Energy quantization for a nonlinear sigma model with critical gravitinos

We study some analytical and geometric properties of a two-dimensional nonlinear sigma model with gravitino which comes from supersymmetric string theory. When the action is 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 quadratic differential. Using it we obtain a Pohozaev type identity and finally we can establish the energy identities along a weakly convergent sequence of fields with uniformly bounded energies.


Introduction
The 2-dimensional nonlinear sigma models constitute important models in quantum field theory. They have not only physical applications, but also geometric implications, and therefore their properties have been the focus of important lines of research. In mathematics, they arise as two-dimensional harmonic maps and pseudo holomorphic curves. In modern physics the basic matter fields are described by vector fields as well as spinor fields, which are coupled by supersymmetries. The base manifolds are two-dimensional, and therefore their conformal and spin structures come into play. From the physics side, in the 1970s a supersymmetric 2dimensional nonlinear sigma model was proposed in [6,14]; the name "supersymmetric" comes from the fact that the action functional is invariant under certain transformations of the matter fields, see for instance [13,18]. From the perspective of geometric analysis, they seem to be natural candidates for a variational approach, and one might expect that the powerful variational methods developed for harmonic maps and pseudo holomorphic curves could be applied 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 naturally involves anti-commuting variables which are not amenable to inequalities, and therefore variational methods cannot be applied, and one rather needs algebraic tools. This would lead to what one may call super harmonic maps. Here, we adopt a different approach. We transform the anti-commuting variables into commuting ones, as in ordinary Riemannian geometry. In particular, the domains of the action functionals are ordinary Riemann surfaces instead of super Riemann surfaces. Then one has more fields to control, not only the maps between Riemannian manifolds and Riemannian metrics, but also their super partners. Such a model was developed and investigated in [22]. Part of the symmetries, including some super symmetries, are inherited, although some essential supersymmetries are hidden or lost. As is known, the symmetries of such functionals are quite important for the analysis, in order to overcome some analytical problems 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 carry out the essential steps of the variational analysis. The analytical key will be a Pohozaev type 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 the Levi-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 1 , see [26,19]. The spinor bundle S is a left module over the Clifford bundle Cl(M, −g) with the Clifford map being denoted by γ : T M → End(S); sometimes it will be simply denoted by a dot. The Clifford relation reads 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 bundle over 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 is surjective, and moreover the following short exact sequence splits: The projection map to the kernel is denoted by Q : S ⊗ T M → S ⊗ T M. More explicitly, in a local oriented orthonormal frame (e α ) of M, a section χ ∈ Γ(S ⊗ T M) can be written as χ α ⊗ e α 2 , and the Q-projection is given by where ω = e 1 · e 2 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 the twisted spinor bundle S ⊗ φ * T N with bundle metric g s ⊗ φ * h and connection ∇ ≡ ∇ S⊗φ * T N , which is also a Dirac bundle, and the Clifford action on this bundle is also denoted by γ or simply a dot. A section of this bundle is called a vector spinor, and it serves as a super partner of 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 ) ∇ eα ψ = e α · ∇ eα ψ. It is elliptic and essentially self-adjoint with respect to the inner product in L 2 (S ⊗ φ * T N). In a local coordinate (y i ) of N, write ψ = ψ i ⊗ φ * ( ∂ ∂y i ), then where / ∂ is the spin Dirac operator on S. For later convention, we set SR(ψ) := ψ l , ψ j gs ψ k ⊗ φ * R N ( and R(ψ) := SR(ψ), ψ gs⊗φ * h . The action functional under consideration is given by From [22] we know that the Euler-Lagrange equations are where S∇R(ψ) = φ * (∇ N R) ijkl ψ i , ψ k gs ψ j , ψ l gs , and One notices that this action functional can actually be defined for (φ, ψ) that possess only little regularity; we only need integrability properties to make the action well defined, that is, φ ∈ W 1,2 (M, N) and ψ ∈ Γ 1,4/3 (S ⊗ φ * T N). The corresponding solutions of (1) in the sense of distributions are called weak solutions. When the Riemannian metric g and the gravitino χ are assumed to be smooth parameters, it is shown in [22] that any weak solution (φ, ψ) is actually smooth. We will show that these solutions have more interesting geometric and analytical properties. Embed (N, h) isometrically into some Euclidean space R K . Then a solution can be represented by a tuple of functions φ = (φ 1 , · · · , φ K ) taking values in R K and a tuple of spinors ψ = (ψ 1 , · · · , ψ K ) where each ψ i is a (pure) spinor and they together satisfy the condition that at each point φ(x) in the image, for any normal vector ν = (ν 1 , · · · , ν K ) Moreover, writing the second fundamental form of the isometric embedding as A = (A i jk ), the Euler-Lagrange equations can be written in the following form (see [22]) Here the V i 's are vector fields on M defined by One should note that there is some ambiguity here, because the second fundamental form maps tangent vectors of the submanifold N to normal vectors, so the lower indices of A i jk should be tangential indices, and the upper ones normal. However, one can extend the second fundamental form to a tubular neighborhood of N in R K such that all the A i jk 's make sense. Alternatively, one can rewrite the extrinsic equations without labeling indices, but we want to derive estimates and see how the second fundamental form A affects the system, hence we adopt this formulation.
This action functional is closely related to Dirac-harmonic maps with curvature term. Actually, if the gravitinos vanish in the model, the action A then reads whose critical points are known as Dirac-harmonic maps with curvature term. These were firstly introduced in [10] and further investigated in [4,5,24]. Furthermore, if the curvature term is also 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 in this 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 the solution spaces, and here especially the rescaled conformal invariance. Lemma 1.1. Let f : ( M ,g) → (M, g) be a conformal diffeomorphism, with f * g = e 2ug , and suppose the spin structure of ( 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 that under the transformation Remark. Furthermore, the following quantities are also invariant under the transformations in the above lemma: where ∇ ≡ ∇ S⊗T M . Also observe that Q is only a linear projection operator, so Qχ enjoys the same analytic properties as χ. In our model, most time it is only the Q-part of χ which is involved, so all the assumptions and conclusions can be made on the Qχ's. The rescaled conformal 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 has to be rescaled in the ordinary way, which gives rise to an additional factor e u , such that the corresponding norms are invariant. For more detailed investigations one can refer to [23] where more symmetry properties of our nonlinear sigma model with gravitinos are analyzed.
Example 1. When the map f is a rescaling by a constant λ on the Euclidean space with the standard Euclidean metric g 0 , then f * g 0 = λ 2 g 0 and (f −1 ) * is a rescaling by λ −1 . In this case the gravitino χ transforms to √ λBχ α ⊗ e α , where e α is a standard basis for (R 2 , g 0 ).
For a given pair (φ, ψ) and a domain U ⊂ M, the energy of this pair (φ, ψ) on U is defined to be and when U is the entire manifold we write E(φ, ψ) omitting U. Similarly, the energy of the map φ resp. the vector spinor ψ on U is defined by From the previous lemma we know that they are rescaling invariant. We will show that whenever the local energy of a solution is small, then some higher derivatives of this solution can be controlled by its energy and some appropriate norm of the gravitino; this is known as the small energy regularity. On the other hand, similar to the theories for harmonic maps and Dirac-harmonic maps, the energy of a solution should not be globally small, that is, when the domain is a closed surface, in particular the standard sphere, because too small energy forces the solution to be trivial. That is, there are energy gaps between the trivial and nontrivial solutions 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 critical with respect to variations. As shown in [23], this is equivalent to the vanishing of the corresponding supercurrent. Then we will see in Section 3 that the energy-momentum tensor, defined using a local orthonormal frame (e α ) by ψ, e α · ∇ e β ψ + e β · ∇ eα ψ − ψ, / D g ψ g αβ + e η · e α · χ η ⊗ φ * e β + e η · e β · χ η ⊗ φ * e α , ψ + 4 (1 ⊗ φ * )Qχ, ψ g αβ + |Qχ| 2 |ψ| 2 g αβ + 1 6 R(ψ)g αβ e α ⊗ e β , is symmetric, traceless and divergence free, see Proposition 3.3. Hence it gives rise to a holomorphic quadratic differential, see Proposition 3.4. In a local conformal coordinate z = x + iy, this differential reads where in a local chart χ = χ x ⊗ ∂ x + χ y ⊗ ∂ y and Consequently we can establish a Pohozaev type identity for our model in Section 4. This will be the key ingredient for the analysis in the sequel.
Theorem 1.2. (Pohozaev identity.) Let (φ, ψ) be a smooth solution of (1) on B * 1 := B 1 \{0} with χ being a critical gravitino which is smooth on B 1 . Assume that (φ, ψ) has finite energy on B 1 . Then for any 0 < r < 1, In Section 4 we also prove that isolated singularities are removable, using a result from the Appendix 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 1, 4 3 to some smooth limit χ, a subsequence can be extracted which converges weakly in W 1,2 × L 4 to a solution defined on (M, g), and by a rescaling argument, known as the blow-up procedure, we can get some solutions with vanishing gravitinos, i.e. Dirac harmonic maps with curvature term, defined on the standard sphere S 2 with target manifold (N, h), known as "bubbles". Moreover, the energies pass 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 respect to smooth critical gravitinos χ k which converge in W 1, 4 3 to a smooth limit χ, and assume their energies are uniformly bounded: Then passing to a subsequence if necessary, the sequence (φ k , ψ k ) converges weakly in the space W 1,2 (M, N) × L 4 (S ⊗ R K ) to a smooth solution (φ, ψ) with respect to χ. Moreover, the blow-up set is a finite (possibly empty) set of points {p 1 , . . . , p I }, and correspondingly a finite set (possibly empty) of Dirac-harmonic maps with curvature term (σ l i , ξ l i ) defined on S 2 with target manifold (N, h), for l = 1, . . . , L i and i = 1, . . . , I, such that the following energy identities hold: The proof will be given in Section 5. Although these conclusions are similar to those for harmonic 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.

Small energy regularity and energy gap property
In this section we consider the behavior of solutions with small energies.
2.1. First we show the small energy regularities. Recall that for harmonic maps and Diracharmonic maps and its variants [30,8,5,24], it suffices to assume that the energy on a local domain is small. However, as we will see soon, here we have to assume that the gravitinos are also small. For the elliptic estimates used here, one can refer to [3,16,11], or more adapted versions in [1].
) Consider the local model defined on the Euclidean unit disk B 1 ⊂ R 2 and the target manifold is a submanifold (N, h) ֒→ R K with second fundamental form A. For any p 1 ∈ (1, 4 3 ) and p 2 ∈ (1, 2) there exists an ε 1 = ε 1 (A, p 1 , p 2 ) ∈ (0, 1) such that if the gravitino χ and a solution (φ, ψ) of (1) satisfy then for any U ⋐ B 1 , the following estimates hold: Remark. Note that if the second fundamental form A vanishes identically, then N is a totally geodesic submanifold of the Euclidean space R K , hence there is no curvatures on N and the model then is easy and not of interest. So we will assume that A = 0, and without loss of generality, we assume |A| ≡ A ≥ 1. For some C(p) depending on the value of p to be chosen later, the small barrier constant ε 1 will be required to satisfy where |∇A| ≡ ∇A . These restrictions will be explained in the proof.
Remark. Note also that since the domain is the Euclidean disk B 1 , the connection ∇ is actually equivalent to ∇ s .
Consider the L p -norm (where p ∈ (1, 2)) of the left hand side: is bounded by some constant C ′ = C ′ (U 1 , p). Since ηψ vanishes on the boundary and / ∂ is an elliptic operator of order one, we have Then from together with the fact provided that (6) is satisfied. Now consider the map φ. The equations for ηφ are Notice Since ηV i ∈ L 4p 4−p (B 1 ), it follows from the L p theory of Laplacian operators that . Then From [22], Z(A, ∇A) ≤ |A||∇A|. Thus the L 4p 4+p norm of ∆v can thus be estimated by As before assume that ∇η Since ηφ = u + v, combining (9) and (10), we get By the small energy assumption, and Sobolev embedding, this implies that The estimates (7) and (11), together with the small energy assumption, imply that for any p ∈ (1, 2), Note that as p ր 2, 4p 4−p ր 4. Thus, ηφ is almost a W 1,4 map and ηψ is almost a W 1,2 vector spinor. Now χ ∈ W 1, 4 3 , thus in the equations for the map φ, the divergence terms can be reconsidered. Take another cutoff function, still denoted by η, such that 0 ≤ η ≤ 1, η| U 2 ≡ 1, and supp η ⊂ U 1 . Then ηφ satisfies equations of the same form as (8), and div( is under control by (12). Recalling (8) we have the estimate By the smallness assumptions and the L p theory for Laplacian operator (here p = 8 7 ) we get .
One can check that similar estimates hold for ηφ W 1,p (B 1 ) for any p ∈ (1, 4 3 ). This accomplishes the proof.
Recall the Sobolev embeddings Thus we see that the map φ is Hölder continuous with . In particular, when the energies of (φ, ψ) and certain norms of the gravitino are small, say smaller than ε (where ε ≤ ε 1 ), the 1 4 -Hölder norm of the map in the interior is also small, with the estimate The existence of harmonic spinors is closely related to the topological and Riemannian structures. Examples of closed surfaces which don't admit harmonic spinors include S 2 with arbitrary Riemannian metric and the torus T 2 with a nontrivial spin structure, and many others. For more 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 1 < p < ∞, there holds See e.g. [11] for a proof 3 . As / ∂ is an elliptic operator of first order, one has ∇ s ψ a 3 There they show a proof for p = 4 3 , but it is easily generalized to a general p ∈ (1, ∞).
Remark. Observe that although the estimates here are similar to those in the proof of small energy regularities, they come from a different point of view. There we have to take cutoff functions to make the boundary terms vanish in order that the local elliptic estimates are applicable 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.

Critical gravitino and energy-momentum tensor
In this section we consider the energy-momentum tensor along a solution to (1). We will see that it gives rise to a holomorphic quadratic differential when the gravitino is critical, which is needed for the later analysis.
From now on we assume that the gravitino χ is also critical for the action functional with respect to variations; that is, for any smooth family (χ t ) t of gravitinos with χ 0 = χ, it holds that d dt t=0 A(φ, ψ; g, χ t ) = 0.
Therefore, we can put the energy-momentum tensor into the following form: This form relates closely to the energy-momentum tensor for Dirac-harmonic maps in [8, Section 3] and that for Dirac-harmonic maps with curvature term in [24,Section 4], which also have the following nice properties. Such computations have been provided in [4, Section 3], but since certain algebraic aspects are different here, we need to spell out the computations in detail.
Proof. It remains to show that T is covariantly conserved. Let x ∈ M and take the normal coordinate at x such that ∇e α (x) = 0. We will show that ∇ eα T αβ (x) = 0. At the point x, making use of the Euler-Lagrange equations, one can calculate as follows.
As in the harmonic map case, such a 2-tensor then corresponds to a holomorphic quadratic differential 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) dz 2 := (T 11 − iT 12 )(dx + i dy) 2 , with T 11 and T 12 now being the coefficients of the energy-momentum tensor T in the local coordinate, that is, Here we have abbreviated the gravitino terms as F αβ 's: where χ = χ x ⊗ ∂ x + χ y ⊗ ∂ y in a local chart.
Proposition 3.4. The quadratic differential T (z) dz 2 is well-defined and holomorphic.
Proof. The well-definedness is straightforward and the holomorphicity follows from Proposition 3.3.

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, we can locate the solution on the punctured Euclidean unit disk B * 1 ≡ B 1 \{0}. Using the quadratic holomorphic differential derived in the previous section, we obtain the Pohozaev type formulae containing gravitino terms in Theorem 1.2. When the gravitino vanishes, they will reduce to the 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 Proof of Theorem 1.2. By definition we have Note that |∇ψ| ≤ C(|∇ s ψ| + |ψ||∇φ|). Apply the Young inequality once again to obtain From the initial assumptions we known that φ ∈ W 1,2 (B * 1 , N), ψ ∈ L 4 (B * 1 ) and χ is smooth in B 1 , thus by Theorem 6.1, (φ, ψ) is actually a weak solution on the whole disk B 1 . Using the ellipticity of the Dirac operator, ψ belongs to W 1, 4 3 loc (B 1 ). Therefore |T (z)| is integrable on the disk B r for any r < 1. Recall from Proposition 3.4 that T (z) is a holomorphic function defined 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 0 < r < 1, it holds that |z|=r zT (z) dz = 0. One can compute that in polar coordinate z = re iθ , The identity ψ, / Dψ = R(ψ)/3 along a critical ψ implies Finally, it suffices to note that Integrating (5) with respect to the radius, we get Meanwhile note that in polar coordinate (r, θ), This can be combined with Theorem 1.2 to give estimates on each component of the gradient of the map φ; in particular, R(ψ) + F 11 cos 2θ + F 12 sin 2θ dx.
Next we consider the isolated singularities of a solution. We show they are removable provided the gravitino is critical and does not have a singularity there, and the energy of the solution is finite. Different from Dirac-harmonic maps in [8,Theorem 4.6] and those with curvature term in [24, Theorem 6.1] (ses also [5,Theorem 3.12]), we obtain this result using the regularity theorems of weak solutions. Thus we have to show first that weak solutions can be extended over an isolated point in a punctured neighborhood. This is achieved in the Appendix.  (φ, ψ) be a smooth solution defined on the punctured disk B * 1 ≡ B 1 \{0}. If χ is a smooth critical gravitino on B 1 and if (φ, ψ) has finite energy on B * 1 , then (φ, ψ) extends to a smooth solution on B 1 . Proof. From Theorem 6.1 in the Appendix we know that (φ, ψ) is also a weak solution on the whole disk B 1 . By taking a smaller disc centered at the origin and rescaling as above, one may assume that E(φ, ψ; B 1 ) and χ W 1, 4 3 (B 1 ) are sufficiently small. From the result in [22] we then see that (φ, ψ) is actually smooth in B 1/2 (0). In addition to the assumption, we see that it is a smooth solution on the whole disk.

Energy identity
In this section we consider the compactness of the critical points space, i.e. the space of solutions 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 is useful for the proof of energy identities. Let 0 < 2r 2 < r 1 < 1.
where C 0 ≥ 1 is a universal constant which doesn't depend on r 1 and r 2 .
Proof. Under a rescaling by 1/r 1 , the domain A r 2 ,r 1 changes to B 1 \B r 0 where r 0 = r 2 /r 1 . By rescaling invariance it suffices to prove it on B 1 \B r 0 . Choose a cutoff function η r 0 such that η r 0 = 1 in B 1 \B 2r 0 , η r 0 = 0 in B r 0 , and that |∇η r 0 | ≤ C/r 0 . Similarly as in the previous sections, the equations for η r 0 ψ read Using [8,Lemma 4.7], we can estimate Lemma 4.7]. This implies that . Using the Sobolev embedding theorem, we obtain the estimate on B 1 \B r 0 , and scaling back, we get the desired result with C 0 = 2C ′ 0 .
Thanks to the invariance under rescaled conformal transformations, the estimate in Lemma 5.1 can be applied to any conformally equivalent domain, in particular we will apply it on cylinders 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 2 ,r 1 with critical gravitino. Then Here C 1 ≥ 1 is some universal constant.
Proof. Make a rescaling as in Lemma 5.1. Choose a function q(r) on B 1 which is piecewise linear in log r with for r 0 ≤ 2 −m ≤ 1, and q(r 0 ) is defined to be the average of φ on the circle of radius r 0 . Then q is harmonic in A m := { 1 2 m < r < 1 2 m−1 } ⊂ B 1 \B r 0 and in the annulus near the boundary {x ∈ R 2 |x| = r 0 }. Note that where V is given by (4) and f is an abbreviation for Since q(r 0 ) is the average of φ over ∂B r 0 we see that By the equation of (q − φ), These together imply that Recall the Pohozaev formulae (5) or its consequence (24), and note that they hold also on the annulus domains. Note also that Therefore we get From this it follows that Then we rescale back to A r 2 ,r 1 . The universal constant C 1 can be taken to be 16, for instance.
Finally we can show the energy identities, Theorem 1.3. The corresponding ones for Diracharmonic maps with curvature term were obtained in [24], following the scheme of [15,7] and using a method which is based on a type of three circle lemma. Here we apply a method in the 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 Pohozaev type 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 subsequence converging weakly in W 1,2 ×L 4 to a limit (φ, ψ) which is a weak solution with respect to χ. Also the boundedness of energies implies that the blow-up set S consists of only at most finitely many points (possibly empty). If S = ∅, then the sequence converges strongly and the conclusion follows directly. Now we assume it is not empty, say S = {p 1 , . . . , p I }. Moreover, using the small energy regularities and compact Sobolev embeddings, by a covering argument similar to that in [30] we see that there is a subsequence converging strongly in the W 1,2 × L 4 -topology on the subset (M\ ∪ I i=1 B δ (p i )) for any δ > 0. When the limit gravitino χ is smooth, by the regularity theorems in [22] together with the removable singularity theorem 4.1 we see that (φ, ψ) is indeed a smooth solution with respect to χ.
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 and on M\ I i=1 B δ i , the sequence (φ k , ψ k ) converges strongly to (φ, ψ) in W 1,2 × L 4 . Thus, to show the energy identities, it suffices to prove that there exist solutions (σ l i , ξ l i ) of (1) with vanishing gravitinos (i.e. Dirac-harmonic maps with curvature term) defined on the standard 2-sphere S 2 , 1 ≤ l ≤ L i , such that This will hold if we prove for each i = 1, · · · , I, First we consider the case that there is only one bubble at the blow-up point p = p 1 . Then what we need to prove is that there exists a solution (σ 1 , ξ 1 ) with vanishing gravitino such that For each (φ k , ψ k ), we choose λ k such that and then choose x k ∈ B δ (p) such that Passing to a subsequence if necessary, we may assume that λ k → 0 and x k → p as k → ∞. Denoteφ . Then (φ k ,ψ k ) is a solution with respect toχ k on the unit disk B 1 (0), and by the rescaled conformal invariance of the energies, Recall that the χ k 's are assumed to converge in W 1,4/3 norm. Due to the rescaled conformal invariance in Lemma 1.1, we have, for any fixed R > 0, as k → ∞. It follows thatχ k converges to 0.
Since we assumed that there is only one bubble, the sequence (φ k ,ψ k ) strongly converge to some (φ,ψ) in W 1,2 (B R , N) × L 4 (B R , S × R K ) for any R ≥ 1. Indeed, this is clearly true for R ≤ 1 because of the small energy regularities, and if for some R 0 ≥ 1, the convergence on B R 0 is not strong, then the energies would concentrate at some point outside the unit disk, and by rescaling a second nontrivial bubble would be obtained, contradicting the assumption that there is only one bubble. Thus, since R can be arbitrarily large, we get a nonconstant (because energy ≥ ε 1 2 ) solution on R 2 . By stereographic projection we obtain a nonconstant solution on S 2 \{N} with energy bounded by Λ and with zero gravitino. Thanks to the removable singularity theorem for Dirac-harmonic maps with curvature term (apply Theorem 4.1 with χ ≡ 0 or see [24,Theorem 6.1]), we actually have a nontrivial solution on S 2 . This is the first bubble at the blow-up point p.
Now consider the neck domain It suffices to show that Note that the strong convergence assumption on χ k 's implies that (26) lim 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 the polar coordinate around x k . Consider the maps f k : (R × S 1 , (t, θ), g = dt 2 + dθ 2 ) → (R 2 , (r k , θ k ), ds 2 = dr 2 k + r 2 k dθ 2 k ) given by f k (t, θ) = (e −t , θ). Then f −1 k (A(δ, R; k)) = (− log δ, − log λ k R) × S 1 ≡ P k (δ, R) ≡ P k . After a translation in the R direction, the domains P k converge to the cylinder R × S 1 . It is known that f k is conformal . Thus a solution defined in a neighborhood of x k is transformed to a solution defined on part of the cylinder via and that by the remark after Lemma 1.1, for any R ∈ (0, ∞), which follows from (26). For any fixed T > 0, observe that (φ k , ψ k , χ k ) converges strongly to (φ, ψ, χ) on the annulus domain B δ (p)\B δe −T (p), which implies that (Φ k , Ψ k , X k ) converges strongly to (Φ, Ψ, X) on Let 0 < ε < ε 1 be given. Because of E(φ, ψ) ≤ Λ and (27), there exists a δ > 0 small such that E(φ, ψ; B δ (p)) < ε 2 and such that for large k. Thus for the T given above, there is a k(T ) > 0 such that for k > k(T ), In a similar way, we denote T k ≡ | log λ k R| and Q T,k ≡ [T k − T, T k ] × S 1 . Then for k large enough, For the part in between [T 0 + T, T k − T ], we claim that there is a k(T ) such that for k ≥ k(T ), To prove this claim we will follow the arguments as in the case of harmonic maps in [15] and Dirac-harmonic maps in [8]. Suppose this is false, then there exists a sequence {t k } such that t k → ∞ as k → ∞ and Because of the energies near the ends are small by (29) and (30), we know that t k −T 0 , 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 holds that From (27) we see thatX k go to 0 in W 1, 4 3 loc . Due to the bounded energy assumption we may assume that (Φ k ,Ψ k ) converges weakly to some (Φ ∞ ,Ψ ∞ ) in W 1,2 loc × L 4 loc (R × S 1 ), passing to a subsequence if necessary. Moreover, by a similar argument as before, the convergence is strong except near at most finitely many points. If this convergence is strong on R × S 1 , we obtain a nonconstant solution with respect to zero gravitino on the whole of R × S 1 , hence, by a conformal transformation, a Dirac-harmonic map with curvature term on S 2 \{N, S} with finite energy. The removable singularity theorem then ensures a nontrivial solution on S 2 , 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 0 , θ 0 ) at which the sequence blows up, giving rise to another nontrivial solution with zero gravitino on S 2 , again contradicting L = 1. Therefore (31) has to hold.
Applying a finite decomposition argument similar to [34,35], we can divide P k into finitely many parts where N is a uniform integer, and on each part the energy of (Φ k , Ψ k ) is bounded by δ = ( 1 8C 0 C 1 C(A) ) 2 where we put C(A) := |A|(|A| + |∇A|). Actually, since E(Φ k , Ψ k ; P k ) ≤ Λ, we know that it can be always divided into at most N = [Λ/δ] + 1 parts such that on each part the energy is not more than δ. We will use the notation P n k = [T n−1 k , T n k ] × S 1 ,P n k = [T n−1 k − 1, T n k ] × S 1 , and ∆P n k =P n k − P n k . With Lemma 5.1 on the annuli, we get , where we have used the fact that QX k L 4 (P k ) can be very small when we take k large and δ small, because of (27). Note that on ∆P n k the energies of (Φ k , Ψ k ) are bounded by ε. Moreover, since on [T n k − 1/2, T n k + 1/2] × S 1 the small energy assumption holds, thus the boundary terms above are also controlled by Cε due to the small regularity theorems. Therefore, combining with (28), we get It remains to control the energy of Φ k on P n k . We divide P n k into smaller parts such that on each of them the energy of Φ k is smaller than ε. Then the small regularity theorems imply that |φ k − q k | ≤ C * √ ε (which may be assumed to be less than 1), see (13). Then applying Lemma 5.2 (transformed onto the annuli) on each small part and summing up the inequalities, one sees that Using an argument similar to the above one, and combining with (32), we see that with C(Λ) being a uniform constant independent of k, n, N and the choice of ε. Therefore, on the neck domains, P k |∇Φ k | 2 + |Ψ k | 4 dx = N n=1 P n k |∇Φ k | 2 + |Ψ k | 4 dx ≤ CN ε 1 3 .
As N is uniform (independent of ε and k) and ε can be arbitrarily small, thus (25) follows, and this accomplishes the proof for the case where there is only one bubble.
When there are more bubbles, we apply an induction argument on the number of bubbles in a standard way, see [15] for the details. The proof is thus finished.
We remark that the conclusion clearly holds when the gravitino χ is fixed. Then as Theorem 1.3 shows, a sequence of solutions with bounded energies will contain a weakly convergent subsequence and at certain points this subsequence blows up to give some bubbles. In the language of Teichmüller theory [32], the solution space can be compactified by adding some boundaries, which consists of the Dirac-harmonic maps with curvature term on two-dimensional spheres. This is in particular true when the sequence of gravitinos is assumed to be uniformly small in the C 1 norm, which is of interest when one wants to consider perturbations of the zero gravitinos.

Appendix
In this appendix we show that a weak solution to a system with coupled first and second order elliptic equations on the punctured unit disk can be extended as a weak solution on the whole unit disk, when the system satisfies some natural conditions. This is observed for elliptic systems of second order in the two-dimensional calculus of variations, see [20,Appendix], and we generalize it in the following form.
As before, we denote the unit disk in R 2 by B 1 and the punctured unit disk by B * 1 = B 1 \{0}. Let S denote the trivial spinor bundle over B 1 .
That is, when the growth condition (34) is satisfied, any weak solution to (33) on the punctured disk B * 1 is also a weak solution on the whole disk.
Therefore, the first equation of (35) holds.
Next we show that the second equation of (35) also holds. Indeed, by assumption