂∂y j (cid:17) , then ( β t ⊗ ) ψ = β t ( ψ j ) ⊗ φ ∗ (cid:18) ∂∂y j (cid:19) is a section of S g t ⊗ φ ∗ T N . Recall the definition of the twisted Dirac operator (4), /D g t ( β t ⊗ ) ψ = /∂ g t ( β t ψ j ) ⊗ φ ∗ (cid:18) ∂∂y j (cid:19) + γ t ( E α ( t )) β t ψ j ⊗ φ ∗ (cid:18) ∇ T NT φ ( E α ) ∂∂y j (cid:19) YMMETRIES AND CONSERVATION LAWS 15 and consequently (cid:10) ( β t ⊗ ) ψ, /D g t ( β t ⊗ ) ψ (cid:11) g s ( t ) ⊗ φ ∗ h = g s ( t ) (cid:0) β t ψ i , /∂ g t ( β t ψ j ) (cid:1) h ij ( φ )+ g s ( t ) (cid:0) β t ψ i , γ t ( E α ) β t ψ j (cid:1) h (cid:18) ∂∂y i , ∇ T NT φ ( E α ) ∂∂y j (cid:19) = g s ( ψ i , β − t /∂ g t ( β t ψ j )) h ij ( φ ) + g s ( ψ i , γ ( e α ) ψ j ) h (cid:18) ∂∂y i , ∇ T NT φ ( E α ) ∂∂y j (cid:19) = g s ( ψ i , /∂ g t ψ j ) h ij ( φ ) + g s ( ψ i , γ ( e α ) ψ j ) h (cid:18) ∂∂y i , ∇ T NT φ ( E α ) ∂∂y j (cid:19) . Taking derivative with respect to t gives dd t (cid:12)(cid:12)(cid:12) t =0 II = dd t (cid:12)(cid:12)(cid:12) t =0 (cid:10) ( β t ⊗ ) ψ, /D g t ( β t ⊗ ) ψ (cid:11) g s ( t ) ⊗ φ ∗ h = g s (cid:18) ψ i , dd t (cid:12)(cid:12)(cid:12) t =0 /∂ g t ψ j (cid:19) h ij ( φ )+ g s ( ψ i , γ ( e α ) ψ j ) h (cid:18) ∂∂y i , dd t (cid:12)(cid:12)(cid:12) t =0 ∇ T NT φ ( E α ) ∂∂y j (cid:19) = g s (cid:18) ψ i , − γ ( e α ) ∇ sK ( e α ) ψ j (cid:19) h ij ( φ )+ g s ( ψ i , γ ( e α ) ψ j ) h (cid:18) ∂∂y i , ∇ T NT φ ( − K ( e α )) ∂∂y j (cid:19) = − K βα (cid:18) g s ( ψ i , ∇ se β ψ j ) h ij ( φ ) + g s ( ψ i , γ ( e α ) ψ j ) h (cid:18) ∂∂y i , ∇ T NT φ ( e β ) ∂∂y j (cid:19)(cid:19) = − K βα D ψ, γ ( e α ) e ∇ e β ψ E g s ⊗ φ ∗ h = − K βα (cid:28) ψ, (cid:0) γ ( e α ) e ∇ e β ψ + γ ( e β ) e ∇ e α ψ (cid:1)(cid:29) g s ⊗ φ ∗ h . where the second equality follows from the fact that for any vector field X ∈ Γ( T M ) , g s ( ψ i , γ ( X ) ψ j ) h ij ( φ ) = 0 due to the skew-adjointness of the Clifford multiplications, and the last equality holdssince K is symmetric.(iii) Next we consider the third summand of the action functional, which mixes all the fieldstogether. Actually, since − h ( ⊗ φ ∗ ) Q t ( β t ⊗ b t ) χ, ( β t ⊗ ) ψ i g s ( t ) ⊗ φ ∗ h = 2 (cid:28) γ t ( E α ) γ t ( E β ) β t χ α ⊗ φ ∗ ( E β ) , β t ψ j ⊗ φ ∗ (cid:18) ∂∂y j (cid:19)(cid:29) g s ( t ) ⊗ φ ∗ h = 2 g s ( t ) (cid:0) β t ( γ ( e α ) γ ( e β ) χ α ) , β t ψ j (cid:1) h ( φ ) (cid:18) T φ ( E β ) , ∂∂y j (cid:19) = 2 g s (cid:0) γ ( e α ) γ ( e β ) χ α , ψ j (cid:1) h ( φ ) (cid:18) T φ ( b t e β ) , ∂∂y j (cid:19) , we have dd t (cid:12)(cid:12)(cid:12) t =0 III = − t (cid:12)(cid:12)(cid:12) t =0 h ( ⊗ φ ∗ ) Q t ( β t ⊗ b t ) χ, ( β t ⊗ ) ψ i g s ( t ) ⊗ φ ∗ h = 2 g s (cid:0) γ ( e α ) γ ( e β ) χ α , ψ j (cid:1) h ( φ ) (cid:18) dd t (cid:12)(cid:12)(cid:12) t =0 T φ ( b t e β ) , ∂∂y j (cid:19) = − K βα · h γ ( e η ) γ ( e α ) χ η ⊗ φ ∗ e β + γ ( e η ) γ ( e β ) χ η ⊗ φ ∗ e α , ψ i g s ⊗ φ ∗ h (iv) For the fourth term we conclude as in (8) that dd t (cid:12)(cid:12)(cid:12) t =0 IV = − dd t (cid:12)(cid:12)(cid:12) t =0 | Q t ( β t ⊗ b t ) χ | g s ( t ) ⊗ g t | ( β t ⊗ ) ψ | g s ( t ) ⊗ φ ∗ h = dd t (cid:12)(cid:12)(cid:12) t =0 | Qχ | g s ⊗ g | ψ | g s ⊗ φ ∗ h = 0 (v) The curvature term is analogous to (9): dd t (cid:12)(cid:12)(cid:12) t =0 V = −
16 dd t (cid:12)(cid:12)(cid:12) t =0 R(( β t ⊗ ) ψ ) = −
16 dd t (cid:12)(cid:12)(cid:12) t =0 R( ψ ) = 0 (vi) We still need to consider the change in the volume form. As in Section 2.1, we havethat dd t (cid:12)(cid:12)(cid:12) t =0 d vol g t = 12 Tr g ( K ) d vol g = 12 K βα δ αβ d vol g Consequently we have Z M (I + II + III + IV + V) dd t d vol g t (cid:12)(cid:12)(cid:12) t =0 = 12 Z M (I + II + III + IV + V) (cid:12)(cid:12)(cid:12) t =0 K βα δ αβ d vol g . Summing the contributions from (i) to (vi) up we obtain dd t (cid:12)(cid:12)(cid:12) t =0 A ( φ, ( β t ⊗ ) ψ ; g t , ( β t ⊗ b t ) χ ) = − Z M K βα T αβ d vol g = − Z M (cid:28) ∂g t ∂t (cid:12)(cid:12)(cid:12) t =0 , T (cid:29) d vol g . Here the inner product under integral is the induced one on symmetric covariant 2-tensors, T αβ = g αη T ηβ and T = T αβ e α ⊗ e β , where T αβ = 2 h φ ∗ e α , φ ∗ e β i φ ∗ h + 12 D ψ, γ ( e α ) e ∇ e β ψ + γ ( e β ) e ∇ e α ψ E g s ⊗ φ ∗ h + h γ ( e η ) γ ( e α ) χ η ⊗ φ ∗ e β + γ ( e η ) γ ( e β ) χ η ⊗ φ ∗ e α , ψ i g s ⊗ φ ∗ h − (cid:18) | d φ | g ∨ ⊗ φ ∗ h + h ψ, /D g ψ i − h ( ⊗ φ ∗ ) Qχ, ψ i − | Qχ | | ψ | −
16 R( ψ ) (cid:19) g αβ . (12)Being a quantity rising from the variation of a symmetric 2-tensor, the energy-momentum tensoris naturally symmetric, as the expression clearly shows.One can verify that the energy-momentum tensor is in general not traceless. This is due tothe fact that the action functional is not invariant under conformal transformations on g , thatis, for g ′ = e u g in general A ( φ, ( β ⊗ ) ψ ; g ′ , ( β ⊗ b ) χ ) = A ( φ, ψ ; g, χ ) . Instead, the action functional is invariant under the rescaled conformal invariance, see (7). Asin the case of the Dirac action the conservation law corresponding to the rescaled conformal
YMMETRIES AND CONSERVATION LAWS 17 invariance prescribes the trace of the energy-momentum tensor: t (cid:12)(cid:12)(cid:12) t =0 A (cid:16) φ, e − tu ( β t ⊗ ) ψ ; e tu g, e − tu ( β t ⊗ b t ) χ (cid:17) = Z M (cid:28) − uψ, EL ( ψ ) (cid:29) − h ug, T i + (cid:28) − uχ, J (cid:29) d vol g = − Z M u (cid:18) Tr g ( T ) + h ψ, EL ( ψ ) i + 12 h χ, J i (cid:19) d vol g As the integral has to vanish for all functions u , we conclude Tr g ( T ) = − h ψ, EL ( ψ ) i − h χ, J i , where EL ( ψ ) denotes the Euler–Lagrange equation for ψ computed in [18]. Notice, if theEuler–Lagrange equation for ψ is satisfied and either χ or J vanish, then T is actually traceless.In that case T can be identified with a smooth section of T ∗ M ⊗ C T ∗ M , that is a quadraticdifferential. We will later show that under certain conditions T is actually a holomorphicquadratic differential.3.4. Diffeomorphism invariance.
It will be shown that the super action functional is invari-ant under diffeomorphisms if the fields are “pulled back” in an appropriate way. Let f ∈ Diff( M ) and consider the following diffeomorphism transformations : φ φ ′ := φ ◦ f,ψ = ψ j ⊗ φ ∗ (cid:18) ∂∂y j (cid:19) ψ ′ := F − ◦ ψ j ◦ f ⊗ ( φ ◦ f ) ∗ (cid:18) ∂∂y j (cid:19) ,g g ′ := g f ,χ = χ α ⊗ e α χ ′ := F − ◦ χ α ◦ f ⊗ ( T f ) − e α , (13)where F : ( S g f , g fs ) → ( S g , g s ) is the isomorphism introduced in Section 2.3. Then we claim that A ( φ ′ , ψ ′ ; g ′ , χ ′ ) = A ( φ, ψ ; g, χ ) . To see this, suppose that under the diffeomorphism f , y x = f ( y ) . Then as in the harmonicmap case | d φ ′ | g ′∨ ⊗ φ ′∗ h ( y ) = | d φ | g ∨ ⊗ φ ∗ h ( x ) . For those terms involving spinors, we note that for any spinor σ ∈ Γ( S ) ,(14) F − ( γ ( e α ) σ ) f ( y ) = γ ′ (cid:0) ( T f ) − y e α (cid:1) F − ( σ ) f ( y ) where γ ′ denotes the Clifford multiplications with respect to the metric g ′ = g f . From this wewill see that the other terms are also invariant:(i) First consider the Dirac term /D g ′ ψ ′ ( y ) = ( /∂ g ′ ) y F − | y ( ψ k ◦ f ) y ⊗ φ ′∗ (cid:18) ∂∂y k (cid:19) + γ ′ ( T f − ( e α )) F − | y ( ψ k ◦ f ) y ⊗ ∇ T N ( T φ ′ )( T f ) − e α (cid:18) ∂∂y k (cid:19) = F − | y ( /∂ g ) x ψ k ( x ) ⊗ φ ′∗ (cid:18) ∂∂y k (cid:19) + F − | y (cid:0) γ ( e α ) ψ k ( x ) (cid:1) ⊗ ∇ T NT φ ( e α ) (cid:18) ∂∂y k (cid:19) . Thus h ψ ′ , /D g ′ ψ ′ i ( y )= g fs | y (cid:16) F − | y ( ψ j ◦ f ) y , F − | y ( /∂ g ) x ψ k ( x ) (cid:17) h ( φ ′ ( y )) jk + g fs | y (cid:16) F − | y ( ψ j ◦ f ) y , F − | y (cid:0) γ ( e α ) ψ k ( x ) (cid:1)(cid:17) h | φ ′ ( y ) (cid:18) ∂∂y j , ∇ T NT φ ( e α ) ∂∂y k (cid:19) = g s | x (cid:0) ψ j ( x ) , ( /∂ g ) x ψ k ( x ) (cid:1) h ( φ ( x )) jk + g s | x (cid:0) ψ j ( x ) , γ ( e α ) ψ k ( x ) (cid:1) h | φ ( x ) (cid:18) ∂∂y j , ∇ T NT φ ( e α ) (cid:18) ∂∂y k (cid:19)(cid:19) = h ψ, /D g ψ i ( x ) . Moreover, | ψ ′ ( y ) | g fs ⊗ φ ′∗ h = g fs | y (cid:16) F − | y ( ψ j ◦ f ) y , F − | y ( ψ k ◦ f ) y (cid:17) h ( φ ′ ( y )) jk = g s | x (cid:0) ψ j ( x ) , ψ k ( x ) (cid:1) h ( φ ( x )) jk = | ψ ( x ) | g s ⊗ φ ∗ h and R( ψ ′ ) | y = R Nijkl ( φ ′ ( y )) g fs | y (cid:16) F − | y ( ψ i ◦ f ) y , F − | y ( ψ k ◦ f ) y (cid:17) × g fs | y (cid:16) F − | y ( ψ j ◦ f ) y , F − | y ( ψ l ◦ f ) y (cid:17) = R N ( φ ( x )) ijkl g s | x (cid:0) ψ i ( x ) , ( /∂ g ) x ψ k ( x ) (cid:1) g s | x (cid:0) ψ j ( x ) , ( /∂ g ) x ψ l ( x ) (cid:1) = R( ψ ) | x . (ii) For the gravitino, from (14) it follows that Q ′ χ ′ ( y ) = F − | y ( γ ( e α ) γ ( e β ) χ α ) | f ( y ) ⊗ ( T f ) − e α . Hence we have | Q ′ χ ′ ( y ) | g fs ⊗ g t = | Qχ ( x ) | g s ⊗ g . For the mixed term, note that ( ⊗ φ ′∗ ) Q ′ χ ′ ( y ) = γ ′ (cid:0) ( T f ) − y e α (cid:1) γ ′ (cid:0) ( T f ) − y e β (cid:1) F − | y χ α ( f ( y )) ⊗ φ ′∗ ( T f ) − e α = F − | f ( y ) ( γ ( e α ) γ ( e β ) χ α ) f ( y ) ⊗ φ ∗ e α . Then it is immediate that h ( ⊗ φ ′∗ ) Q ′ χ ′ ( y ) , ψ ′ ( y ) i g fs ⊗ φ ′∗ h = h ( ⊗ φ ∗ ) Qχ ( x ) , ψ ( x ) i g s ⊗ φ ∗ h . Therefore, by the change of variable formula, the diffeomorphism invariance of A is confirmed. YMMETRIES AND CONSERVATION LAWS 19
The symmetry of diffeomorphism invariance will give another conservation law. Actually, let X ∈ Γ( T M ) generate a global flow ( f t ) as in Section 2.3. Then we have t (cid:12)(cid:12)(cid:12) t =0 A ( φ t , ψ t ; g t , χ t )= Z M − (cid:28) dd t (cid:12)(cid:12)(cid:12) t =0 φ t , EL ( φ ) (cid:29) + 2 D ∇ S g ⊗ φ ∗ t T N∂ t (cid:0) ( β − t ⊗ ) ψ t (cid:1) (cid:12)(cid:12)(cid:12) t =0 , EL ( ψ ) E d vol g + Z M − (cid:28) dd t (cid:12)(cid:12)(cid:12) t =0 g t , T (cid:29) + (cid:28) dd t (cid:12)(cid:12)(cid:12) t =0 χ t , J (cid:29) d vol g = Z M − h φ ∗ ( X ) , EL ( φ ) i + 2 h δψ ( X ) , EL ( ψ ) i d vol g + Z M − hL X g, T i + D L S g ⊗ T MX χ, J E d vol g . (15)Note that for the vector spinor we have to take the covariant derivative to obtain the variationfield which is abbreviated as δψ ( X ) in the last formula, which is the same approach we tookin [18]. Here the Lie derivative on S g ⊗ T M is the one defined in [2]; that is, under a localorthonormal frame ( e α ) , for χ = χ α ⊗ e α , L S g ⊗ T MX χ := dd t (cid:12)(cid:12)(cid:12) t =0 β − t F − t ( χ α ◦ f t ) ⊗ b − t ( T f t ) − ( e α ◦ f t ) . Notice, that on the vector part of χ the above t -derivative will not result in the ordinary Liederivative on tangent vectors, see also [2]. From Section 4 one knows that Z M hL X g, T i d vol g = − Z M h X, div g ( T ) i d vol g and Z M hL S g ⊗ T MX χ, J i d vol g = Z M h X, div χ ( J ) i d vol g . Therefore, along solutions of the Euler–Lagrange equations, the following identity holds: div g ( T ) + div χ ( J ) = 0 , where the formal divergence operator div χ is defined in Section 4. Remark.
If the gravitino vanishes, then from (15) we know that, along solutions of the Euler–Lagrange equations, Z M − hL X g, T i d vol g = Z M h X, div g T i d vol g . This tells us that T is divergence-free, and hence the energy-momentum tensor corresponds toa holomorphic quadratic differential.Before going to the discussion on super symmetry, we summarize the results obtained up tonow in the following theorem. Theorem 2.
Consider the super action functional defined by (5) .(1) The total variation formula is δ A = Z M − h δφ, EL ( φ ) i + 2 h δψ, EL ( ψ ) i − h δg, T i + h δχ, J i d vol g , where the Euler–Lagrange equations are given in (6) and the energy-momentum tensor T is given by (12) and the supercurrent J is given by (11) .(2) This action functional is invariant under rescaled conformal transformations: A (cid:16) φ, e − u ( β ⊗ ) ψ ; e u g, e − u ( β ⊗ b ) χ (cid:17) = A ( φ, ψ ; g, χ ) . Consequently, Tr g ( T ) = − h ψ, EL ( ψ ) i − h χ, J i , even off shell.(3) This action is invariant under super Weyl transformations. That is, for any ζ ∈ Γ( S g ⊗ T M ) , it holds that A ( φ, ψ ; g, χ + P ζ ) = A ( φ, ψ ; g, χ ) . Consequently, P J = 0 , orequivalently, J is a smooth section of S g ⊗ C T M .(4) This action functional is invariant under diffeomorphisms. That is, it is invariant underthe transformation (13) for f ∈ Diff( M ) . Consequently, along solutions of the Euler–Lagrange Equations (6) , the coupled conservation law holds div g ( T ) + div χ ( J ) = 0 . (5) If either χ = 0 or J = 0 , then along solutions of the Euler–Lagrange equations, theenergy-momentum tensor T is symmetric, traceless and divergence-free. Hence it corre-sponds to a holomorphic quadratic differential on M . Degenerate super symmetry.
The action functional (5) is motivated from the actionfunctional of two-dimensional super symmetric sigma models [3] also called the super conformalaction functional in [17]. The major reason for the introduction of those models was supersymmetry. As was argued in [20] super symmetry requires anti-commutative variables. Hence afull super symmetry cannot be expected for the action functional (5). Surprisingly, the followingspecial case of super symmetry, which we will call degenerate super symmetry, persists:
Proposition 2.
Let q be a section of S g . The action functional (5) is invariant under thefollowing infinitesimal transformations δφ = h q, ψ i g s δψ = − γ (grad φ ) qδg = 0 δχ = ( ∇ s q ) ♯ where ( ∇ s q ) ♯ ≡ ∇ se α q ⊗ e α ∈ Γ( S g ⊗ T M ) , given that χ = 0 and the following holds: (16) Z M D ψ, R N (cid:16) h q, ψ i g s , φ ∗ e α (cid:17) γ ( e α ) ψ E + D S ∇ R ( ψ ) , h q, ψ i g s E − h SR ( ψ ) , γ (grad φ ) q i d vol g = 0 The last condition is in particular fulfilled if the target manifold N is flat.Remark. The degenerate super symmetry transformations of Proposition 2 coincide up to thesign of δψ and δχ with the super symmetry transformations given in [19] for the super conformalaction functional. The sign difference is necessary to compensate for the use of the oppositeClifford algebra in this work. YMMETRIES AND CONSERVATION LAWS 21
Proof.
The variation of A ( φ, ψ ; g, χ ) is given by δ A ( φ, ψ ; g, χ ) = Z M g αβ φ ∗ h (cid:0) ∇ φ ∗ T Ne α h q, ψ i , φ ∗ e β (cid:1) − (cid:10) γ ( e α ) q ⊗ φ ∗ e α , /Dψ (cid:11) − D ψ, /D ( γ ( e α ) q ⊗ φ ∗ e α ) + R N (cid:16) h q, ψ i g s , φ ∗ e α (cid:17) γ ( e α ) ψ E + 2 h γ ( e α ) γ ( e β ) ∇ e α q ⊗ φ ∗ e β , ψ i− (cid:16)D S ∇ R ( ψ ) , h q, ψ i g s E − h SR ( ψ ) , γ (grad φ ) q i (cid:17) d vol Here we have used δ (cid:0) /Dψ (cid:1) = /D ( δψ ) + R N ( δφ, φ ∗ e α ) γ ( e α ) ψ,δ (cid:0) R N ( ψ ) (cid:1) = h S ∇ R ( ψ ) , δφ i + 4 h SR ( ψ ) , δψ i , compare [18, Section 4.1, (2) and (5)]. If we now use (16), the variation of the action reducesto: δ A ( φ, ψ ; g, χ ) = Z M g αβ φ ∗ h ( ∇ e α h q, ψ i , φ ∗ e β ) − (cid:10) γ ( e α ) q ⊗ φ ∗ e α , /Dψ (cid:11) − (cid:10) ψ, /D ( γ ( e α ) q ⊗ φ ∗ e α ) (cid:11) + 2 h γ ( e α ) γ ( e β ) ∇ e α q ⊗ φ ∗ e β , ψ i d vol g = Z M g αβ ( h∇ e α q ⊗ φ ∗ e β , ψ i + h q ⊗ φ ∗ e β , ∇ e α ψ i ) + (cid:10) γ ( e β ) γ ( e α ) q ⊗ φ ∗ e α , ∇ e β ψ (cid:11) − (cid:10) γ ( e β ) (cid:0) γ (cid:0) ∇ e β e α (cid:1) q ⊗ φ ∗ e α + γ ( e α ) ∇ e β q ⊗ φ ∗ e α + γ ( e α ) q ⊗ ∇ e β φ ∗ e α (cid:1) , ψ (cid:11) + 2 h γ ( e α ) γ ( e β ) ∇ e α q ⊗ φ ∗ e β , ψ i d vol g = Z M − h γ ( e β ) γ ( e α ) ∇ e α q ⊗ φ ∗ e β , ψ i − h γ ( e β ) γ ( e α ) q ⊗ ∇ e α φ ∗ e β , ψ i− h γ ( e β ) γ ( e α ) q ⊗ φ ∗ e β , ∇ e α ψ i − (cid:10) γ ( e β ) γ (cid:0) ∇ e β e α (cid:1) q ⊗ φ ∗ e β , ψ (cid:11) + (cid:10) γ ( e β ) γ ( e α ) q ⊗ (cid:0) ∇ e α φ ∗ e β − ∇ e β φ ∗ e α (cid:1) , ψ (cid:11) d vol g = Z M − e α h γ ( e β ) γ ( e α ) q ⊗ φ ∗ e β , ψ i + h ( γ ( ∇ e α e β ) γ ( e α ) + γ ( e β ) γ ( ∇ e α e α )) q ⊗ φ ∗ e β , ψ i− (cid:10) γ ( e β ) γ (cid:0) ∇ e β e α (cid:1) q ⊗ φ ∗ e β , ψ (cid:11) + (cid:10) γ ( e β ) γ ( e α ) q ⊗ (cid:0) ∇ e α φ ∗ e β − ∇ e β φ ∗ e α (cid:1) , ψ (cid:11) d vol g = 0 . Here in the last step we have used the torsion-freeness of the connection. (cid:3)
Theorem 3.
Suppose as in Proposition 2 that χ = 0 and condition (16) is fulfilled. In additionwe assume that φ and ψ fulfill the Euler–Lagrange equations. Then div g J = 0 . In particular,the supercurrent J can be identified with a holomorphic section of T M ⊗ C S .Proof. In the particular case that φ and ψ fulfill the Euler–Lagrange equations, we know thatfor all super symmetry parameters q : δ A ( φ, ψ ; g, χ ) = Z M h δχ, J i d vol = Z M D ( ∇ s q ) ♯ , J E d vol = − Z M h q, div g J i d vol. (cid:3) In the remainder of this subsection we give an application of the degenerate super symmetryto the action functional of Dirac harmonic maps, with or without curvature term. Recallthat the functional of Dirac-harmonic maps with curvature term in [6] can be obtained fromthe functional (5) by setting the gravitino to zero. It is a Corollary of Proposition 2 that thefunctional of Dirac-harmonic maps with or without curvature terms also has a degenerate supersymmetry.
Corollary 1 (Degenerate super symmetry of the functional of Dirac-harmonic maps withcurvature term) . Let q ∈ Γ( S g ) be a twistor spinor. The functional of Dirac-harmonic mapswith curvature term is invariant under the following infinitesimal transformations δφ = h q, ψ i g s δψ = − γ (grad φ ) q provided that the curvature condition (16) holds. Proof.
The calculation proceeds as in the proof of Proposition 2 with the additional conditionthat the term that describes the variation of the gravitino needs to be zero. The term of δ A ( φ, ψ ; g, χ ) that arises from the variation of the gravitino is Z M (cid:10) γ ( e α ) γ ( e β ) ∇ se α q ⊗ φ ∗ e β , ψ (cid:11) d vol g = − Z M D(cid:16) ∇ se β q + γ ( e β ) /∂ g q (cid:17) ⊗ φ ∗ e β , ψ E d vol g . This term vanishes if q is a twistor spinor, i.e. for all vector field X it holds ∇ sX q + 12 γ ( X ) /∂ g q = 0 . (cid:3) Remark.
Notice that it is particular to the two-dimensional setup that the condition to be atwistor spinor or a holomorphic spinor are identical. Hence, Corollary 1 does not come as asurprise, as also for super Riemann surfaces a holomorphic super symmetry leaves the -partof the gravitino invariant, see [19, Chapter 11.1].Recall that the Dirac-harmonic map functional in [5] does not include the curvature term ofthe target manifold. But note that a curvature term will arise when taking variations. Corollary 2 (Degenerate super symmetry of the Dirac-harmonic map functional) . Let q ∈ Γ( S g ) be a twistor spinor. The functional of Dirac-harmonic maps is invariant under thefollowing infinitesimal transformations δφ = h q, ψ i g s δψ = − γ (grad φ ) q provided that the following curvature condition(17) Z M h ψ, R ( h q, ψ i g s , φ ∗ e α ) γ ( e α ) ψ i g s ⊗ φ ∗ h d vol g = 0 . holds.In [5] two seemingly unrelated critical Dirac-harmonic maps have been constructed: Thetrivial solution ( φ , , where φ is a harmonic map and the twistor spinor solution ( φ , ψ ) where ψ = − γ ( e α ) q ⊗ φ ∗ e α is constructed from a twistor spinor q and the harmonic map φ .We will now show that those two solutions are related via degenerate super symmetry.Consider the family defined on the time interval [0 , given by φ t = φ ψ t = − tγ ( e α ) q ⊗ φ ∗ e α YMMETRIES AND CONSERVATION LAWS 23 for a twistor spinor q ∈ Γ( S g ) . The family ( φ t , ψ t ) interpolates between the trivial and thetwistor spinor solution via a family of degenerate super symmetries. Indeed for every t = τ wehave dd t (cid:12)(cid:12)(cid:12)(cid:12) t = τ φ t = 0 = h q, ψ τ i dd t (cid:12)(cid:12)(cid:12)(cid:12) t = τ ψ t = − γ ( e α ) q ⊗ φ ∗ e α = − γ (grad φ ) q. Fortunately the condition (17) is fulfilled along this family, and hence we conclude that ( φ t , ψ t ) iscritical for all t ∈ [0 , . Consequently, we have a critical family of degenerate super symmetriesand the twistor spinor solution should be considered equivalent to the trivial solution.We expect that more non-trivial critical Dirac-harmonic maps can be constructed with thehelp of super symmetry. The difficulty lies in the construction of suitable families ( φ t , ψ t ) .3.6. Conclusion.
We have shown that the functional of Dirac-harmonic maps with gravitinois invariant under the rescaled conformal transformations, super Weyl transformations anddiffeomorphisms. The energy-momentum tensor and the supercurrent of the action have beencalculated. We found that the rescaled conformal invariance prescribes the trace of the energy-momentum tensor, whereas super Weyl invariance assures that J is a section of S g ⊗ C T M off shell. The diffeomorphism invariance leads on shell to a coupled differential equation ofdivergence type involving the energy-momentum tensor and the supercurrent.Furthermore, we have shown that in the case of vanishing gravitino and under certain con-ditions on the curvature of the target manifold a degenerate super symmetry leaves the actionfunctional invariant infinitesimally. The restricting conditions on the curvature and of the van-ishing gravitino cannot be lifted in the setting of Dirac-harmonic maps but only in the world ofsuper geometry. The degenerate super symmetry leads on shell to a divergence-free supercur-rent. Hence, in that case the energy-momentum tensor can be identified with a holomorphicquadratic differential and the supercurrent with an holomorphic section of
T M ⊗ C S g .Finally, we want to mention that the functional A also possesses a U (1) -gauge symmetry. Aparticular special case of this gauge symmetry is A ( φ, ψ ; g, χ ) = A ( φ, − ψ ; g, − χ ) . Appendix
As we frequently use the divergence operators on various fields, we make a review in thissection and explain their relations to Lie derivatives and the Cauchy–Riemann operators.4.1. Recall that on a Riemannian manifold ( M, g ) with local coordinates ( x α ) , the divergenceof a vector field X = X α ∂∂x α is a function on M defined by div g X = 1 √ det g ∂∂x α (cid:16)p det gX α (cid:17) where det g = det ( g αβ ) . In terms of the Levi-Civita connection ∇ LC the divergence operator isexpressed by div g ( X ) = Tr g ( ∇ X ) ∈ C ∞ ( M ) . In this way the divergence operator is the negative L -adjoint operator of the gradient operator:for any f ∈ C ∞ ( M ) and any X ∈ Γ( T M ) , Z M h X, grad( f ) i d vol g = Z M h− div g X, f i d vol g . The divergence operator on symmetric two-tensors is defined in a similar way. Let k ∈ Γ(Sym( T ∗ M ⊗ T ∗ M )) , the divergence operator on k is the one-form given by div g k = Tr g ( ∇ k ) := dim M X α =1 ∇ e α k ( e α , · ) ∈ Γ( T ∗ M ) , where ( e α ) is a local orthonormal frame. In local coordinates ( x α ) , k = k αβ d x α ⊗ d x β with k αβ = k βα and let K = K αβ d x β ⊗ ∂∂x α ∈ End(
T M ) be the associated (1 , -tensor, div g ( k ) = (cid:18) √ det g ∂∂x α (cid:16)p det gK αβ (cid:17) − g αη ∂g ηγ ∂x β K γα (cid:19) d x β . Then we claimed that, for any vector field X ∈ Γ( T M ) ,(18) Z M hL X g, k i d vol g = − Z M h X, div g k i d vol g . Actually, using the local expression above, Z M (div g k )( X ) d vol g = Z M (cid:18) √ det g ∂∂x α (cid:16)p det gK αβ (cid:17) − g αη ∂g ηγ ∂x β K γα (cid:19) X β d vol g = Z M ∂∂x α (cid:16)p det gK αβ X β (cid:17) d x − Z M K αβ ∂X β ∂x α + 12 g αη ∂g ηγ ∂x β K γα X β d vol g = Z M div g ( K ( X )) d vol g − Z M h k, L X g i d vol g . Since M is closed, the divergence theorem implies the first summand in the last line vanishes.Hence the claim is confirmed.Recall that a symmetric, traceless and divergence-free two-tensor corresponds to a holomor-phic quadratic differential on a Riemann surface, see e.g. [23, Section 2.4].4.2. Now we consider the divergence operators on spinor fields. Recall from [2] that on thespinor bundle S g , the Lie derivative with respect to X ∈ Γ( T M ) on a spinor σ ∈ Γ( S g ) isrelated to the spin connection ∇ s via L SX σ = ∇ sX σ − γ (d X ♭ ) σ, where X ♭ denotes the dual one-form of the vector field X and the 2-form d X ♭ acts via Cliffordmultiplication. We would like to define a divergence operator on spinor fields such that aformula of the same type as (18) holds. This is achieved in the following lemma. Lemma 1.
Let ( M, g ) be a Riemann surface, with almost complex structure J M ∈ Aut(
T M ) .For ρ , σ ∈ Γ( S g ) , define the divergence of ρ with respect to σ as div σ ( ρ ) = h∇ s σ, ρ i ♯ + 14 J M grad (cid:0) h γ ( ω ) σ, ρ i (cid:1) ∈ Γ( T M ) , where ω stands for the volume element in the Clifford bundle. The following holds for all vectorfields X ∈ Γ( T M ) Z M hL SX σ, ρ i d vol g = Z M h X, div σ ( ρ ) i d vol g . YMMETRIES AND CONSERVATION LAWS 25
Proof.
Take local isothermal coordinates ( x α ) and write X = X α ∂∂x α . Then X ♭ = X α g αβ d x β ≡ X β d x β . Since d X ♭ = (cid:18) ∂X ∂x − ∂X ∂x (cid:19) d x ∧ d x , one sees that γ (d X ♭ ) = (cid:18) ∂X ∂x − ∂X ∂x (cid:19) √ det g γ ( ω ) . Thus, Z M h γ (d X ♭ ) σ, ρ i d vol g = Z M (cid:28)(cid:18) ∂X ∂x − ∂X ∂x (cid:19) √ det g γ ( ω ) σ, ρ (cid:29) p det g d x = Z M ∂X ∂x h γ ( ω ) σ, ρ i − ∂X ∂x h γ ( ω ) σ, ρ i d x = Z M − X ∂∂x ( h γ ( ω ) σ, ρ i ) + X ∂∂x ( h γ ( ω ) σ, ρ i ) d x = Z M (cid:10) ∗ X ♭ , d h γ ( ω ) σ, ρ i (cid:11) d vol g = Z M (cid:10) J M X, grad( h γ ( ω ) σ, ρ i ) (cid:11) d vol g = Z M (cid:10) X, − J M grad( h γ ( ω ) σ, ρ i ) (cid:11) d vol g . Therefore, Z M hL SX σ, ρ i d vol g = Z M h∇ sX σ, ρ i − h γ (d X ♭ ) σ, ρ i d vol g = Z M (cid:10) X, h∇ s σ, ρ i ♯ + 14 J M grad( h γ ( ω ) σ, ρ i ) (cid:11) d vol g . (cid:3) Remark.
In contrast to the divergence operators defined for the vector fields and for the sym-metric 2-tensors, this divergence operator div σ ( ρ ) doesn’t involve derivatives of ρ , but onlyderivatives of σ . In this sense, it is only a formal “divergence” operator.4.3. Next we also need to consider the divergence operators defined on the tensor productbundle S g ⊗ T M . Let ϕ = ϕ α ⊗ e α ∈ Γ( S g ⊗ T M ) be a gravitino. In a local orthonormal frame ( e α ) , the g -divergence operator is div g ( ϕ ) := Tr g (cid:16) b ∇ ϕ (cid:17) = X α D b ∇ e α ϕ, e α E ∈ Γ( S g ) , where we use b ∇ to denote the connection of S g ⊗ T M . For any spinor field q ∈ Γ( S g ) , usingintegration by parts, Z M h q, div g ϕ i d vol g = Z M D q, h b ∇ e α ϕ, e α i E d vol g = Z M D q ⊗ e α , b ∇ e α ϕ E d vol g = − Z M (cid:10) ∇ se α q ⊗ e α , ϕ (cid:11) d vol g = − Z M D ( ∇ s q ) ♯ , ϕ E d vol g Note that
P ϕ = 0 implies that ϕ is a smooth section of the complex vector bundle S g ⊗ C T M . Ifin addition div g ϕ = 0 , then ϕ is then holomorphic, which is to say, ϕ ∨ ≡ ϕ α ⊗ e α ∈ Γ( S ∨ g ⊗ T ∗ M ) is a holomorphic section.We also need the χ -divergence of ϕ for a gravitino field χ ∈ Γ( S g ⊗ T M ) . It is formallydefined to make the following identity hold: Z M hL S g ⊗ T MX χ, ϕ i d vol g = Z M h X, div χ ϕ i d vol g This can be assured using the Riesz representation theorem.
References [1] Christian Bär. On harmonic spinors. Acta Physica Polonica Series B 29:859–870, 1998[2] Jean-Pierre Bourguignon and Paul Gauduchon. Spineurs, opérateurs de Dirac et variations de métriques.Communications in mathematical Physics 144(3): 581–599, 1992.[3] L. Brink, Paolo Di Vecchia and Paul Howe. A locally supersymmetric and reparametrization invariantaction for the spinning string. Physics Letters B 65(5):471–474, 1976.[4] Qun Chen, Jürgen Jost, Jiayu Li and Guofang Wang. Regularity theorems and energy identities forDirac-harmonic maps. Mathematische Zeitschrift 251(1):61–84, 2005.[5] Qun Chen, Jürgen Jost, Jiayu Li and Guofang Wang. Dirac-harmonic maps. Mathematische Zeitschrift254(2):409–432, 2006.[6] Qun Chen, Jürgen Jost and Guofang Wang. Liouville theorems for Dirac-harmonic maps. Journal ofMathematical Physics 48:113517, 2008.[7] Qun Chen, Jürgen Jost and Guofang Wang. Nonlinear Dirac equations on Riemann surfaces. Annals ofGlobal Analysis and Geometry 33(3):253–270, 2008.[8] Pierre Deligne et al. Quantum fields and strings: a course for mathematicians. American MathematicalSociety, Providence, 1999.[9] Stanley Deser and Bruno Zumino. A complete action for the spinning string. Physics Letters B 65(4):369–373, 1976.[10] James Eells and Joseph H Sampson. Harmonic mappings of Riemannian manifolds. American Journalof Mathematics 86(1): 109–160, 1964.[11] I.M. Gelfand and S.V. Fomin. Calculus of variations. Translated and Edited by Richard A. Silverman.Dover Publications, Inc., 1963.[12] Nicolas Ginoux. The Dirac spectrum. Springer, Berlin, 2009.[13] Nigel Hitchin. Harmonic spinors. Advances in Mathematics 14(1): 1–55; 1974.[14] Jürgen Jost and Xianqing Li-Jost. Calculus of variations. Cambridge University Press, 1999.[15] Jürgen Jost. Geometry and physics. Springer, Berlin, 2009.[16] Jürgen Jost. Compact Riemann Surfaces, 3rd Ed. Springer, Berlin, 2006.[17] Jürgen Jost, Enno Keßler and Jürgen Tolksdorf. Super Riemann surfaces, metrics, and gravitinos. 2014,arXiv:1412.5146.
YMMETRIES AND CONSERVATION LAWS 27 [18] Jürgen Jost, Enno Keßler, Jürgen Tolksdorf, Ruijun Wu and Miaomiao Zhu. Regularity of solutions ofthe nonlinear sigma model with gravitino. 2016, arXiv:1610.02289.[19] Enno Keßler. The super conformal action functional on super Riemann surfaces. Ph.D. thesis, Univer-sität Leipzig, 2017.[20] Enno Keßler and Jürgen Tolksdorf. The functional of super Riemann surfaces–a “semi-classical” survey”.Vietnam Journal of Mathematics 44(1): 215–229, 2016.[21] H. Blaine Lawson and Marie-Louise Michelsohn. Spin geometry. Princeton University Press, New Jersey,1989.[22] Stephan Maier. Generic metrics and connections on
Spin - and
Spin c -manifolds. Communications inMathematical Physics 188(2): 407–437, 1997.[23] Anthony Tromba. Teichmüller theory in Riemannian geometry. Birkhäuser, 2012. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig,Germany
E-mail address : [email protected] Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig,Germany
E-mail address : [email protected]
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig,Germany
E-mail address : [email protected]
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig,Germany
E-mail address : [email protected]
School of Mathematical Sciences, Shanghai Jiao Tong University, Dongchuan Road 800,200240 Shanghai, P.R.China
E-mail address ::