Energy quantization for a nonlinear sigma model with critical gravitinos
aa r X i v : . [ m a t h . DG ] F e b ENERGY QUANTIZATION FOR A NONLINEAR SIGMA MODELWITH CRITICAL GRAVITINOS
JÜRGEN JOST, RUIJUN WU AND MIAOMIAO ZHU
Abstract.
We study some analytical and geometric properties of a two-dimensional nonlinearsigma model with gravitino which comes from supersymmetric string theory. When the actionis critical w.r.t. variations of the various fields including the gravitino, there is a symmetric,traceless and divergence-free energy-momentum tensor, which gives rise to a holomorphic qua-dratic differential. Using it we obtain a Pohozaev type identity and finally we can establish theenergy identities along a weakly convergent sequence of fields with uniformly bounded energies. Introduction
The 2-dimensional nonlinear sigma models constitute important models in quantum fieldtheory. They have not only physical applications, but also geometric implications, and there-fore their properties have been the focus of important lines of research. In mathematics, theyarise as two-dimensional harmonic maps and pseudo holomorphic curves. In modern physicsthe basic matter fields are described by vector fields as well as spinor fields, which are coupledby supersymmetries. The base manifolds are two-dimensional, and therefore their conformaland spin structures come into play. From the physics side, in the 1970s a supersymmetric 2-dimensional nonlinear sigma model was proposed in [6, 14]; the name “supersymmetric” comesfrom the fact that the action functional is invariant under certain transformations of the mat-ter fields, see for instance [13, 18]. From the perspective of geometric analysis, they seemto be natural candidates for a variational approach, and one might expect that the powerfulvariational methods developed for harmonic maps and pseudo holomorphic curves could be ap-plied here as well. However, because of the various spinor fields involved, new difficulties arise.The geometric aspects have been developed in mathematical terms in [25], but this naturallyinvolves anti-commuting variables which are not amenable to inequalities, and therefore vari-ational methods cannot be applied, and one rather needs algebraic tools. This would lead towhat one may call super harmonic maps. Here, we adopt a different approach. We transformthe anti-commuting variables into commuting ones, as in ordinary Riemannian geometry. Inparticular, the domains of the action functionals are ordinary Riemann surfaces instead of superRiemann surfaces. Then one has more fields to control, not only the maps between Riemannianmanifolds and Riemannian metrics, but also their super partners. Such a model was developedand investigated in [22]. Part of the symmetries, including some super symmetries, are inher-ited, although some essential supersymmetries are hidden or lost. As is known, the symmetriesof such functionals are quite important for the analysis, in order to overcome some analyticalproblems that arise as we are working in a limiting situation of the Palais-Smale condition.Therefore, here we shall develop a setting with a large symmetry group. This will enable us to
Date : October 12, 2018.
Key words and phrases. nonlinear sigma-model, Dirac-harmonic map, gravitino, supercurrent, Pohozaevidentity, energy identity.The third author was supported in part by National Science Foundation of China (No. 11601325). carry out the essential steps of the variational analysis. The analytical key will be a Pohozaevtype identity.We will follow the notation conventions of [22], which are briefly recalled in the following. Let ( M, g ) be an oriented closed Riemannian surface with a fixed spin structure, and let S → M be a spinor bundle, of real rank four, associated to the given spin structure. Note that theLevi-Civita connection ∇ M on M and the Riemannian metric g induce a spin connection ∇ s on S in a canonical way and a spin metric g s which is a fiberwise real inner product , see[26, 19]. The spinor bundle S is a left module over the Clifford bundle Cl( M, − g ) with theClifford map being denoted by γ : T M → End( S ) ; sometimes it will be simply denoted by adot. The Clifford relation reads γ ( X ) γ ( Y ) + γ ( Y ) γ ( X ) = − g ( X, Y ) , ∀ X, Y ∈ X ( M ) . The Clifford action is compatible with the spinor metric and the spin connection, making S into a Dirac bundle in the sense of [26]. Therefore, the bundle S ⊗ T M is also a Dirac bundleover M , and a section χ ∈ Γ( S ⊗ T M ) is taken as a super partner of the Riemannian metric,and called a gravitino. The Clifford multiplication gives rise to a map δ γ : S ⊗ T M → S , where δ γ ( s ⊗ v ) = γ ( v ) s = v · s for s ∈ Γ( S ) and v ∈ Γ( T M ) , and extending linearly. This map issurjective, and moreover the following short exact sequence splits: → ker → S ⊗ T M δ γ −→ S → . The projection map to the kernel is denoted by Q : S ⊗ T M → S ⊗ T M . More explicitly, ina local oriented orthonormal frame ( e α ) of M , a section χ ∈ Γ( S ⊗ T M ) can be written as χ α ⊗ e α , and the Q -projection is given by Qχ := − γ ( e β ) γ ( e α ) χ β ⊗ e α = 12 (cid:0) ( χ + ω · χ ) ⊗ e − ω · ( χ + ω · χ ) ⊗ e (cid:1) , where ω = e · e is the real volume element in the Clifford bundle.Let ( N, h ) be a compact Riemannian manifold and φ : M → N a map. One can consider thetwisted spinor bundle S ⊗ φ ∗ T N with bundle metric g s ⊗ φ ∗ h and connection e ∇ ≡ ∇ S ⊗ φ ∗ T N ,which is also a Dirac bundle, and the Clifford action on this bundle is also denoted by γ orsimply a dot. A section of this bundle is called a vector spinor, and it serves as a super partnerof the map φ in this model. The twisted spin Dirac operator /D is defined in the canonical way:let ( e α ) be a local orthonormal frame of M , then for any vector spinor ψ ∈ Γ( S ⊗ φ ∗ T N ) , define /Dψ := γ ( e α ) e ∇ e α ψ = e α · e ∇ e α ψ. It is elliptic and essentially self-adjoint with respect to the inner product in L ( S ⊗ φ ∗ T N ) . Ina local coordinate ( y i ) of N , write ψ = ψ i ⊗ φ ∗ ( ∂∂y i ) , then /Dψ = /∂ψ i ⊗ φ ∗ (cid:18) ∂∂y i (cid:19) + γ ( e α ) ψ i ⊗ φ ∗ (cid:18) ∇ NT φ ( e α ) ∂∂y i (cid:19) , where /∂ is the spin Dirac operator on S . For later convention, we set SR ( ψ ) := h ψ l , ψ j i g s ψ k ⊗ φ ∗ (cid:18) R N ( ∂∂y k , ∂∂y l ) ∂∂y j (cid:19) = R ijkl ( φ ) h ψ l , ψ j i ψ k ⊗ φ ∗ (cid:18) ∂∂y i (cid:19) , Here we take the real rather than the Hermitian one used in some previous works on Dirac-harmonic maps(with or without curvature term), as clarified in [22]. Here and in the sequel, the summation convention is always used.
NERGY QUANTIZATION 3 and R( ψ ) := h SR ( ψ ) , ψ i g s ⊗ φ ∗ h .The action functional under consideration 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( ψ ) d vol g , From [22] we know that the Euler–Lagrange equations are τ ( φ ) = 12 R( ψ, e α · ψ ) φ ∗ e α − S ∇ R ( ψ ) − ( h∇ se β ( e α · e β · χ α ) , ψ i g s + h e α · e β · χ α , e ∇ e β ψ i g s ) ,/Dψ = | Qχ | ψ + 13 SR ( ψ ) + 2( ⊗ φ ∗ ) Qχ, (1)where S ∇ R ( ψ ) = φ ∗ ( ∇ N R ) ijkl h ψ i , ψ k i g s h ψ j , ψ l i g s , and R( ψ, e α · ψ ) φ ∗ e α = h ψ k , e α · ψ l i g s e α ( φ j ) φ ∗ (cid:18) R (cid:18) ∂∂y k , ∂∂y l (cid:19) ∂∂y j (cid:19) = R ijkl h ψ k , ∇ φ j · ψ l i ⊗ φ ∗ (cid:18) ∂∂y i (cid:19) . One notices that this action functional can actually be defined for ( φ, ψ ) that possess onlylittle regularity; we only need integrability properties to make the action well defined, that is, φ ∈ W , ( M, N ) and ψ ∈ Γ , / ( S ⊗ φ ∗ T N ) . The corresponding solutions of (1) in the sense ofdistributions are called weak solutions. When the Riemannian metric g and the gravitino χ areassumed to be smooth parameters, it is shown in [22] that any weak solution ( φ, ψ ) is actuallysmooth. We will show that these solutions have more interesting geometric and analyticalproperties. Embed ( N, h ) isometrically into some Euclidean space R K . Then a solution can berepresented by a tuple of functions φ = ( φ , · · · , φ K ) taking values in R K and a tuple of spinors ψ = ( ψ , · · · , ψ K ) where each ψ i is a (pure) spinor and they together satisfy the condition thatat each point φ ( x ) in the image, for any normal vector ν = ( ν , · · · , ν K ) ∈ T ⊥ φ ( x ) N ⊂ T φ ( x ) R K , K X i =1 ψ i ( x ) ν i ( φ ( x )) = 0 . Moreover, writing the second fundamental form of the isometric embedding as A = ( A ijk ) , theEuler–Lagrange equations can be written in the following form (see [22]) ∆ φ i = A ijk h∇ φ j , ∇ φ k i + A ijm A mkl h ψ j , ∇ φ k · ψ l i + Z i ( A, ∇ A ) jklm h ψ j , ψ l ih ψ k , ψ m i − div V i − A ijk h V j , ∇ φ k i , (2) /∂ψ i = − A ijk ∇ φ j · ψ k + | Qχ | ψ i + 13 A ijm A mkl (cid:0) h ψ k , ψ l i ψ j − h ψ j , ψ k i ψ l (cid:1) − e α · ∇ φ i · χ α . (3)Here the V i ’s are vector fields on M defined by(4) V i = h e α · e β · χ α , ψ i i e β . One should note that there is some ambiguity here, because the second fundamental formmaps tangent vectors of the submanifold N to normal vectors, so the lower indices of A ijk should be tangential indices, and the upper ones normal. However, one can extend the second JÜRGEN JOST, RUIJUN WU AND MIAOMIAO ZHU fundamental form to a tubular neighborhood of N in R K such that all the A ijk ’s make sense.Alternatively, one can rewrite the extrinsic equations without labeling indices, but we wantto derive estimates and see how the second fundamental form A affects the system, hence weadopt this formulation.This action functional is closely related to Dirac-harmonic maps with curvature term. Actu-ally, if the gravitinos vanish in the model, the action A then reads L c ( φ, ψ ) = Z M | d φ | + h ψ, /Dψ i −
16 R( ψ ) d vol g , whose critical points are known as Dirac-harmonic maps with curvature term. These were firstlyintroduced in [10] and further investigated in [4, 5, 24]. Furthermore, if the curvature term isalso omitted, then we get the Dirac-harmonic map functional which was introduced in [7, 8]and further explored from the perspective of geometric analysis in e.g. [35, 36, 37, 33, 9, 27, 31].From the physical perspective, they constitute a simplified version of the model considered inthis paper, and describe the behavior of the nonlinear sigma models in degenerate cases.The symmetries of this action functional always play an important role in the study of thesolution spaces, and here especially the rescaled conformal invariance. Lemma 1.1.
Let f : ( f M , ˜ g ) → ( M, g ) be a conformal diffeomorphism, with f ∗ g = e u ˜ g , andsuppose the spin structure of ( f M , ˜ g ) is isomorphic to the pullback of the given one of ( M, g ) .There is an identification B : S → ˜ S which is an isomorphism and fiberwise isometry such thatunder the transformation φ ˜ φ := φ ◦ f,ψ ˜ ψ := e u ( B ⊗ φ ∗ T N ) ψ,χ ˜ χ := e u ( B ⊗ ( f − ) ∗ ) χ,g ˜ g, each summand of the action functional stays invariant, and also Z M | ψ | d vol g = Z f M | ˜ ψ | d vol ˜ g . Remark.
Furthermore, the following quantities are also invariant under the transformations inthe above lemma: Z M | χ | d x, Z M | e ∇ ψ | d x, Z M | b ∇ χ | d x, where b ∇ ≡ ∇ S ⊗ T M . Also observe that Q is only a linear projection operator, so Qχ enjoysthe same analytic properties as χ . In our model, most time it is only the Q -part of χ whichis involved, so all the assumptions and conclusions can be made on the Qχ ’s. The rescaledconformal invariance with respect to ψ was shown in [17], and see also [8]. As for the gravitino χ , the spinor part has to be rescaled in the same way as ψ , while the tangent vector part hasto be rescaled in the ordinary way, which gives rise to an additional factor e u , such that thecorresponding norms are invariant. For more detailed investigations one can refer to [23] wheremore symmetry properties of our nonlinear sigma model with gravitinos are analyzed. Example . When the map f is a rescaling by a constant λ on the Euclidean space with thestandard Euclidean metric g , then f ∗ g = λ g and ( f − ) ∗ is a rescaling by λ − . In this casethe gravitino χ transforms to √ λBχ α ⊗ e α , where e α is a standard basis for ( R , g ) . NERGY QUANTIZATION 5
For a given pair ( φ, ψ ) and a domain U ⊂ M , the energy of this pair ( φ, ψ ) on U is definedto be E ( φ, ψ ; U ) := Z U | d φ | + | ψ | d vol g , and when U is the entire manifold we write E ( φ, ψ ) omitting U . Similarly, the energy of themap φ resp. the vector spinor ψ on U is defined by E ( φ ; U ) := Z U | d φ | d vol g , resp. E ( ψ ; U ) := Z U | ψ | d vol g . From the previous lemma we know that they are rescaling invariant. We will show that wheneverthe local energy of a solution is small, then some higher derivatives of this solution can becontrolled by its energy and some appropriate norm of the gravitino; this is known as thesmall energy regularity. On the other hand, similar to the theories for harmonic maps andDirac-harmonic maps, the energy of a solution should not be globally small, that is, when thedomain is a closed surface, in particular the standard sphere, because too small energy forcesthe solution to be trivial. That is, there are energy gaps between the trivial and nontrivialsolutions of (1) on some closed surfaces. These will be shown in Section 2.To proceed further we restrict to some special gravitinos, i.e. those gravitinos that are crit-ical with respect to variations . As shown in [23], this is equivalent to the vanishing of thecorresponding supercurrent . Then we will see in Section 3 that the energy-momentum tensor,defined using a local orthonormal frame ( e α ) by T = (cid:8) h φ ∗ e α , φ ∗ e β i − | d φ | g αβ + 12 D ψ, e α · e ∇ e β ψ + e β · e ∇ e α ψ E − h ψ, /D g ψ i g αβ + h e η · e α · χ η ⊗ φ ∗ e β + e η · e β · χ η ⊗ φ ∗ e α , ψ i + 4 h ( ⊗ φ ∗ ) Qχ, ψ i g αβ + | Qχ | | ψ | g αβ + 16 R( ψ ) g αβ (cid:9) e α ⊗ e β , is symmetric, traceless and divergence free, see Proposition 3.3. Hence it gives rise to a holo-morphic quadratic differential, see Proposition 3.4. In a local conformal coordinate z = x + iy ,this differential reads T ( z ) d z := ( T − iT )(d x + i d y ) , with T = (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂x (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂y (cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:16) h ψ, γ ( ∂ x ) e ∇ ∂ x ψ i − h ψ, γ ( ∂ y ) e ∇ ∂ y ψ i (cid:17) + F ,T = (cid:28) ∂φ∂x , ∂φ∂y (cid:29) φ ∗ h + h ψ, γ ( ∂ x ) e ∇ ∂ y ψ i + F , where in a local chart χ = χ x ⊗ ∂ x + χ y ⊗ ∂ y and F = 2 h− χ x ⊗ φ ∗ ( ∂ x ) − γ ( ∂ x ) γ ( ∂ y ) χ y ⊗ φ ∗ ( ∂ x ) , ψ i + 2 h ( ⊗ φ ∗ ) Qχ, ψ i g ( ∂ x , ∂ x ) ,F = 2 h− χ x ⊗ φ ∗ ( ∂ y ) − γ ( ∂ x ) γ ( ∂ y ) χ y ⊗ φ ∗ ( ∂ y ) , ψ i . Consequently we can establish a Pohozaev type identity for our model in Section 4. This willbe the key ingredient for the analysis in the sequel.
JÜRGEN JOST, RUIJUN WU AND MIAOMIAO ZHU
Theorem 1.2. (Pohozaev identity.)
Let ( φ, ψ ) be a smooth solution of (1) on B ∗ := B \{ } with χ being a critical gravitino which is smooth on B . Assume that ( φ, ψ ) has finite energyon B . Then for any < r < , Z π (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12)(cid:12)(cid:12) − r (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂θ (cid:12)(cid:12)(cid:12)(cid:12) d θ = Z π −h ψ, γ ( ∂ r ) e ∇ ∂ r ψ i + 16 R( ψ ) − ( F cos 2 θ + F sin 2 θ ) d θ = Z π (cid:28) ψ, r γ ( ∂ θ ) e ∇ ∂ θ ψ (cid:29) −
16 R( ψ ) − ( F cos 2 θ + F sin 2 θ ) d θ. (5)In Section 4 we also prove that isolated singularities are removable, using a result from theAppendix and the regularity theorem in [22].Finally, for a sequence of solutions ( φ k , ψ k ) with uniformly bounded energies defined on ( M, g ) with respect to critical gravitinos χ k which converge in W , to some smooth limit χ ,a subsequence can be extracted which converges weakly in W , × L to a solution definedon ( M, g ) , and by a rescaling argument, known as the blow-up procedure, we can get somesolutions with vanishing gravitinos, i.e. Dirac harmonic maps with curvature term, defined onthe standard sphere S with target manifold ( N, h ) , known as “bubbles”. Moreover, the energiespass to the limit, i.e. the energy identities hold. Theorem 1.3. (Energy identities.)
Let ( φ k , ψ k ) be a sequence of solutions of (1) with respectto smooth critical gravitinos χ k which converge in W , to a smooth limit χ , and assume theirenergies are uniformly bounded: E ( φ k , ψ k ) ≤ Λ < ∞ . Then passing to a subsequence if necessary, the sequence ( φ k , ψ k ) converges weakly in thespace W , ( M, N ) × L ( S ⊗ R K ) to a smooth solution ( φ, ψ ) with respect to χ . Moreover,the blow-up set S := \ r> (cid:26) p ∈ M (cid:12)(cid:12)(cid:12) lim inf k →∞ Z B r ( p ) |∇ φ k | + | ψ k | d vol g ≥ ε (cid:27) is a finite (possibly empty) set of points { p , . . . , p I } , and correspondingly a finite set (possiblyempty) of Dirac-harmonic maps with curvature term ( σ li , ξ li ) defined on S with target mani-fold ( N, h ) , for l = 1 , . . . , L i and i = 1 , . . . , I , such that the following energy identities hold: lim k →∞ E ( φ k ) = E ( φ ) + I X i =1 L i X l =1 E ( σ li ) , lim k →∞ E ( ψ k ) = E ( ψ ) + I X i =1 L i X l =1 E ( ξ li ) . The proof will be given in Section 5. Although these conclusions are similar to those forharmonic maps and Dirac-harmonic maps and some of its variants in e.g. [20, 29, 7, 35, 24],one has to pay special attentions to the critical gravitinos.2.
Small energy regularity and energy gap property
In this section we consider the behavior of solutions with small energies.
NERGY QUANTIZATION 7
Theorem 2.1. ( ε -Regularity theorem.) Consider the local model defined on the Euclideanunit disk B ⊂ R and the target manifold is a submanifold ( N, h ) ֒ → R K with second funda-mental form A . For any p ∈ (1 , ) and p ∈ (1 , there exists an ε = ε ( A, p , p ) ∈ (0 , such that if the gravitino χ and a solution ( φ, ψ ) of (1) satisfy E ( φ, ψ ; B ) = Z B |∇ φ | + | ψ | d x ≤ ε , Z B | χ | + | b ∇ χ | d x ≤ ε , then for any U ⋐ B , the following estimates hold: k φ k W ,p ( U ) ≤ C (cid:0) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k W , ( B ) (cid:1) , k ψ k W ,p ( U ) ≤ C (cid:0) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k W , ( B ) (cid:1) , where C = C ( p , p , U, N ) > .Remark. Note that if the second fundamental form A vanishes identically, then N is a totallygeodesic submanifold of the Euclidean space R K , hence there is no curvatures on N and themodel then is easy and not of interest. So we will assume that A = 0 , and without loss ofgenerality, we assume | A | ≡ k A k ≥ . For some C ( p ) depending on the value of p to be chosenlater, the small barrier constant ε will be required to satisfy C ( p ) | A | √ ε ≤ , C ( p ) | A ||∇ A | ε / ≤ , (6)where |∇ A | ≡ k∇ A k . These restrictions will be explained in the proof. Remark.
Note also that since the domain is the Euclidean disk B , the connection b ∇ is actuallyequivalent to ∇ s . Proof of Theorem 2.1.
Since N is taken as a compact submanifold of R K , we may assume thatit is contained in a ball of radius C N in R K , which implies k φ k L ∞ ≤ C N . Moreover, as we aredealing with a local solution ( φ, ψ ) , we may assume that R B φ d x = 0 , so that the Poincaréinequalities hold: for any p ∈ [1 , ∞ ] , k φ k L p ( B ) ≤ C p k∇ φ k L p ( B ) . Let ( U k ) k ≥ be a sequence of nonempty disks such that B ⋑ U ⋑ U ⋑ U ⋑ · · · . Take a smooth cutoff function η : B → R such that ≤ η ≤ , η | U ≡ , and supp η ⊂ B .Then ηψ satisfies /∂ ( ηψ i ) = ∇ η · ψ i + η /∂ψ i = ∇ η · ψ i − A ijk ∇ φ j · ( ηψ k ) + | Qχ | ( ηψ i )+ 13 A ijm A mkl (cid:0) h ψ k , ψ l i ( ηψ j ) − h ψ j , ψ k i ( ηψ l ) (cid:1) − e α · ∇ ( ηφ i ) · χ α + e α · φ i ∇ η · χ α . JÜRGEN JOST, RUIJUN WU AND MIAOMIAO ZHU
Then one has | /∂ ( ηψ ) | ≤|∇ η || ψ | + | A ||∇ φ | · | ηψ | + | Qχ | | ηψ | + | A | | ψ | · | ηψ | + | Qχ ||∇ ( ηφ ) | + | φ ||∇ η || Qχ | . Consider the L p -norm (where p ∈ (1 , ) of the left hand side: k /∂ ( ηψ ) k L p ( B ) ≤k∇ η k L p − p ( B ) k ψ k L ( B ) + | A |k∇ φ k L ( B ) k ηψ k L p − p ( B ) + k Qχ k L ( B ) k ηψ k L p − p ( B ) + | A | k ψ k L ( B ) k ηψ k L p − p ( B ) + k Qχ k L ( B ) k∇ ( ηφ ) k L p − p ( B ) + C N k∇ η k L p − p ( B ) k Qχ k L ( B ) . Assume that k∇ η k L p − p ( B ) is bounded by some constant C ′ = C ′ ( U , p ) . Since ηψ vanishes onthe boundary and /∂ is an elliptic operator of order one, we have k ηψ k L p − p ( B ) ≤ C ( p ) k∇ s ( ηψ ) k L p ( B ) ≤ C ( p ) k /∂ ( ηψ ) k L p ( B ) . Then from k /∂ ( ηψ ) k L p ( B ) ≤ (cid:16) | A |k∇ φ k L ( B ) + k Qχ k L ( B ) + | A | k ψ k L ( B ) (cid:17) k ηψ k L p − p ( B ) + k Qχ k L ( B ) k∇ ( ηφ ) k L p − p ( B ) + C N C ′ (cid:0) k ψ k L ( B ) + k Qχ k L ( B ) (cid:1) together with the fact | e ∇ ( ηψ ) | ≤ |∇ s ( ηψ ) | + | A || ηψ ||∇ φ | , it follows that k e ∇ ( ηψ ) k L p ( B ) ≤ C ( p ) (cid:18) k Qχ k L ( B ) k∇ ( ηφ ) k L p − p ( B ) + C N C ′ (cid:0) k ψ k L ( B ) + k Qχ k L ( B ) (cid:1)(cid:19) (7)provided that (6) is satisfied.Now consider the map φ . The equations for ηφ are ∆( ηφ i ) = η ∆ φ i + 2 h∇ η, ∇ φ i i + (∆ η ) φ i = η (cid:0) A ijk h∇ φ j , ∇ φ k i + A ijm A mkl h ψ j , ∇ φ k · ψ l i + Z i ( A, ∇ A ) jklm h ψ j , ψ l ih ψ k , ψ m i− div V i − A ijk h V j , ∇ φ k i (cid:1) + 2 h∇ η, ∇ φ i i + (∆ η ) φ i . Using η ∇ φ i = ∇ ( ηφ i ) − φ i ( ∇ η ) , we can rewrite it as ∆( ηφ i ) = A ijk h∇ φ j , ∇ ( ηφ k ) i + A ijm A mkl h ψ j , ∇ ( ηφ k ) · ψ l i + Z i ( A, ∇ A ) jklm h ψ j , ψ l ih ψ k , ηψ m i− div( ηV i ) − A ijk h V j , ∇ ( ηφ k ) i + 2 h∇ η, ∇ φ i i + (∆ η ) φ i − A ijk h∇ φ j , φ k ∇ η i − A ijm A mkl h ψ j , φ k ∇ η · ψ l i + h∇ η, V i i + A ijk h V j , φ k ∇ η i . (8)Notice that ηφ i ∈ C ∞ ( B ) . Split it as ηφ i = u i + v i , where u i ∈ C ∞ ( B ) uniquely solves (seee.g. [12, Chap. 8]) ∆ u i = − div( ηV i ) . NERGY QUANTIZATION 9
Since ηV i ∈ L p − p ( B ) , it follows from the L p theory of Laplacian operators that(9) k u k W , p − p ( B ) ≤ C ( p ) k Qχ k L ( B ) k ηψ k L p − p ( B ) . Then v i ∈ C ∞ ( B ) satisfies ∆ v i =∆( ηφ i ) − ∆ u i = A ijk h∇ φ j , ∇ ( ηφ k ) i + A ijm A mkl h ψ j , ∇ ( ηφ k ) · ψ l i + Z i ( A, ∇ A ) jklm h ψ j , ψ l ih ψ k , ηψ m i − A ijk h V j , ∇ ( ηφ k ) i + 2 h∇ η, ∇ φ i i + (∆ η ) φ i − A ijk h∇ φ j , φ k ∇ η i − A ijm A mkl h ψ j , φ k ∇ η · ψ l i + h∇ η, V i i + A ijk h V j , φ k ∇ η i . From [22], k Z ( A, ∇ A ) k ≤ | A ||∇ A | . Thus the L p p norm of ∆ v can thus be estimated by k ∆ v k L p p ( B ) ≤ | A |k∇ φ k L ( B ) k∇ ( ηφ ) k L p − p ( B ) + | A | k ψ k L ( B ) k∇ ( ηφ ) k L p − p ( B ) + | A ||∇ A |k ψ k L ( B ) k ηψ k L p − p ( B ) + | A |k Qχ k L ( B ) k ψ k L ( B ) k∇ ( ηφ ) k L p − p ( B ) +2 k∇ η k L p − p ( B ) k∇ φ k L ( B ) + k ∆ η k L p − p ( B ) k φ k L ( B ) + | A |k∇ φ k L ( B ) k φ ∇ η k L p − p ( B ) + | A | k ψ k L ( B ) k φ ∇ η k L p − p ( B ) + k∇ η k L p − p ( B ) k Qχ k L ( B ) k ψ k L ( B ) + | A |k φ ∇ η k L p − p ( B ) k Qχ k L ( B ) k ψ k L ( B ) . As before assume that k∇ η k L p − p ( B ) and k ∆ η k L p − p ( B ) are bounded by C ′ = C ′ ( U , p ) . Collect-ing the terms, we get k ∆ v k L p p ( B ) ≤ (cid:0) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + | A |k Qχ k L ( B ) k ψ k L ( B ) (cid:1) k∇ ( ηφ ) k L p − p ( B ) + | A ||∇ A |k ψ k L ( B ) k ηψ k L p − p ( B ) + C ′ C N (cid:0) k∇ φ k L ( B ) + k φ k L ( B ) + | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k L ( B ) k ψ k L ( B ) + | A |k Qχ k L ( B ) k ψ k L ( B ) (cid:1) . By Sobolev embedding, k v k W , p − p ( B ) ≤ C ( p ) (cid:0) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k L ( B ) (cid:1) k∇ ( ηφ ) k L p − p ( B ) + C ( p ) | A ||∇ A |k ψ k L ( B ) k ηψ k L p − p ( B ) + 4 C ( p ) C ′ C N (cid:0) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k L ( B ) (cid:1) . (10)Since ηφ = u + v , combining (9) and (10), we get k ηφ k W , p − p ( B ) ≤ C ( p ) (cid:0) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k L ( B ) (cid:1) k∇ ( ηφ ) k L p − p ( B ) + C ( p ) | A ||∇ A |k ψ k L ( B ) k ηψ k L p − p ( B ) + C ( p ) k Qχ k L ( B ) k ηψ k L p − p ( B ) +4 C ( p ) C ′ C N (cid:0) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k L ( B ) (cid:1) . (11)By the small energy assumption, and Sobolev embedding, this implies that k ηφ k W , p − p ( B ) ≤ C ( p ) (cid:0) | A ||∇ A |k ψ k L ( B ) + k Qχ k L ( B ) (cid:1) k ηψ k W ,p ( B ) + 8 C ( p ) C ′ C N (cid:0) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k L ( B ) (cid:1) . The estimates (7) and (11), together with the small energy assumption, imply that for any p ∈ (1 , ,(12) k ηφ k W , p − p ( B ) + k ηψ k W ,p ( B ) ≤ C ( p, η, N ) (cid:0) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k L ( B ) (cid:1) . Note that as p ր , p − p ր . Thus, ηφ is almost a W , map and ηψ is almost a W , vectorspinor.Now χ ∈ W , , thus in the equations for the map φ , the divergence terms can be reconsidered.Take another cutoff function, still denoted by η , such that ≤ η ≤ , η | U ≡ , and supp η ⊂ U .Then ηφ satisfies equations of the same form as (8), and div( ηV i ) ∈ L p ( B ) for any p ∈ [1 , ) .For example, we take p = , then k div( ηV i ) k L ( B ) ≤ C ( η ) k ψ k W , ( U ) k Qχ k W , ( B ) , and note that k ψ k W , ( U ) is under control by (12). Recalling (8) we have the estimate k ∆( ηφ ) k L ( B ) ≤ | A |k∇ φ k L ( B ) k∇ ( ηφ ) k L ( B ) + | A | k ψ k L ( B ) k∇ ( ηφ ) k L ( B ) + | A ||∇ A |k ψ k L ( B ) k ηψ k L ( B ) + | A |k Qχ k L ( B ) k ψ k L ( B ) k∇ ( ηφ ) k L ( B ) + k div( ηV ) k L ( B ) + 2 k∇ η k L ( B ) k∇ φ k L ( B ) + k ∆ η k L ( B ) k φ k L ( B ) + | A |k∇ φ k L ( B ) k φ ∇ η k L ( B ) + | A | k ψ k L ( B ) k φ ∇ η k L ( B ) + k∇ η k L ( B ) k Qχ k L ( B ) k ψ k L ( B ) + | A |k φ ∇ η k L ( B ) k Qχ k L ( B ) k ψ k L ( B ) . As before we assume k∇ η k L ( B ) and k ∆ η k L ( B ) are bounded by C ′′ = C ′′ ( U , U ) . Then k ∆( ηφ ) k L ( B ) ≤ (cid:16) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k L ( B ) (cid:17) k∇ ( ηφ ) k L ( B ) + | A ||∇ A |k ψ k L ( B ) k ηψ k L ( B ) + C ( η ) k ψ k W , ( U ) k Qχ k W , ( B ) + 4 C N C ′′ (cid:16) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k L ( B ) (cid:17) . By the smallness assumptions and the L p theory for Laplacian operator (here p = ) we get k ηφ k W , ( B ) ≤ C ( p, U , N ) (cid:16) | A |k∇ φ k L ( B ) + | A | k ψ k L ( B ) + k Qχ k W , ( B ) (cid:17) . One can check that similar estimates hold for k ηφ k W ,p ( B ) for any p ∈ (1 , ) . This accomplishesthe proof. (cid:3) Recall the Sobolev embeddings W , ( B ) ֒ → W , ( B ) ֒ → C / ( B ) . Thus we see that the map φ is Hölder continuous with k ηφ k C / ( B ) ≤ C k ηφ k W , ( B ) . In particular, when the energies of ( φ, ψ ) and certain norms of the gravitino are small, saysmaller than ε (where ε ≤ ε ), the -Hölder norm of the map in the interior is also small, withthe estimate(13) k φ k C / ( U ) ≤ C ( N, U, | A | ) √ ε. NERGY QUANTIZATION 11
Proposition 2.2. (Energy gap property.)
Suppose that ( φ, ψ ) is a solution to (1) de-fined on an oriented closed surface ( M, g ) with target manifold ( N, h ) . Suppose that the spinorbundle S → ( M, g ) doesn’t admit any nontrivial harmonic spinors. Then there exists an ε = ε ( M, g, A ) ∈ (0 , such that if (14) E ( φ, ψ ) + k Qχ k W , ( M ) ≤ ε , then ( φ, ψ ) has to be a trivial solution. The existence of harmonic spinors is closely related to the topological and Riemannian struc-tures. Examples of closed surfaces which don’t admit harmonic spinors include S with arbitraryRiemannian metric and the torus T with a nontrivial spin structure, and many others. Formore information on harmonic spinors one can refer to [17, 2]. Proof of Proposition 2.2.
When the spinor bundle S doesn’t admit nontrivial harmonic spinors,the Dirac operator is “invertible”, in the sense that for any < p < ∞ , there holds k σ k L p ( M ) ≤ C ( p ) k /∂σ k L p ( M ) , ∀ σ ∈ Γ( S ) . See e.g. [11] for a proof . As /∂ is an elliptic operator of first order, one has k∇ s ψ a k L ( M ) ≤ C (cid:16) k /∂ψ a k L ( M ) + k ψ a k L ( M ) (cid:17) , ≤ a ≤ K. It follows that(15) k ψ k W , ( M ) ≤ C k /∂ψ k L ( M ) + | A |k ψ k L ( M ) k∇ φ k L ( M ) . From (3) one gets k /∂ψ k L ( M ) ≤| A |k∇ φ k L ( M ) k ψ k L ( M ) + k Qχ k L ( M ) k ψ k L ( M ) + | A | k ψ k L ( M ) k ψ k L ( M ) + k Qχ k L ( M ) k∇ φ k L ( M ) . Since (14) holds, using (15) one obtains(16) k ψ k W , ( M ) ≤ C (cid:0) k Qχ k L ( M ) + k ψ k L ( M ) (cid:1) k∇ φ k L ( M ) . Next we deal with the map φ . From (2) it follows that k ∆ φ k L ( M ) ≤| A |k∇ φ k L ( M ) k∇ φ k L ( M ) + | A | k ψ k L ( M ) k∇ φ k L ( M ) + | A ||∇ A |k ψ k L ( M ) k ψ k L ( M ) + (cid:16) k b ∇ Qχ k L ( M ) + C k Qχ k L ( M ) (cid:17) k ψ k L ( M ) + k Qχ k L ( M ) k e ∇ ψ k L ( M ) + | A |k Qχ k L ( M ) k ψ k L ( M ) k∇ φ k L ( M ) . Combining with (14) this gives k∇ φ k L ( M ) ≤ Cε k ψ k W , ( M ) ≤ Cε (cid:0) k Qχ k L ( M ) + k ψ k L ( M ) (cid:1) k∇ φ k L ( M ) . There they show a proof for p = , but it is easily generalized to a general p ∈ (1 , ∞ ) . Therefore, when ε is sufficiently small, this implies ∇ φ ≡ , that is, φ = const. Then (16) saysthat ψ is also trivial. (cid:3) Remark.
Observe that although the estimates here are similar to those in the proof of smallenergy regularities, they come from a different point of view. There we have to take cutofffunctions to make the boundary terms vanish in order that the local elliptic estimates areapplicable without boundary terms. Here, on the contrary, we rely on the hypothesis that S doesn’t admit nontrivial harmonic spinors to obtain the estimate (16) which is a global property.3. Critical gravitino and energy-momentum tensor
In this section we consider the energy-momentum tensor along a solution to (1). We will seethat it gives rise to a holomorphic quadratic differential when the gravitino is critical, which isneeded for the later analysis.From now on we assume that the gravitino χ is also critical for the action functional withrespect to variations; that is, for any smooth family ( χ t ) t of gravitinos with χ = χ , it holdsthat dd t (cid:12)(cid:12)(cid:12) t =0 A ( φ, ψ ; g, χ t ) = 0 . One can conclude from this by direct calculation that the supercurrent J = J α ⊗ e α vanishes(or see [23]), where J α = 2 h φ ∗ e β , e β · e α · ψ i φ ∗ h + | ψ | e β · e α · χ β . Equivalently it can be formulated as | ψ | e β · e α · χ β = − h φ ∗ e β , e β · e α · ψ i φ ∗ h , ∀ α. Recall that Qχ = − e β · e α · χ β ⊗ e α . Thus(17) | ψ | Qχ = − | ψ | e β · e α · χ β ⊗ e α = h φ ∗ e β , e β · e α · ψ i φ ∗ h ⊗ e α . It follows that | Qχ | | ψ | = h χ, | Qχ | χ i φ ∗ h = h χ η ⊗ e η , h φ ∗ e β , e β · e α · ψ i φ ∗ h ⊗ e α i g s ⊗ g = h χ α ⊗ φ ∗ e β , e β · e α · ψ i g s ⊗ φ ∗ h = h e α · e β · χ α ⊗ φ ∗ e β , ψ i g s ⊗ φ ∗ h = − h ( ⊗ φ ∗ ) Qχ, ψ i . Since the Euler–Lagrange equations for ψ are(18) /Dψ = 13 SR ( ψ ) + | Qχ | ψ + 2( ⊗ φ ∗ ) Qχ, so if ψ is critical, i.e. the above equation (18) holds, then h ψ, /Dψ i = 13 h SR ( ψ ) , ψ i = 13 R( ψ ) . Therefore the following relation holds: h ψ, e · e ∇ e ψ i = −h ψ, e · e ∇ e ψ i + 13 R( ψ ) . NERGY QUANTIZATION 13
Lemma 3.1.
For any φ and ψ , and for any β , e β ( | Qχ | | ψ | ) = 2 h∇ se β ( e α · e η · χ α ) ⊗ φ ∗ e η , ψ i + 2 | Qχ | h ψ, e ∇ e β ψ i . Proof.
Since e β ( | Qχ | | ψ | ) = e β ( | Qχ | ) | ψ | + 2 | Qχ | h ψ, e ∇ e β ψ i , it suffices to compute e β ( | Qχ | ) | ψ | . Note that e β ( | Qχ | ) = e β h χ, Qχ i = − e β h χ α , e η · e α · χ η i = − (cid:0) h∇ se β χ α , e η · e α · χ η i + h χ α , e η · e α · ∇ se β χ η i (cid:1) = −h∇ se β χ α , e η · e α · χ η i . Therefore, by virtue of (17), e β ( | Qχ | ) | ψ | = −h∇ se β χ α , | ψ | e η · e α · χ η i = 2 (cid:10) ∇ se β χ α , h φ ∗ e η , e η · e α · ψ i φ ∗ h (cid:11) g s = 2 h∇ se β χ α ⊗ φ ∗ e η , e η · e α · ψ i = 2 h∇ se β ( e α · e η · χ α ) ⊗ φ ∗ e η , ψ i . The desired equality follows. (cid:3)
Lemma 3.2.
For any φ and ψ , (19) h ( ⊗ φ ∗ ) Qχ, ω · ψ i = 0 , where ω = e · e is the volume element.Proof. Since | ψ | ( ⊗ φ ∗ ) Qχ = − | ψ | e η · e α · χ η ⊗ φ ∗ e α = h φ ∗ e η , e η · e α · ψ i φ ∗ h ⊗ φ ∗ e α , We have | ψ | h ( ⊗ φ ∗ ) Qχ, e · e · ψ i = hh φ ∗ e η , e η · e α · ψ i φ ∗ h ⊗ φ ∗ e α , e · e · ψ i = (cid:10) h φ ∗ e η , e η · e α · ψ i φ ∗ h , h φ ∗ e α , e · e · ψ i φ ∗ h (cid:11) g s . According to the Clifford relation it holds that (cid:10) h φ ∗ e η , e η · e α · ψ i φ ∗ h , h φ ∗ e α , e · e · ψ i φ ∗ h (cid:11) g s = (cid:10) h φ ∗ e η , e · e · e η · e α · ψ i φ ∗ h , h φ ∗ e α , ψ i φ ∗ h (cid:11) g s = (cid:10) h φ ∗ e η , e η · e α · e · e · ψ i φ ∗ h , h φ ∗ e α , ψ i φ ∗ h (cid:11) g s = (cid:10) h φ ∗ e η , e · e · ψ i φ ∗ h , h φ ∗ e α , e α · e η · ψ i φ ∗ h (cid:11) g s = − (cid:10) h φ ∗ e α , e · e · ψ i φ ∗ h , h φ ∗ e η , e η · e α · ψ i φ ∗ h (cid:11) g s . It follows that | ψ | h ( ⊗ φ ∗ ) Qχ, e · e · ψ i = 0 . At any point x ∈ M , if ψ ( x ) = 0 , then (19) holds; and if | ψ ( x ) | 6 = 0 , then by the calculationsabove (19) also holds. This finishes the proof. (cid:3) Remark.
More explicitly (19) is equivalent to(20) h e · e · χ ⊗ φ ∗ e + χ ⊗ φ ∗ e − χ ⊗ φ ∗ e + e · e · χ ⊗ φ ∗ e , ψ i = 0 . From [23] we know the energy-momentum tensor is given by T = T αβ e α ⊗ e β where T αβ =2 h φ ∗ e α , φ ∗ e β i φ ∗ h − | d φ | g αβ + 12 D ψ, e α · e ∇ e β ψ + e β · e ∇ e α ψ E g s ⊗ φ ∗ h − h ψ, /D g ψ i g αβ + h e η · e α · χ η ⊗ φ ∗ e β + e η · e β · χ η ⊗ φ ∗ e α , ψ i g s ⊗ φ ∗ h + 4 h ( ⊗ φ ∗ ) Qχ, ψ i g αβ + | Qχ | | ψ | g αβ + 16 R( ψ ) g αβ . (21)Suppose that ( φ, ψ ) satisfies the Euler–Lagrange equations (1) and that the supercurrent J vanishes. Then T αβ =2 h φ ∗ e α , φ ∗ e β i − | d φ | g αβ + 12 h ψ, e α · e ∇ e β ψ + e β · e ∇ e α ψ i − h ψ, /Dψ i g αβ + h e η · e α · χ η ⊗ φ ∗ e β + e η · e β · χ η ⊗ φ ∗ e α , ψ i − h e θ · e η · χ θ ⊗ φ ∗ e η , ψ i g αβ . Clearly T is symmetric and traceless. We will show it is also divergence free. Before this werewrite it into a suitable form. Multiplying ω = e · e to both sides of equations (18), we get e · e ∇ e ψ − e · e ∇ e ψ = 13 ω · SR ( ψ ) + | Qχ | ω · ψ + 2 ω · ( ⊗ φ ∗ ) Qχ.
Note that the right hand side is perpendicular to ψ : h ψ, ω · SR ( ψ ) i = R ijkl h ψ j , ψ l ih ψ i , ω · ψ k i = 0 , | Qχ | h ψ, ω · ψ i = 0 , h ω · ( ⊗ φ ∗ ) Qχ, ψ i = − h ( ⊗ φ ∗ ) Qχ, ω · ψ i = 0 . Hence h ψ, e · e ∇ e ψ i − h ψ, e · e ∇ e ψ i = 0 . Consequently, h ψ, e α · e ∇ e β ψ + e β · e ∇ e α ψ i = h ψ, e α · e ∇ e β ψ i . Moreover, by (20), h e η · e · χ η ⊗ φ ∗ e , ψ i − h e η · e · χ η ⊗ φ ∗ e , ψ i = h− χ ⊗ φ ∗ e − e · e · χ ⊗ φ ∗ e − e · e · χ ⊗ φ ∗ e + χ ⊗ φ ∗ e , ψ i = 0 . Therefore, we can put the energy-momentum tensor into the following form: T αβ =2 h φ ∗ e α , φ ∗ e β i − | d φ | g αβ + h ψ, e α · e ∇ e β ψ i − h ψ, /Dψ i g αβ + 2 h e η · e α · χ η ⊗ φ ∗ e β , ψ i − h e θ · e η · χ θ ⊗ φ ∗ e η , ψ i g αβ . (22)This form relates closely to the energy-momentum tensor for Dirac-harmonic maps in [8, Section3] and that for Dirac-harmonic maps with curvature term in [24, Section 4], which also havethe following nice properties. Such computations have been provided in [4, Section 3], but sincecertain algebraic aspects are different here, we need to spell out the computations in detail. Proposition 3.3.
Let ( φ, ψ, χ ) be critical. Then the tensor T given by (21) or equivalently (22) is symmetric, traceless, and covariantly conserved.Proof. It remains to show that T is covariantly conserved. Let x ∈ M and take the normalcoordinate at x such that ∇ e α ( x ) = 0 . We will show that ∇ e α T αβ ( x ) = 0 . At the point x ,making use of the Euler–Lagrange equations, one can calculate as follows. NERGY QUANTIZATION 15 • ∇ e α (2 h φ ∗ e α ,φ ∗ e β i − | d φ | g αβ )=2 h∇ e α ( φ ∗ e α ) , φ ∗ e β i + 2 h φ ∗ e α , ∇ e α ( φ ∗ e β ) i − h φ ∗ e α , ∇ e β ( φ ∗ e α ) i =2 h τ ( φ ) , φ ∗ e β i = h R( ψ, e α · ψ ) φ ∗ e α , φ ∗ e β i − h S ∇ R ( ψ ) , φ ∗ e β i− h∇ se α ( e η · e α · χ η ) ⊗ φ ∗ e β , ψ i − h e η · e α · χ η ⊗ φ ∗ e β , e ∇ e α ψ i . • ∇ e α ( h ψ, e α · e ∇ e β ψ i − h ψ, /Dψ i g αβ )= h e ∇ e α ψ, e α · e ∇ e β ψ i + h ψ, e α · e ∇ e α e ∇ e β ψ i − h e ∇ e β ψ, /Dψ i − h ψ, e ∇ e β /Dψ i = − h /Dψ, e ∇ e β ψ i + h ψ, /D e ∇ e β ψ i − h e ∇ e β ψ, /Dψ i − h ψ, e ∇ e β /Dψ i = − h /Dψ, e ∇ e β ψ i + h ψ, /D e ∇ e β ψ − e ∇ e β /Dψ i . Note that /D e ∇ e β ψ − e ∇ e β /Dψ = e α · Ric S ( e α , e β ) ψ + R( φ ∗ e α , φ ∗ e β )= 12 Ric( e β ) ψ + R( φ ∗ e α , φ ∗ e β ) e α · ψ, and that h ψ, Ric( e β ) ψ i = 0 . Hence one has ∇ e α ( h ψ, e α · e ∇ e β ψ i − h ψ, /Dψ i g αβ )= − h| Qχ | ψ + 13 SR ( ψ ) + 2( ⊗ φ ∗ ) Qχ, e ∇ e β ψ i + h ψ, R( φ ∗ e α , φ ∗ e β ) e α · ψ i = − | Qχ | h ψ, e ∇ e β ψ i − h SR ( ψ ) , e ∇ e β ψ i − h ( ⊗ φ ∗ ) Qχ, e ∇ e β ψ i− h R( ψ, e α · ψ ) φ ∗ e α , φ ∗ e β i . • ∇ e α (cid:18)
16 R( ψ ) g αβ (cid:19) = 16 h S ∇ R ( ψ ) , φ ∗ e β i + 23 h SR ( ψ ) , e ∇ e β ψ i . • ∇ e α (cid:0) h e η · e α · χ η ⊗ φ ∗ e β , ψ i − δ αβ h e η · e η · χ η ⊗ φ ∗ e η , ψ i (cid:1) = 2 h∇ se α ( e η · e α · χ η ) ⊗ φ ∗ e β , ψ i + 2 h e η · e α · χ η ⊗ ∇ e α ( φ ∗ e β ) , ψ i + 2 h e η · e α · χ η ⊗ φ ∗ e β , e ∇ e α ψ i − ∇ e β (cid:0) h e η · e α · χ η ⊗ φ ∗ e α , ψ i (cid:1) . Summarize these terms and use the previous lemmata to get ∇ e α T αβ = − | Qχ | h ψ, e ∇ e β ψ i − h ( ⊗ φ ∗ ) Qχ, e ∇ e β ψ i + 2 h e η · e α · χ η ⊗ ∇ e α ( φ ∗ e β ) , ψ i − ∇ e β (cid:0) h e η · e α · χ η ⊗ φ ∗ e α , ψ i = 2 h∇ se β ( e α · e η · χ η ) ⊗ φ ∗ e η , ψ i − ∇ e β (cid:0) h e η · e α · χ η ⊗ φ ∗ e α , ψ i (cid:1) + 2 h e η · e α · χ η ⊗ φ ∗ e α , e ∇ e β ψ i + 2 h e η · e α · χ η ⊗ ∇ e β ( φ ∗ e α ) , ψ i− ∇ e β (cid:0) h e η · e α · χ η ⊗ φ ∗ e α , ψ i (cid:1) = 0 . This accomplishes the proof. (cid:3)
As in the harmonic map case, such a 2-tensor then corresponds to a holomorphic quadraticdifferential on M . For the case of Dirac-harmonic maps (with or without curvature terms), see[8, 24] and [4]. More precisely, in a local isothermal coordinate z = x + iy , set T ( z ) d z := ( T − iT )(d x + i d y ) , with T and T now being the coefficients of the energy-momentum tensor T in the localcoordinate, that is, T = (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂x (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂y (cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:16) h ψ, γ ( ∂ x ) e ∇ ∂ x ψ i − h ψ, γ ( ∂ y ) e ∇ ∂ y ψ i (cid:17) + F ,T = (cid:28) ∂φ∂x , ∂φ∂y (cid:29) φ ∗ h + h ψ, γ ( ∂ x ) e ∇ ∂ y ψ i + F . Here we have abbreviated the gravitino terms as F αβ ’s: F = 2 h− χ x ⊗ φ ∗ ( ∂ x ) − γ ( ∂ x ) γ ( ∂ y ) χ y ⊗ φ ∗ ( ∂ x ) , ψ i + 2 h ( ⊗ φ ∗ ) Qχ, ψ i g ( ∂ x , ∂ x ) ,F = 2 h− χ x ⊗ φ ∗ ( ∂ y ) − γ ( ∂ x ) γ ( ∂ y ) χ y ⊗ φ ∗ ( ∂ y ) , ψ i , (23)where χ = χ x ⊗ ∂ x + χ y ⊗ ∂ y in a local chart. Proposition 3.4.
The quadratic differential T ( z ) d z is well-defined and holomorphic.Proof. The well-definedness is straightforward and the holomorphicity follows from Proposition3.3. (cid:3) Pohozaev identity and removable singularities
In this section we show that a solution of (1) with finite energy admits no isolated poles,provided that the gravitino is critical. As the singularities under consideration are isolated, wecan locate the solution on the punctured Euclidean unit disk B ∗ ≡ B \{ } . Using the quadraticholomorphic differential derived in the previous section, we obtain the Pohozaev type formulaecontaining gravitino terms in Theorem 1.2. When the gravitino vanishes, they will reduce tothe Pohozaev identities for Dirac-harmonic maps with curvature term, see e.g. [24, Lemma 5.3]and also [5, Lemma 3.11] where a somewhat different identity is derived.Recall that the F αβ ’s are given in (23) and they can be controlled via Young inequality by | F αβ | ≤ C |∇ φ || ψ || χ | ≤ C ( |∇ φ | + | ψ | + | χ | ) . Proof of Theorem 1.2.
By definition we have | T ( z ) | ≤ C (cid:16) |∇ φ | + | e ∇ ψ || ψ | + | F αβ | (cid:17) . Note that |∇ ψ | ≤ C ( |∇ s ψ | + | ψ ||∇ φ | ) . Apply the Young inequality once again to obtain | T ( z ) | ≤ C (cid:16) |∇ φ | + | ψ | + |∇ s ψ | + | χ | (cid:17) . From the initial assumptions we known that φ ∈ W , ( B ∗ , N ) , ψ ∈ L ( B ∗ ) and χ is smooth in B , thus by Theorem 6.1, ( φ, ψ ) is actually a weak solution on the whole disk B . Using theellipticity of the Dirac operator, ψ belongs to W , loc ( B ) . Therefore | T ( z ) | is integrable on thedisk B r for any r < . Recall from Proposition 3.4 that T ( z ) is a holomorphic function defined NERGY QUANTIZATION 17 on the punctured disk. Hence, it has a pole at the origin of order at most one. In particular, zT ( z ) is holomorphic in the whole disk. Then by Cauchy theorem, for any < r < , it holdsthat R | z | = r zT ( z ) d z = 0 . One can compute that in polar coordinate z = re iθ , r Re ( z T ( z )) = (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12)(cid:12)(cid:12) − r (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂θ (cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:18) h ψ, γ ( ∂ r ) ∇ ∂ r ψ i − (cid:28) ψ, r γ ( ∂ θ ) ∇ ∂ θ ψ (cid:29)(cid:19) + F cos 2 θ + F sin 2 θ. The identity h ψ, /Dψ i = R( ψ ) / along a critical ψ implies (cid:18) h ψ, γ ( ∂ r ) ∇ ∂ r ψ i − (cid:28) ψ, r γ ( ∂ θ ) ∇ ∂ θ ψ (cid:29)(cid:19) = h ψ, γ ( ∂ r ) ∇ ∂ r ψ i −
16 R( ψ )= − (cid:28) ψ, r γ ( ∂ θ ) ∇ ∂ θ ψ (cid:29) + 16 R( ψ ) . Finally, it suffices to note that Im (cid:18)Z | z | = r zT ( z ) d z (cid:19) = r Z π Re ( z T ( z )) d θ. (cid:3) Integrating (5) with respect to the radius, we get Z B (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12)(cid:12)(cid:12) − r (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂θ (cid:12)(cid:12)(cid:12)(cid:12) d x = Z B −h ψ, γ ( ∂ r ) ∇ ∂ r ψ i + 16 R( ψ ) − ( F cos 2 θ + F sin 2 θ ) d x = Z B (cid:28) ψ, r γ ( ∂ θ ) ∇ ∂ θ ψ (cid:29) −
16 R( ψ ) − ( F cos 2 θ + F sin 2 θ ) d x. Meanwhile note that in polar coordinate ( r, θ ) , |∇ φ | = (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12)(cid:12)(cid:12) + 1 r (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂θ (cid:12)(cid:12)(cid:12)(cid:12) . This can be combined with Theorem 1.2 to give estimates on each component of the gradientof the map φ ; in particular,(24) Z B r (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂θ (cid:12)(cid:12)(cid:12)(cid:12) d x = 12 Z B |∇ φ | + h ψ, γ ( ∂ r ) ∇ ∂ r ψ i −
16 R( ψ ) + F cos 2 θ + F sin 2 θ d x. Next we consider the isolated singularities of a solution. We show they are removable providedthe gravitino is critical and does not have a singularity there, and the energy of the solution isfinite. Different from Dirac-harmonic maps in [8, Theorem 4.6] and those with curvature termin [24, Theorem 6.1] (ses also [5, Theorem 3.12]), we obtain this result using the regularitytheorems of weak solutions. Thus we have to show first that weak solutions can be extendedover an isolated point in a punctured neighborhood. This is achieved in the Appendix.
Theorem 4.1. (Removable singularity.)
Let ( φ, ψ ) be a smooth solution defined on thepunctured disk B ∗ ≡ B \{ } . If χ is a smooth critical gravitino on B and if ( φ, ψ ) has finiteenergy on B ∗ , then ( φ, ψ ) extends to a smooth solution on B .Proof. From Theorem 6.1 in the Appendix we know that ( φ, ψ ) is also a weak solution on thewhole disk B . By taking a smaller disc centered at the origin and rescaling as above, one mayassume that E ( φ, ψ ; B ) and k χ k W , ( B ) are sufficiently small. From the result in [22] we then see that ( φ, ψ ) is actually smooth in B / (0) . In addition to the assumption, we see that it is asmooth solution on the whole disk. (cid:3) Energy identity
In this section we consider the compactness of the critical points space, i.e. the space ofsolutions of (1). In the end we will prove the main result, the energy identities in Theorem 1.3.As in [35, Lemma 3.2] we establish the following estimate for ψ on annulus domains, which isuseful for the proof of energy identities. Let < r < r < . Lemma 5.1.
Let ψ be a solution of (3) defined on A r ,r ≡ B r \ B r . Then k e ∇ ψ k L ( B r \ B r ) + k ψ k L ( B r \ B r ) ≤ C (cid:16) | A |k∇ φ k L ( A r ,r ) + k Qχ k L ( A r ,r ) + | A | k ψ k L ( A r ,r ) (cid:17) k ψ k L ( A r ,r ) + C k Qχ k L ( A r ,r ) k∇ φ k L ( A r ,r ) + C k ψ k L ( B r \ B r ) + Cr k e ∇ ψ k L ( ∂B r ) + Cr k ψ k L ( ∂B r ) , where C ≥ is a universal constant which doesn’t depend on r and r .Proof. Under a rescaling by /r , the domain A r ,r changes to B \ B r where r = r /r . Byrescaling invariance it suffices to prove it on B \ B r . Choose a cutoff function η r such that η r = 1 in B \ B r , η r = 0 in B r , and that |∇ η r | ≤ C/r . Similarly as in the previous sections,the equations for η r ψ read /∂ (cid:0) η r ψ i (cid:1) = η r (cid:18) − A ijk ∇ φ j · ψ k + | Qχ | ψ i + 13 A ijm A mkl (cid:0) h ψ k , ψ l i ψ j − h ψ j , ψ k i ψ l (cid:1)(cid:19) − η r e α · ∇ φ i · χ α + ∇ η r · ψ i . Using [8, Lemma 4.7], we can estimate k η r ψ k W , ( B ) ≤ C ′ | A | (cid:13)(cid:13) η r |∇ φ || ψ | (cid:13)(cid:13) L ( B ) + C ′ (cid:13)(cid:13) η r | Qχ | | ψ | (cid:13)(cid:13) L ( B ) + C ′ | A | (cid:13)(cid:13) η r | ψ | (cid:13)(cid:13) L ( B ) + C ′ (cid:13)(cid:13) η r |∇ φ || Qχ | (cid:13)(cid:13) L ( B ) + C ′ (cid:13)(cid:13) |∇ η r || ψ | (cid:13)(cid:13) L ( B ) + C ′ k η r ψ k W , ( ∂B ) , where the constant C ′ is also from [8, Lemma 4.7]. This implies that k ψ k W , ( B \ B r ) ≤ C ′ | A |k∇ φ k L ( B \ B r ) k ψ k L ( B \ B r ) + C ′ k Qχ k L ( B \ B r ) k ψ k L ( B \ B r ) + C ′ | A | k ψ k L ( B \ B r ) + C ′ k Qχ k L ( B \ B r ) k∇ φ k L ( B \ B r ) + C ′ k∇ η r k L ( B r \ B r ) k ψ k L ( B r \ B r ) + C ′ k η r ψ k W , ( ∂B ) ≤ C ′ (cid:16) | A |k∇ φ k L ( B \ B r ) + k Qχ k L ( B \ B r ) + | A | k ψ k L ( B \ B r ) (cid:17) k ψ k L ( B \ B r ) + C ′ k Qχ k L ( B \ B r ) k∇ φ k L ( B \ B r ) + C ′ k ψ k L ( B r \ B r ) + C ′ k η r ψ k W , ( ∂B ) . Using the Sobolev embedding theorem, we obtain the estimate on B \ B r , and scaling back,we get the desired result with C = 2 C ′ . (cid:3) NERGY QUANTIZATION 19
Thanks to the invariance under rescaled conformal transformations, the estimate in Lemma5.1 can be applied to any conformally equivalent domain, in particular we will apply it oncylinders later.Similarly we can estimate the energies of the map φ satisfying (1) on the annulus domains,in the same flavor as for Dirac-harmonic maps, see e.g. [35, Lemma 3.3]. Lemma 5.2.
Let ( φ, ψ ) be a solution of (1) defined on A r ,r with critical gravitino. Then Z B r \ B r |∇ φ | d x ≤ C Z B r \ B r | A | | ψ | + | e ∇ ψ | + | Qχ | | ψ | d x + C Z ∂ ( B r \ B r ) ( q − φ ) (cid:18) h V, ∂∂r i − ∂φ∂r (cid:19) d s + C sup B r \ B r | q − φ | Z B r \ B r | A | |∇ φ | + | A | ( | A | + |∇ A | ) | ψ | + | ψ | | Qχ | d x. Here C ≥ is some universal constant.Proof. Make a rescaling as in Lemma 5.1. Choose a function q ( r ) on B which is piecewiselinear in log r with q ( 12 m ) = 12 π Z π φ ( 12 m , θ ) d θ, for r ≤ − m ≤ , and q ( r ) is defined to be the average of φ on the circle of radius r . Then q is harmonic in A m := { m < r < m − } ⊂ B \ B r and in the annulus near the boundary { x ∈ R (cid:12)(cid:12) | x | = r } . Note that ∆( q − φ ) = − ∆ φ = − A ( φ )( ∇ φ, ∇ φ ) + div V − f, where V is given by (4) and f is an abbreviation for f i ≡ A ijm A mkl h ψ j , ∇ φ k · ψ l i + Z i ( A, ∇ A ) jklm h ψ j , ψ l ih ψ k , ψ m i − A ijk h V j , ∇ φ k i . Using Green’s formula we get Z B \ B r | d q − d φ | d x = − Z B \ B r ( q − φ )∆( q − φ ) d x + Z ∂ ( B \ B r ) ( q − φ ) ∂∂r ( q − φ ) d s. Since q ( r ) is the average of φ over ∂B r we see that Z ∂ ( B \ B r ) ( q − φ ) ∂∂r ( q − φ ) d s = − Z ∂ ( B \ B r ) ( q − φ ) ∂φ∂r d s. By the equation of ( q − φ ) , − Z B \ B r ( q − φ )∆( q − φ ) d x = Z B \ B r ( q − φ ) ( A ( φ )( ∇ φ, ∇ φ ) + f ) − ( q − φ ) div V d x = Z B \ B r ( q − φ ) ( A ( φ )( ∇ φ, ∇ φ ) + f ) + h∇ ( q − φ ) , V i d x + Z ∂ ( B \ B r ) ( q − φ ) h V, ∂∂r i d s. These together imply that Z B \ B r | d q − d φ | d x ≤ Z B \ B r q − φ ) ( A ( φ )( ∇ φ, ∇ φ ) + f ) + | V | d x + Z ∂ ( B \ B r ) q − φ ) (cid:18) h V, ∂∂r i − ∂φ∂r (cid:19) d s. Recall the Pohozaev formulae (5) or its consequence (24), and note that they hold also on theannulus domains. Note also that Z B \ B r | d q − d φ | d x ≥ Z B \ B r r (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂θ (cid:12)(cid:12)(cid:12)(cid:12) d x. Therefore we get Z B \ B r |∇ φ | + h ψ, γ ( ∂ r ) e ∇ ∂ r ψ i −
16 R( ψ ) + F cos 2 θ + F sin 2 θ d x ≤ Z B \ B r q − φ ) ( A ( φ )( ∇ φ, ∇ φ ) + f ) + | V | d x + Z ∂ ( B \ B r ) q − φ ) (cid:18) h V, ∂∂r i − ∂φ∂r (cid:19) d s. From this it follows that Z B \ B r |∇ φ | d x ≤ Z B \ B r | A | | ψ | + | e ∇ ψ | + 32 | Qχ | | ψ | d x + Z ∂ ( B \ B r ) q − φ ) (cid:18) h V, ∂∂r i − ∂φ∂r (cid:19) d s +16 sup B \ B r | q − φ | Z B \ B r | A | |∇ φ | + | A | ( | A | + |∇ A | ) | ψ | + | ψ | | Qχ | d x. Then we rescale back to A r ,r . The universal constant C can be taken to be , for instance. (cid:3) Finally we can show the energy identities, Theorem 1.3. The corresponding ones for Dirac-harmonic maps with curvature term were obtained in [24], following the scheme of [15, 7] andusing a method which is based on a type of three circle lemma. Here we apply a method inthe same spirit as those in [34, 35]. Since we have no control of higher derivatives of gravitinos,the strong convergence assumption on gravitinos is needed here. We remark that the Pohozaevtype identity established in Theorem 1.2 is crucial in the proof of this theorem.
Proof of Theorem 1.3.
The uniform boundedness of energies implies that there is a subsequenceconverging weakly in W , × L to a limit ( φ, ψ ) which is a weak solution with respect to χ . Alsothe boundedness of energies implies that the blow-up set S consists of only at most finitely manypoints (possibly empty). If S = ∅ , then the sequence converges strongly and the conclusionfollows directly. Now we assume it is not empty, say S = { p , . . . , p I } . Moreover, using thesmall energy regularities and compact Sobolev embeddings, by a covering argument similar tothat in [30] we see that there is a subsequence converging strongly in the W , × L -topologyon the subset ( M \ ∪ Ii =1 B δ ( p i )) for any δ > .When the limit gravitino χ is smooth, by the regularity theorems in [22] together with theremovable singularity theorem 4.1 we see that ( φ, ψ ) is indeed a smooth solution with respectto χ . NERGY QUANTIZATION 21
Since M is compact and blow-up points are only finitely many, we can find small disks B δ i being small neighborhood of each blow-up point p i such that B δ i ∩ B δ j = ∅ whenever i = j andon M \ S Ii =1 B δ i , the sequence ( φ k , ψ k ) converges strongly to ( φ, ψ ) in W , × L .Thus, to show the energy identities, it suffices to prove that there exist solutions ( σ li , ξ li ) of(1) with vanishing gravitinos (i.e. Dirac-harmonic maps with curvature term) defined on thestandard 2-sphere S , ≤ l ≤ L i , such that I X i =1 lim δ i → lim k →∞ E ( φ k ; B δ i ) = I X i =1 L i X l =1 E ( σ li ) , I X i =1 lim δ i → lim k →∞ E ( ψ k ; B δ i ) = I X i =1 L i X l =1 E ( ξ li ) . This will hold if we prove for each i = 1 , · · · , I , lim δ i → lim k →∞ E ( φ k ; B δ i ) = L i X l =1 E ( σ li ) , lim δ i → lim k →∞ E ( ψ k ; B δ i ) = L i X l =1 E ( ξ li ) . First we consider the case that there is only one bubble at the blow-up point p = p . Thenwhat we need to prove is that there exists a solution ( σ , ξ ) with vanishing gravitino such that lim δ → lim k →∞ E ( φ k ; B δ ) = E ( σ ) , lim δ → lim k →∞ E ( ψ k ; B δ ) = E ( ξ ) . For each ( φ k , ψ k ) , we choose λ k such that max x ∈ D δ ( p ) E ( φ k , ψ k ; B λ k ( x )) = ε , and then choose x k ∈ B δ ( p ) such that E ( φ k , ψ k ; B λ k ( x k )) = ε . Passing to a subsequence if necessary, we may assume that λ k → and x k → p as k → ∞ .Denote ˜ φ k ( x ) = φ k ( x k + λ k x ) , ˜ ψ k ( x ) = λ k ψ k ( x k + λ k x ) , ˜ χ k = λ k χ k ( x k + λ k x ) . Then ( ˜ φ k , ˜ ψ k ) is a solution with respect to ˜ χ k on the unit disk B (0) , and by the rescaledconformal invariance of the energies, E ( ˜ φ k , ˜ ψ k ; B (0)) = E ( φ k , ψ k ; B λ k ( x k )) = ε < ε ,E ( ˜ φ k , ˜ ψ k ; B R (0)) = E ( φ k , ψ k ; B λ k R ( x k )) ≤ Λ . Recall that the χ k ’s are assumed to converge in W , / norm. Due to the rescaled conformalinvariance in Lemma 1.1, we have, for any fixed R > , Z B R (0) | ˜ χ k | + | b ∇ ˜ χ k | d x = Z B λkR ( x k ) | χ k | + | b ∇ χ k | d vol g → as k → ∞ . It follows that ˜ χ k converges to . Since we assumed that there is only one bubble, the sequence ( ˜ φ k , ˜ ψ k ) strongly converge tosome ( ˜ φ, ˜ ψ ) in W , ( B R , N ) × L ( B R , S × R K ) for any R ≥ . Indeed, this is clearly true for R ≤ because of the small energy regularities, and if for some R ≥ , the convergence on B R is not strong, then the energies would concentrate at some point outside the unit disk, andby rescaling a second nontrivial bubble would be obtained, contradicting the assumption thatthere is only one bubble. Thus, since R can be arbitrarily large, we get a nonconstant (becauseenergy ≥ ε ) solution on R . By stereographic projection we obtain a nonconstant solution on S \{ N } with energy bounded by Λ and with zero gravitino. Thanks to the removable singularitytheorem for Dirac-harmonic maps with curvature term (apply Theorem 4.1 with χ ≡ or see[24, Theorem 6.1]), we actually have a nontrivial solution on S . This is the first bubble at theblow-up point p .Now consider the neck domain A ( δ, R ; k ) := { x ∈ R | λ k R ≤ | x − x k | ≤ δ } . It suffices to show that lim R →∞ lim δ → lim k →∞ E ( φ k , ψ k ; A ( δ, R ; k )) = 0 . (25)Note that the strong convergence assumption on χ k ’s implies that(26) lim δ → lim k →∞ Z A ( δ,R ; k ) | χ k | + | b ∇ χ k | d x ≤ lim δ → Z B δ ( p ) | χ | + | b ∇ χ | d x = 0 , by, say, Lebesgue’s dominated convergence theorem.To show (25), it may be more intuitive to transform them to a cylinder. Let ( r k , θ k ) be thepolar coordinate around x k . Consider the maps f k : ( R × S , ( t, θ ) , g = d t + d θ ) → ( R , ( r k , θ k ) , d s = d r k + r k d θ k ) given by f k ( t, θ ) = ( e − t , θ ) . Then f − k ( A ( δ, R ; k )) = ( − log δ, − log λ k R ) × S ≡ P k ( δ, R ) ≡ P k .After a translation in the R direction, the domains P k converge to the cylinder R × S . It isknown that f k is conformal f ∗ k (d r k + r k d θ k ) = e − t (d t + d θ ) . Thus a solution defined in a neighborhood of x k is transformed to a solution defined on part ofthe cylinder via Φ k ( x ) := φ k ◦ f k ( x ) , Ψ k ( x ) := e − t Bψ k ◦ f k ( x ) , X k ( x ) := e − t Bχ k ◦ f k ( x ) , where B is the isomorphism given in Lemma 1.1. Note that E (Φ k , Ψ k ; P k ) = E ( φ k , ψ k ; A ( δ, R ; k )) ≤ Λ , and that by the remark after Lemma 1.1, for any R ∈ (0 , ∞ ) ,(27) lim δ → lim k →∞ Z P k ( δ,R ) | X k | + | b ∇ X k | d x = lim δ → Z A ( δ,R ; k ) | χ k | + | b ∇ χ k | d x = 0 , which follows from (26).For any fixed T > , observe that ( φ k , ψ k , χ k ) converges strongly to ( φ, ψ, χ ) on the annulusdomain B δ ( p ) \ B δe − T ( p ) , which implies that (Φ k , Ψ k , X k ) converges strongly to (Φ , Ψ , X ) on P T ≡ [ T , T + T ] × S , where T = − log δ and Φ( x ) := φ ◦ f ( x ) , Ψ( x ) := e − t Bψ ◦ f ( x ) , X ( x ) := e − t Bχ ◦ f ( x ) , where f ( t, θ ) = ( e − t , θ ) . NERGY QUANTIZATION 23
Let < ε < ε be given. Because of E ( φ, ψ ) ≤ Λ and (27), there exists a δ > small suchthat E ( φ, ψ ; B δ ( p )) < ε and such that(28) Z B δ ( x k ) | χ k | + | b ∇ χ k | d x < ε for large k . Thus for the T given above, there is a k ( T ) > such that for k > k ( T ) ,(29) E (Φ k , Ψ k ; P T ) < ε. In a similar way, we denote T k ≡ | log λ k R | and Q T,k ≡ [ T k − T, T k ] × S . Then for k largeenough,(30) E (Φ k , Ψ k ; Q T,k ) < ε. For the part in between [ T + T, T k − T ] , we claim that there is a k ( T ) such that for k ≥ k ( T ) ,(31) Z [ t,t +1] × S |∇ Φ k | + | Ψ k | d x < ε, ∀ t ∈ [ T , T k − . To prove this claim we will follow the arguments as in the case of harmonic maps in [15] andDirac-harmonic maps in [8]. Suppose this is false, then there exists a sequence { t k } such that t k → ∞ as k → ∞ and Z [ t k ,t k +1] × S |∇ Φ k | + | Ψ k | d x ≥ ε. Because of the energies near the ends are small by (29) and (30), we know that t k − T , T k − t k →∞ . Thus by a translation from t to t − t k , we get solutions ( ˜Φ k , ˜Ψ k ; ˜ X k ) , and for all k it holdsthat Z [0 , × S |∇ ˜Φ k | + | ˜Ψ k | d x ≥ ε. From (27) we see that ˜ X k go to in W , loc . Due to the bounded energy assumption we mayassume that ( ˜Φ k , ˜Ψ k ) converges weakly to some ( ˜Φ ∞ , ˜Ψ ∞ ) in W , loc × L loc ( R × S ) , passing toa subsequence if necessary. Moreover, by a similar argument as before, the convergence isstrong except near at most finitely many points. If this convergence is strong on R × S , weobtain a nonconstant solution with respect to zero gravitino on the whole of R × S , hence,by a conformal transformation, a Dirac-harmonic map with curvature term on S \{ N, S } withfinite energy. The removable singularity theorem then ensures a nontrivial solution on S ,contradicting the assumption that L = 1 . On the other hand if the sequence ( ˜Φ k , ˜Ψ k ; ˜ X k ) does not converge strongly to ( ˜Φ ∞ , ˜Ψ ∞ ; 0) , then we may find some point ( t , θ ) at which thesequence blows up, giving rise to another nontrivial solution with zero gravitino on S , againcontradicting L = 1 . Therefore (31) has to hold.Applying a finite decomposition argument similar to [34, 35], we can divide P k into finitelymany parts P k = N [ n =1 P nk , P nk := [ T n − k , T nk ] × S , T k = T , T Nk = T k , where N is a uniform integer, and on each part the energy of (Φ k , Ψ k ) is bounded by δ =( C C C ( A ) ) where we put C ( A ) := | A | ( | A | + |∇ A | ) . Actually, since E (Φ k , Ψ k ; P k ) ≤ Λ , weknow that it can be always divided into at most N = [Λ /δ ] + 1 parts such that on each partthe energy is not more than δ . We will use the notation P nk = [ T n − k , T nk ] × S , ¯ P nk = [ T n − k − , T nk ] × S , and ∆ P nk = ¯ P nk − P nk . With Lemma 5.1 on the annuli, we get k Ψ k k L ( P nk ) + k e ∇ Ψ k k L ( P nk ) ≤ C (cid:16) | A |k∇ Φ k k L ( ¯ P nk ) + k QX k k L ( ¯ P nk ) + | A | k Ψ k k L ( ¯ P nk ) (cid:17) k Ψ k k L ( ¯ P nk ) + C k QX k k L ( ¯ P nk ) k∇ Φ k k L ( ¯ P nk ) + C k Ψ k k L (∆ P nk ) + C k e ∇ Ψ k k L ( T nk × S ) + C k Ψ k k L ( T nk × S ) ≤ k Ψ k k L ( P nk ) + 14 k Ψ k k L (∆ P nk ) + C k QX k k L ( ¯ P nk ) k∇ Φ k k L ( P nk ) + C k QX k k L ( ¯ P nk ) k∇ Φ k k L (∆ P nk ) + C k Ψ k k L (∆ P nk ) + C k e ∇ Ψ k k L ( T nk × S ) + C k Ψ k k L ( T nk × S ) , where we have used the fact that k QX k k L ( P k ) can be very small when we take k large and δ small, because of (27). Note that on ∆ P nk the energies of (Φ k , Ψ k ) are bounded by ε . Moreover,since on [ T nk − / , T nk + 1 / × S the small energy assumption holds, thus the boundary termsabove are also controlled by Cε due to the small regularity theorems. Therefore, combiningwith (28), we get(32) k Ψ k k L ( P nk ) + k e ∇ Ψ k k L ( P nk ) ≤ C (Λ) ε . It remains to control the energy of Φ k on P nk . We divide P nk into smaller parts such that oneach of them the energy of Φ k is smaller than ε . Then the small regularity theorems implythat | φ k − q k | ≤ C ∗ √ ε (which may be assumed to be less than 1), see (13). Then applyingLemma 5.2 (transformed onto the annuli) on each small part and summing up the inequalities,one sees that Z P nk |∇ Φ k | d x ≤ C C ( A ) C ∗ √ ε Z P nk |∇ Φ k | + | Ψ k | + | QX k | | Ψ k | d x + CC ∗ √ ε Z ∂P nk | QX k || Ψ k | + |∇ Φ k | d s + C Z P nk | Ψ k | + | e ∇ Ψ k | + | QX k | | Ψ k | d x. Using an argument similar to the above one, and combining with (32), we see that Z P nk |∇ Φ k | d x ≤ C (Λ) ε , with C (Λ) being a uniform constant independent of k , n , N and the choice of ε . Therefore, onthe neck domains, Z P k |∇ Φ k | + | Ψ k | d x = N X n =1 Z P nk |∇ Φ k | + | Ψ k | d x ≤ C N ε . As N is uniform (independent of ε and k ) and ε can be arbitrarily small, thus (25) follows, andthis accomplishes the proof for the case where there is only one bubble. NERGY QUANTIZATION 25
When there are more bubbles, we apply an induction argument on the number of bubbles ina standard way, see [15] for the details. The proof is thus finished. (cid:3)
We remark that the conclusion clearly holds when the gravitino χ is fixed. Then as Theorem1.3 shows, a sequence of solutions with bounded energies will contain a weakly convergentsubsequence and at certain points this subsequence blows up to give some bubbles. In thelanguage of Teichmüller theory [32], the solution space can be compactified by adding someboundaries, which consists of the Dirac-harmonic maps with curvature term on two-dimensionalspheres. This is in particular true when the sequence of gravitinos is assumed to be uniformlysmall in the C norm, which is of interest when one wants to consider perturbations of the zerogravitinos. 6. Appendix
In this appendix we show that a weak solution to a system with coupled first and secondorder elliptic equations on the punctured unit disk can be extended as a weak solution on thewhole unit disk, when the system satisfies some natural conditions. This is observed for ellipticsystems of second order in the two-dimensional calculus of variations, see [20, Appendix], andwe generalize it in the following form.As before, we denote the unit disk in R by B and the punctured unit disk by B ∗ = B \{ } .Let S denote the trivial spinor bundle over B . Theorem 6.1.
Suppose that φ ∈ W , ( B ∗ , R K ) , ψ ∈ L ( B ∗ , S ⊗ R K ) , χ ∈ L ( B , S ⊗ R ) , andthey satisfy the system on B ∗ ∆ φ = F ( x, φ, ∇ φ, ψ, χ ) + div x ( V ) ,/∂ψ = G ( x, φ, ∇ φ, ψ, χ ) , (33) in the sense of distributions; i.e. for any u ∈ W , ∩ L ∞ ( B ∗ , R K ) and any v ∈ W , ( B ∗ , S ⊗ R K ) ,it holds that Z B ∗ h∇ φ, ∇ u i d x = − Z B ∗ h F ( x, φ, ∇ φ, ψ, χ ) , u i d x + Z B ∗ h V ( x, φ, ∇ φ, ψ, χ ) , ∇ u i d x, Z B ∗ h ψ, /∂v i d x = Z B ∗ h G ( x, φ, ∇ φ, ψ, χ ) , v i d x. Moreover, assume that the following growth condition is satisfied: | F ( x, t, p, q, s ) | + | V ( x, t, p, q, s ) | + | G ( x, t, p, q, s ) | ≤ C (cid:0) | p | + | q | + | s | (cid:1) . (34) Then for any η ∈ W , ∩ L ∞ ( B , R K ) and any ξ ∈ W , ( B , S ⊗ R K ) , it also holds that Z B h∇ φ, ∇ η i d x = − Z B h F ( x, φ, ∇ φ, ψ, χ ) , η i d x + Z B h V ( x, φ, ∇ φ, ψ, χ ) , ∇ η i d x, Z B h ψ, /∂ξ i d x = Z B h G ( x, φ, ∇ φ, ψ, χ ) , ξ i d x. (35) That is, when the growth condition (34) is satisfied, any weak solution to (33) on the punctureddisk B ∗ is also a weak solution on the whole disk. Proof.
For m ≥ , define ρ m ( r ) = , for r ≤ m , log(1 /mr ) / log m, for (1 /m ) ≤ r ≤ /m, , for r ≥ /m. Then for any η ∈ W , ∩ L ∞ ( B , R K ) and any ξ ∈ W , ( B , S ⊗ R K ) , set u m ( x ) = (1 − ρ m ( | x | )) η ( x ) ∈ W , ∩ L ∞ ( B ∗ , R K ) ,v m ( x ) = (1 − ρ m ( | x | )) ξ ( x ) ∈ W , ( B ∗ , S ⊗ R K ) . In fact, | − ρ m | ≤ and |∇ ρ m ( | x | ) | = 1log m r ; hence Z B |∇ ρ m ( | x | ) | d x = 2 π (log m ) Z m − m − r r d r = 2 π log m which goes to 0 as m → ∞ . It follows that u m ∈ W , . Recalling the Sobolev embedding indimension two, W , ( B ∗ ) ֒ → L ( B ∗ ) , v m lies in W , ( B ∗ ) .By assumption, Z B ∗ h∇ φ, ∇ u m i d x = − Z B ∗ h F ( x, φ, ∇ φ, ψ, χ ) , u m i d x + Z B ∗ h V ( x, φ, ∇ φ, ψ, χ ) , ∇ u m i d x. Note that F ( x, φ, ∇ φ, ψ, χ ) ∈ L ( B ∗ ) by the growth condition (34) and | u m | ≤ | η | ∈ L ∞ . Since u m converges to η pointwisely almost everywhere, thus by Lebesgue’s dominated convergencetheorem lim m →∞ Z B ∗ h F ( x, φ, ∇ φ, ψ, χ ) , u m i d x = Z B h F ( x, φ, ∇ φ, ψ, χ ) , η i d x. For the other two terms, note that ∇ u m = −∇ ρ m ( | x | ) η ( x ) + (1 − ρ m ( | x | )) ∇ η ( x ) . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ∗ h∇ φ, −∇ ρ m ( | x | ) η ( x ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ φ k L ( B ∗ ) k η k L ∞ ( B ) k∇ ρ m k L ( B ∗ ) → , as m → ∞ , while Z B ∗ h∇ φ, (1 − ρ m ( | x | )) ∇ η i d x → Z B h∇ φ, ∇ η i d x again by Lebesgue’s dominated convergence theorem. Thus lim m →∞ Z B ∗ h∇ φ, ∇ u m i d x = Z B h∇ φ, ∇ η i d x. Similarly lim m →∞ Z B ∗ h V ( x, φ, ∇ φ, ψ, χ ) , ∇ u m i d x = Z B h V ( x, φ, ∇ φ, ψ, χ ) , ∇ η i d x. Therefore, the first equation of (35) holds.
NERGY QUANTIZATION 27
Next we show that the second equation of (35) also holds. Indeed, by assumption Z B ∗ h ψ, /∂v m i d x = Z B ∗ h G ( x, φ, ∇ φ, ψ, χ ) , v m i d x. Now by the growth condition (34), G ( x, φ, ∇ φ, ψ, χ ) ∈ L ( B ) , and by Sobolev embedding ξ ∈ L ( B ) , thus Lebesgue’s dominated convergence theorem implies lim m →∞ Z B ∗ h G ( x, φ, ∇ φ, ψ, χ ) , v m i d x = Z B h G ( x, φ, ∇ φ, ψ, χ ) , ξ i d x. On the other hand, /∂v m = − γ ( ∇ ρ m ( | x | ))) ξ + (1 − ρ m ( | x | ) /∂ξ , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ∗ h ψ, − γ ( ∇ ρ m ( | x | ))) ξ i d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ψ k L ( B ) k ξ k L ( B ) k∇ ρ m k L ( B ) → , as m → ∞ , while Lebesgue’s dominated convergence theorem implies Z B ∗ h ψ, (1 − ρ m ) /∂ξ i d x → Z B h ψ, /∂ξ i d x since /∂ξ ∈ L ( B ) and ψ ∈ L ( B ∗ ) . This accomplishes the proof. (cid:3) References [1] Bernd Ammann. A variational problem in conformal spin geometry. Habilitation (Hamburg University),2003.[2] Christian Bär. 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Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig,Germany
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