Energy quantization for a nonlinear sigma model with critical gravitinos

By Jürgen Jost, Ruijun Wu, and Miaomiao Zhu

Abstract

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.

1. Introduction

The two-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 pseudoholomorphic 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 two-dimensional nonlinear sigma model was proposed in Reference 6Reference 14; the name “supersymmetric” comes from the fact that the action functional is invariant under certain transformations of the matter fields; see for instance Reference 13Reference 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 pseudoholomorphic 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 Reference 25, but this naturally involves anti-commuting variables which are not amenable to inequalities, and therefore variational methods cannot be applied; rather, one 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 Reference 22. Part of the symmetries, including some supersymmetries, 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 notational conventions of Reference 22, which are briefly recalled in the following. Let  be an oriented closed Riemannian surface with a fixed spin structure, and let  be a spinor bundle, of real rank four, associated to the given spin structure. Note that the Levi-Civita connection  on and the Riemannian metric  induce a spin connection  on  in a canonical way and a spin metric  which is a fiberwise real inner product;⁠Footnote1 see Reference 19Reference 26. The spinor bundle  is a left module over the Clifford bundle  with the Clifford map being denoted by ; sometimes it will be denoted simply by a dot. The Clifford relation reads

1

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 Reference 22.

✖

The Clifford action is compatible with the spinor metric and the spin connection, making into a Dirac bundle in the sense of Reference 26. Therefore, the bundle is also a Dirac bundle over , and a section is taken as a super partner of the Riemannian metric and called a gravitino. The Clifford multiplication gives rise to a map , where for and and extending linearly. This map is surjective, and moreover the following short exact sequence splits:

The projection map to the kernel is denoted by . More explicitly, in a local oriented orthonormal frame of , a section can be written as ,⁠Footnote2 and the -projection is given by

2

Here and in what follows, the summation convention is always used.

✖

where is the real volume element in the Clifford bundle.

Let be a compact Riemannian manifold and let be a map. One can consider the twisted spinor bundle with bundle metric and connection , which is also a Dirac bundle, and the Clifford action on this bundle is also denoted by or simply a dot. A section of this bundle is called a vector spinor, and it serves as a super partner of the map in this model. The twisted spin Dirac operator is defined in the canonical way: let be a local orthonormal frame of . Then for any vector spinor , define

It is elliptic and essentially self-adjoint with respect to the inner product in . In a local coordinate of , write . Then

where is the spin Dirac operator on . For later convention, we set

and .

The action functional under consideration is given by

From Reference 22 we know that the Euler–Lagrange equations are

where and

One notices that this action functional can actually be defined for that possess only little regularity. We only need integrability properties to make the action well defined; that is, and . The corresponding solutions of Equation 1 in the sense of distributions are called weak solutions. When the Riemannian metric  and the gravitino are assumed to be smooth parameters, it is shown in Reference 22 that any weak solution is actually smooth. We will show that these solutions have more interesting geometric and analytical properties. Embed isometrically into some Euclidean space . Then a solution can be represented by a tuple of functions  taking values in and a tuple of spinors where each is a (pure) spinor, and together they satisfy the condition that at each point in the image, for any normal vector ,

Moreover, writing the second fundamental form of the isometric embedding as , the Euler–Lagrange equations can be written in the following form (see Reference 22):

Here the ’s are vector fields on locally given by

One should note that there is some ambiguity here, because the second fundamental form maps tangent vectors of the submanifold to normal vectors, so the lower indices of should be tangential indices, and the upper ones normal. However, one can extend the second fundamental form to a tubular neighborhood of in such that all the ’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 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 then reads

whose critical points are known as Dirac-harmonic maps with curvature term. These were first introduced in Reference 10 and further investigated in Reference 4Reference 5Reference 24. Furthermore, if the curvature term is also omitted, then we get the Dirac-harmonic map functional which was introduced in Reference 7Reference 8 and further explored from the perspective of geometric analysis in e.g. Reference 9Reference 27Reference 31Reference 33Reference 35Reference 36Reference 37. 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 be a conformal diffeomorphism, with , and suppose the spin structure of is isomorphic to the pullback of the given one of . There is an identification which is an isomorphism and fiberwise isometry such that under the transformation

each summand of the action functional stays invariant, and also

Remark.

Furthermore, the following quantities are also invariant under the transformations in the above lemma:

where . Also observe that is only a linear projection operator, so enjoys the same analytic properties as . In our model, most of the time it is only the -part of which is involved, so all the assumptions and conclusions can be made on the ’s. This actually reflects the effects of super Weyl symmetry. The rescaled conformal invariance with respect to was shown in Reference 17; see also Reference 8. As for the gravitino , the spinor part has to be rescaled in the same way as , while the tangent vector part has to be rescaled in the ordinary way, which gives rise to an additional factor such that the corresponding norms are invariant. For more detailed investigations one can refer to Reference 23, where more symmetry properties of our nonlinear sigma model with gravitinos are analyzed.

Example 1.

When the map is a rescaling by a constant on the Euclidean space with the standard Euclidean metric , then and is a rescaling by . In this case the gravitino transforms to , where is a standard basis for .

For a given pair and a domain , the energy of this pair on is suggested in Reference 8 to be

and when is the entire manifold we write , omitting . Similarly, the energies of the map , respectively, the vector spinor , on 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, 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, similarly to the theories for harmonic maps and Dirac-harmonic maps, the energy of a solution on spheres should not be globally small because too small energy forces the solution to be trivial. That is, there are energy gaps between the trivial and nontrivial solutions of Equation 1 on the two-sphere with standard round metric. This is true also for some other surfaces, as shown in Section 2. The round sphere is more important to us since it is the model of bubbles.

To proceed further we restrict to some special gravitinos, i.e., those gravitinos that are critical with respect to variations. As shown in Reference 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 by

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 , this differential reads

with

where in a local chart 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 what follows.

Theorem 1.2 (Pohozaev identity).

Let be a smooth solution of Equation 1 on with being a critical gravitino which is smooth on . Assume that has finite energy on . Then for any ,

In Section 4 we also prove that isolated singularities are removable, using a result from the Appendix and the regularity theorem in Reference 22.

Finally, for a sequence of solutions with uniformly bounded energies defined on with respect to critical gravitinos which converge in to some smooth limit , a subsequence can be extracted which converges weakly in to a solution defined on . 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 with target manifold , known as “bubbles”. Moreover, the energies pass to the limit; i.e., the energy identities hold.

Theorem 1.3 (Energy identities).

Let be a sequence of solutions of Equation 1 with respect to smooth critical gravitinos  which converge in to a smooth limit , and assume their energies are uniformly bounded:

Then passing to a subsequence if necessary, the sequence converges weakly in the space  to a smooth solution  with respect to . Moreover, the blow-up set

is a finite (possibly empty) set of points and correspondingly a finite set (possibly empty) of Dirac-harmonic maps with curvature term defined on with target manifold , for and , 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. Reference 7Reference 20Reference 24Reference 29Reference 35, one has to pay special attention to the critical gravitinos.

2. 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 regularity. Recall that for harmonic maps and Dirac-harmonic maps and its variants Reference 5Reference 8Reference 24Reference 30, 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 Reference 3Reference 11Reference 16 or more adapted versions in Reference 1.

Theorem 2.1 (-regularity theorem).

Consider the local model defined on the Euclidean unit disk , and the target manifold is a submanifold with second fundamental form . For any and there exists an such that if the gravitino and a solution of Equation 1 satisfy

then for any , the following estimates hold:

where .

Remark.

Note that if the second fundamental form vanishes identically, then is a totally geodesic submanifold of the Euclidean space . Hence there are no curvatures on , and the model is then reduced to the scalar case and is not of interest in this article. So we will assume that , and without loss of generality, we assume . For some depending on the value of to be chosen later, the small barrier constant will be required to satisfy

where . These restrictions will be explained in the proof.

Remark.

Note also that since the domain is the Euclidean disk , the connection is actually equivalent to .

Proof of Theorem 2.1.

Since is taken as a compact submanifold of , we may assume that it is contained in a ball of radius in , which implies that . Moreover, as we are dealing with a local solution , we may assume that , so that the Poincaré inequalities hold: for any ,

Let be a sequence of nonempty disks such that

Take a smooth cutoff function such that , , and . Then satisfies

Then one has

Consider the -norm (where ) of the left hand side:

Assume that is bounded by some constant . Since vanishes on the boundary and is an elliptic operator of order one, we have

Then from

together with the fact that

it follows that

provided that Equation 6 is satisfied.

Now consider the map . The equations for are

Using , we can rewrite it as

Notice that . Split it as , where uniquely solves (see e.g. Reference 12, Chap. 8)

Since , it follows from the theory of Laplacian operators that

Then satisfies

From Reference 22, . Thus the norm of can thus be estimated by

As before assume that and are bounded by . Collecting the terms, we get

By Sobolev embedding,

Since , combining Equation 9 and Equation 10, we get

By the small energy assumption and Sobolev embedding, this implies that

The estimates Equation 7 and Equation 11, together with the small energy assumption, imply that for any ,

Note that as , . Thus, is almost a map, and is almost a vector spinor.

Now ; thus in the equations for the map , the divergence terms can be reconsidered. Take another cutoff function, still denoted by , such that , , and . Then satisfies equations of the same form as Equation 8, and for any . For example, we take . Then

and note that is under control by Equation 12. Recalling Equation 8 we have the estimate

As before we assume that and are bounded by . Then

By the smallness assumptions and the theory for Laplacian operator (here ) we get

One can check that similar estimates hold for for any . This accomplishes the proof.

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Recall the Sobolev embeddings

Thus we see that the map is Hölder continuous with

In particular, when the energies of and certain norms of the gravitino are small, say smaller than (where ), the -Hölder norm of the map in the interior is also small, with the estimate

2.2

In this subsection we show the existence of energy gaps. For harmonic maps, this is a well-known property. On certain closed surfaces the energy gaps are known to exist for Dirac-harmonic maps (with or without curvature term), and using a similar method here we get the following version with gravitinos; compare with Reference 7, Theorem 3.1, Reference 11, Lemma 4.1, Reference 4, Lemmas 4.8 and 4.9 and Reference 24, Proposition 5.2.

Proposition 2.2 (Energy gap property).

Suppose that is a solution to Equation 1 defined on an oriented closed surface with target manifold . Suppose that the spinor bundle doesn’t admit any nontrivial harmonic spinors. Then there exists an such that if

then has to be a trivial solution.

The existence of harmonic spinors is related to the topology and Riemannian structures, at least in low dimensions and low genera. Examples of closed surfaces which don’t admit harmonic spinors include with arbitrary Riemannian metric and the torus with a nontrivial spin structure, and many others. For more information on harmonic spinors one can refer to Reference 2Reference 17.

Proof of Proposition 2.2.

When the spinor bundle doesn’t admit nontrivial harmonic spinors, the Dirac operator is “invertible”, in the sense that for any , there holds

See e.g. Reference 11 for a proof.⁠Footnote3 As is an elliptic operator of first order, one has

3

There they show a proof for , but it is easily generalized to a general .

✖

It follows that

From Equation 3 one gets

Since Equation 14 holds, using Equation 15 one obtains

Next we deal with the map . From Equation 2 it follows that

Combining with Equation 14 this gives

Therefore, when is sufficiently small, this implies that ; that is, . Then Equation 16 says that is also trivial.

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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 doesn’t admit nontrivial harmonic spinors to obtain the estimate Equation 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 Equation 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 is also critical for the action functional with respect to variations; that is, for any smooth family of gravitinos with , it holds that

One can conclude from this by direct calculation that the supercurrent vanishes (or see Reference 23), where

Equivalently it can be formulated as

Recall that . Thus

It follows that

Since the Euler–Lagrange equations for are

if is critical, i.e., the above equation Equation 18 holds, then

Therefore the following relation holds:

Lemma 3.1.

For any and and for any ,

Proof.

Since

it suffices to compute . Note that

Therefore, by virtue of Equation 17,

The desired equality follows.

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Lemma 3.2.

For any and ,

where is the volume element.

Proof.

Since

we have

According to the Clifford relation it holds that

It follows that

At any point , if , then Equation 19 holds; and if , then by the calculations above Equation 19 also holds. This finishes the proof.

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Remark.

More explicitly Equation 19 is equivalent to

From Reference 23 we know the energy-momentum tensor is given by where

Suppose that satisfies the Euler–Lagrange equations Equation 1 and that the supercurrent vanishes. Then

Clearly is symmetric and traceless. We will show it is also divergence free. Before this we rewrite it into a suitable form. Multiplying on both sides of equations Equation 18, we get

Note that the right hand side is perpendicular to :

Hence . Consequently,

Moreover, by Equation 20,

Therefore, we can put the energy-momentum tensor into the following form:

This form relates closely to the energy-momentum tensors of Dirac-harmonic maps in Reference 8, Section 3 and of Dirac-harmonic maps with curvature term in Reference 24, Section 4, which also have the following nice properties. Such computations have been provided in Reference 4, Section 3, but since certain algebraic aspects are different here, we need to spell out the computations in detail.

Proposition 3.3.

Let be critical. Then the tensor given by Equation 21 or equivalently Equation 22 is symmetric, traceless, and covariantly conserved.

Proof.

It remains to show that is covariantly conserved. Let and take the normal coordinate at such that . We will show that . At the point , making use of the Euler–Lagrange equations, one can calculate as follows.

•

•

Note that

and that . Hence one has

•

•

Summarize these terms and use the previous lemmata to get

This accomplishes the proof.

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As in the harmonic map case, such a 2-tensor then corresponds to a holomorphic quadratic differential on . For the case of Dirac-harmonic maps (with or without curvature terms), see Reference 8Reference 24 and Reference 4. More precisely, in a local isothermal coordinate , set

with and now being the coefficients of the energy-momentum tensor in the local coordinate, that is,

Here we have abbreviated the gravitino terms as ’s:

where in a local chart.

Proposition 3.4.

The quadratic differential is well defined and holomorphic.

Proof.

The well-definedness is straightforward, and the holomorphicity follows from Proposition 3.3.

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4. Pohozaev identity and removable singularities

In this section we show that a solution of Equation 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 . 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. Reference 24, Lemma 5.3 and also Reference 5, Lemma 3.11, where a somewhat different identity is derived.

Recall that the ’s are given in Equation 23 and they can be controlled via Young inequality by

Proof of Theorem 1.2.

By definition we have

Note that . Apply the Young inequality once again to obtain

From the initial assumptions we know that , , and is smooth in . Thus by Theorem 6.1, is actually a weak solution on the whole disk . Using the ellipticity of the Dirac operator, belongs to . Therefore is integrable on the disk for any . Recall from Proposition 3.4 that is a holomorphic function defined on the punctured disk. Hence, it has a pole at the origin of order at most one. In particular, is holomorphic in the whole disk. Then by the Cauchy theorem, for any , it holds that . One can compute that in polar coordinate ,

The identity along a critical implies that

Finally, it suffices to note that

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Integrating Equation 5 with respect to the radius, we get

Meanwhile note that in polar coordinate ,

This can be combined with Theorem 1.2 to give estimates on each component of the gradient of the map ; in particular,

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. Differently from Dirac-harmonic maps in Reference 8, Theorem 4.6 and those with curvature term in Reference 24, Theorem 6.1 (ses also Reference 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.

Theorem 4.1 (Removable singularity).

Let be a smooth solution defined on the punctured disk . If is a smooth critical gravitino on and if has finite energy on , then extends to a smooth solution on .

Proof.

From Theorem 6.1 in the Appendix we know that is also a weak solution on the whole disk . By taking a smaller disc centered at the origin and rescaling as above, one may assume that  and are sufficiently small. From the result in Reference 22 we then see that is actually smooth in . In addition to the assumption, we see that it is a smooth solution on the whole disk.

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5. Energy identity

In this section we consider the compactness of the critical points space, i.e., the space of solutions of Equation 1. In the end we will prove the main result, the energy identities in Theorem 1.3. As in Reference 35, Lemma 3.2 we establish the following estimate for on annulus domains, which is useful for the proof of energy identities. Let .

Lemma 5.1.

Let be a solution of Equation 3 defined on . Then

where is a universal constant which doesn’t depend on and .

Proof.

Under a rescaling by , the domain changes to where . By rescaling invariance it suffices to prove it on . Choose a cutoff function such that in , in , and . Similarly as in the previous sections, the equations for read

Using Reference 8, Lemma 4.7, we can estimate

where the constant is also from Reference 8, Lemma 4.7. This implies that

Using the Sobolev embedding theorem, we obtain the estimate on , and scaling back, we get the desired result with .

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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 satisfying Equation 1 on the annulus domains, in the same flavor as for Dirac-harmonic maps; see e.g. Reference 35, Lemma 3.3.

Lemma 5.2.

Let be a solution of Equation 1 defined on with critical gravitino. Then

Here is some universal constant.

Proof.

Make a rescaling as in Lemma 5.1. Choose a function on which is piecewise linear in with

for , and is defined to be the average of on the circle of radius . Then is harmonic in and in the annulus near the boundary . Note that

where is given by Equation 4 and is an abbreviation for

Using Green’s formula we get

Since is the average of over we see that

By the equation of ,

These together imply that

Recall the Pohozaev formulae Equation 5 or its consequence Equation 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 . The universal constant can be taken to be , for instance.

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Finally we can show the energy identities, Theorem 1.3. The corresponding ones for Dirac-harmonic maps with curvature term were obtained in Reference 24, following the scheme of Reference 7Reference 15 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 Reference 34Reference 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 to a limit which is a weak solution with respect to . Also the boundedness of energies implies that the blow-up set consists of only at most finitely many points (possibly empty). If , then the sequence converges strongly and the conclusion follows directly. Now we assume it is not empty, say . Moreover, using the small energy regularities and compact Sobolev embeddings, by a covering argument similar to that in Reference 30 we see that there is a subsequence converging strongly in the -topology on the subset for any .

When the limit gravitino is smooth, by the regularity theorems in Reference 22 together with the removable singularity Theorem 4.1 we see that is indeed a smooth solution with respect to .

Since is compact and blow-up points are only finitely many, we can find small disks being small neighborhoods of each blow-up point such that whenever , and on , the sequence  converges strongly to in .

Thus, to show the energy identities, it suffices to prove that there exist solutions of Equation 1 with vanishing gravitinos (i.e., Dirac-harmonic maps with curvature term) defined on the standard 2-sphere , , such that

This will hold if we prove for each ,

First we consider the case that there is only one bubble at the blow-up point . Then what we need to prove is that there exists a solution with vanishing gravitino such that

For each , we choose such that

and then choose such that

Passing to a subsequence if necessary, we may assume that and as . Denote

Then is a solution with respect to on the unit disk , and by the rescaled conformal invariance of the energies,

Recall that the ’s are assumed to converge in norm. Due to the rescaled conformal invariance in Lemma 1.1, we have, for any fixed ,

as . It follows that converges to .

Since we assumed that there is only one bubble, the sequence strongly converges to some in for any . Indeed, this is clearly true for because of the small energy regularities; and if for some the convergence on 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 can be arbitrarily large, we get a nonconstant (because energy ) solution on . By stereographic projection we obtain a nonconstant solution on with energy bounded by and with zero gravitino. Thanks to the removable singularity theorem for Dirac-harmonic maps with curvature term (apply Theorem 4.1 with or see Reference 24, Theorem 6.1), we actually have a nontrivial solution on . This is the first bubble at the blow-up point .

Now consider the neck domain

It suffices to show that

Note that the strong convergence assumption on ’s implies that

by, say, Lebesgue’s dominated convergence theorem.

To show Equation 25, it may be more intuitive to transform them to a cylinder. Let be the polar coordinate around . Consider the maps

given by . Then . After a translation in the direction, the domains converge to the cylinder . It is known that is conformal:

Thus a solution defined in a neighborhood of is transformed to a solution defined on part of the cylinder via

where is the isomorphism given in Lemma 1.1. Note that

and that by the remark after Lemma 1.1, for any ,

which follows from Equation 26.

For any fixed , observe that converges strongly to on the annulus domain , which implies that converges strongly to on , where and

where .

Let be given. Because of and Equation 27, there exists a small enough such that and such that

for large . Thus for the given above, there is a such that for ,

In a similar way, we denote and . Then for large enough,

For the part in between , we claim that there is a such that for ,

To prove this claim we will follow the arguments as in the case of harmonic maps in Reference 15 and Dirac-harmonic maps in Reference 8. Suppose this is false. Then there exists a sequence such that as and

Because the energies near the ends are small by Equation 29 and Equation 30, we know that . Thus by a translation from to , we get solutions , and for all it holds that

From Equation 27 we see that go to in . Due to the bounded energy assumption we may assume that converges weakly to some  in , 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 , we obtain a nonconstant solution with respect to zero gravitino on the whole of , hence, by a conformal transformation, a Dirac-harmonic map with curvature term on with finite energy. The removable singularity theorem then ensures a nontrivial solution on , contradicting the assumption that . On the other hand, if the sequence does not converge strongly to , then we may find some point at which the sequence blows up, giving rise to another nontrivial solution with zero gravitino on , again contradicting . Therefore Equation 31 has to hold.

Applying a finite decomposition argument similar to Reference 34Reference 35, we can divide into finitely many parts:

where is a uniform integer, and on each part the energy of is bounded by , where we put . Actually, since , we know that it can always be divided into at most parts such that on each part the energy is not more than .

We will use the notation

and . With Lemma 5.1 on the annuli, we get

where we have used the fact that can be very small when we take large and small, because of Equation 27. Note that on the energies of are bounded by . Moreover, since on the small energy assumption holds, the boundary terms above are also controlled by due to the small regularity theorems. Therefore, combining with Equation 28, we get

It remains to control the energy of on . We divide into smaller parts such that on each of them the energy of is smaller than . Then the small regularity theorems imply that  (which may be assumed to be less than 1); see Equation 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 one above and combining with Equation 32, we see that

with being a uniform constant independent of , , , and the choice of . Therefore, on the neck domains,

As is uniform (independent of and ) and can be arbitrarily small, Equation 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 Reference 15 for the details. The proof is thus finished.

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We remark that the conclusion clearly holds when the gravitino 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 Reference 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 norm, which is of interest when one wants to consider perturbations of the zero gravitinos.

6. 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 Reference 20, Appendix), and we generalize it in the following form.

As before, we denote the unit disk in by and the punctured unit disk by . Let denote the trivial spinor bundle over .

Theorem 6.1.

Suppose that , , and that they satisfy the system on :

in the sense of distributions; i.e., for any and any , it holds that

Moreover, assume that the following growth condition is satisfied:

Then for any and any , it also holds that

That is, when the growth condition Equation 34 is satisfied, any weak solution to Equation 33 on the punctured disk is also a weak solution on the whole disk.

Proof.

For , define

Then for any and any , set

In fact, and

hence

which goes to 0 as . It follows that . Recalling the Sobolev embedding in dimension two, , lies in .

By assumption,

Note that by the growth condition Equation 34 and . Since converges to pointwisely almost everywhere, by Lebesgue’s dominated convergence theorem

For the other two terms, note that . Then

as , while

again by Lebesgue’s dominated convergence theorem. Thus

Similarly

Therefore, the first equation of Equation 35 holds.

Next we show that the second equation of Equation 35 also holds. Indeed, by assumption

Now by the growth condition Equation 34, , and by Sobolev embedding . Thus Lebesgue’s dominated convergence theorem implies that

On the other hand, and

as , while Lebesgue’s dominated convergence theorem implies that

since and . This accomplishes the proof.

■

Mathematical Fragments

Equation (1)
Equation (2)
Equation (3)
Equation (4)
Lemma 1.1.

Let be a conformal diffeomorphism, with , and suppose the spin structure of is isomorphic to the pullback of the given one of . There is an identification which is an isomorphism and fiberwise isometry such that under the transformation

each summand of the action functional stays invariant, and also

Theorem 1.2 (Pohozaev identity).

Let be a smooth solution of Equation 1 on with being a critical gravitino which is smooth on . Assume that has finite energy on . Then for any ,

Theorem 1.3 (Energy identities).

Let be a sequence of solutions of Equation 1 with respect to smooth critical gravitinos  which converge in to a smooth limit , and assume their energies are uniformly bounded:

Then passing to a subsequence if necessary, the sequence converges weakly in the space  to a smooth solution  with respect to . Moreover, the blow-up set

is a finite (possibly empty) set of points and correspondingly a finite set (possibly empty) of Dirac-harmonic maps with curvature term defined on with target manifold , for and , such that the following energy identities hold:

Theorem 2.1 (-regularity theorem).

Consider the local model defined on the Euclidean unit disk , and the target manifold is a submanifold with second fundamental form . For any and there exists an such that if the gravitino and a solution of Equation 1 satisfy

then for any , the following estimates hold:

where .

Remark.

Note that if the second fundamental form vanishes identically, then is a totally geodesic submanifold of the Euclidean space . Hence there are no curvatures on , and the model is then reduced to the scalar case and is not of interest in this article. So we will assume that , and without loss of generality, we assume . For some depending on the value of to be chosen later, the small barrier constant will be required to satisfy

where . These restrictions will be explained in the proof.

Equation (7)
Equation (8)
Equation (9)
Equation (10)
Equation (11)
Equation (12)
Equation (13)
Proposition 2.2 (Energy gap property).

Suppose that is a solution to Equation 1 defined on an oriented closed surface with target manifold . Suppose that the spinor bundle doesn’t admit any nontrivial harmonic spinors. Then there exists an such that if

then has to be a trivial solution.

Equation (15)
Equation (16)
Equation (17)
Equation (18)
Lemma 3.2.

For any and ,

where is the volume element.

Remark.

More explicitly Equation 19 is equivalent to

Equation (21)
Equation (22)
Proposition 3.3.

Let be critical. Then the tensor given by Equation 21 or equivalently Equation 22 is symmetric, traceless, and covariantly conserved.

Equation (23)
Proposition 3.4.

The quadratic differential is well defined and holomorphic.

Equation (24)
Theorem 4.1 (Removable singularity).

Let be a smooth solution defined on the punctured disk . If is a smooth critical gravitino on and if has finite energy on , then extends to a smooth solution on .

Lemma 5.1.

Let be a solution of Equation 3 defined on . Then

where is a universal constant which doesn’t depend on and .

Lemma 5.2.

Let be a solution of Equation 1 defined on with critical gravitino. Then

Here is some universal constant.

Equation (25)
Equation (26)
Equation (27)
Equation (28)
Equation (29)
Equation (30)
Equation (31)
Equation (32)
Theorem 6.1.

Suppose that , , and that they satisfy the system on :

in the sense of distributions; i.e., for any and any , it holds that

Moreover, assume that the following growth condition is satisfied:

Then for any and any , it also holds that

That is, when the growth condition 34 is satisfied, any weak solution to 33 on the punctured disk is also a weak solution on the whole disk.

References

Reference [1]
Bernd Ammann, A variational problem in conformal spin geometry, Habilitation, Hamburg University, 2003.
Reference [2]
C. Bär, On harmonic spinors, Acta Phys. Polon. B 29 (1998), no. 4, 859–869. Gauge theories of gravitation (Jadwisin, 1997). MR1682311,
Show rawAMSref \bib{bar1998harmonic}{article}{ author={B\"{a}r, C.}, title={On harmonic spinors}, note={Gauge theories of gravitation (Jadwisin, 1997)}, journal={Acta Phys. Polon. B}, volume={29}, date={1998}, number={4}, pages={859--869}, issn={0587-4254}, review={\MR {1682311}}, }
Reference [3]
Heinrich G. W. Begehr, Complex analytic methods for partial differential equations: An introductory text, World Scientific Publishing Co., Inc., River Edge, NJ, 1994, DOI 10.1142/2162. MR1314196,
Show rawAMSref \bib{begehr1994complex}{book}{ author={Begehr, Heinrich G. W.}, title={Complex analytic methods for partial differential equations}, subtitle={An introductory text}, publisher={World Scientific Publishing Co., Inc., River Edge, NJ}, date={1994}, pages={x+273}, isbn={981-02-1550-9}, review={\MR {1314196}}, doi={10.1142/2162}, }
Reference [4]
Volker Branding, Some aspects of Dirac-harmonic maps with curvature term, Differential Geom. Appl. 40 (2015), 1–13, DOI 10.1016/j.difgeo.2015.01.008. MR3333092,
Show rawAMSref \bib{branding2015some}{article}{ author={Branding, Volker}, title={Some aspects of Dirac-harmonic maps with curvature term}, journal={Differential Geom. Appl.}, volume={40}, date={2015}, pages={1--13}, issn={0926-2245}, review={\MR {3333092}}, doi={10.1016/j.difgeo.2015.01.008}, }
Reference [5]
Volker Branding, Energy estimates for the supersymmetric nonlinear sigma model and applications, Potential Anal. 45 (2016), no. 4, 737–754, DOI 10.1007/s11118-016-9564-z. MR3558358,
Show rawAMSref \bib{branding2016energy}{article}{ author={Branding, Volker}, title={Energy estimates for the supersymmetric nonlinear sigma model and applications}, journal={Potential Anal.}, volume={45}, date={2016}, number={4}, pages={737--754}, issn={0926-2601}, review={\MR {3558358}}, doi={10.1007/s11118-016-9564-z}, }
Reference [6]
L. Brink, Paolo Di Vecchia and Paul Howe, A locally supersymmetric and reparametrization invariant action for the spinning string, Phys. Lett. B 65 (1976), no. 5, 471–474.
Reference [7]
Qun Chen, Jürgen Jost, Jiayu Li, and Guofang Wang, Regularity theorems and energy identities for Dirac-harmonic maps, Math. Z. 251 (2005), no. 1, 61–84, DOI 10.1007/s00209-005-0788-7. MR2176464,
Show rawAMSref \bib{chen2005regularity}{article}{ author={Chen, Qun}, author={Jost, J\"{u}rgen}, author={Li, Jiayu}, author={Wang, Guofang}, title={Regularity theorems and energy identities for Dirac-harmonic maps}, journal={Math. Z.}, volume={251}, date={2005}, number={1}, pages={61--84}, issn={0025-5874}, review={\MR {2176464}}, doi={10.1007/s00209-005-0788-7}, }
Reference [8]
Qun Chen, Jürgen Jost, Jiayu Li, and Guofang Wang, Dirac-harmonic maps, Math. Z. 254 (2006), no. 2, 409–432, DOI 10.1007/s00209-006-0961-7. MR2262709,
Show rawAMSref \bib{chen2006dirac}{article}{ author={Chen, Qun}, author={Jost, J\"{u}rgen}, author={Li, Jiayu}, author={Wang, Guofang}, title={Dirac-harmonic maps}, journal={Math. Z.}, volume={254}, date={2006}, number={2}, pages={409--432}, issn={0025-5874}, review={\MR {2262709}}, doi={10.1007/s00209-006-0961-7}, }
Reference [9]
Qun Chen, Jürgen Jost, Guofang Wang, and Miaomiao Zhu, The boundary value problem for Dirac-harmonic maps, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 997–1031, DOI 10.4171/JEMS/384. MR3085099,
Show rawAMSref \bib{CJWZ}{article}{ author={Chen, Qun}, author={Jost, J\"{u}rgen}, author={Wang, Guofang}, author={Zhu, Miaomiao}, title={The boundary value problem for Dirac-harmonic maps}, journal={J. Eur. Math. Soc. (JEMS)}, volume={15}, date={2013}, number={3}, pages={997--1031}, issn={1435-9855}, review={\MR {3085099}}, doi={10.4171/JEMS/384}, }
Reference [10]
Q. Chen, J. Jost, and G. Wang, Liouville theorems for Dirac-harmonic maps, J. Math. Phys. 48 (2007), no. 11, 113517, 13, DOI 10.1063/1.2809266. MR2370260,
Show rawAMSref \bib{chen2008liouville}{article}{ author={Chen, Q.}, author={Jost, J.}, author={Wang, G.}, title={Liouville theorems for Dirac-harmonic maps}, journal={J. Math. Phys.}, volume={48}, date={2007}, number={11}, pages={113517, 13}, issn={0022-2488}, review={\MR {2370260}}, doi={10.1063/1.2809266}, }
Reference [11]
Qun Chen, Jürgen Jost, and Guofang Wang, Nonlinear Dirac equations on Riemann surfaces, Ann. Global Anal. Geom. 33 (2008), no. 3, 253–270, DOI 10.1007/s10455-007-9084-6. MR2390834,
Show rawAMSref \bib{chen2008nonlinear}{article}{ author={Chen, Qun}, author={Jost, J\"{u}rgen}, author={Wang, Guofang}, title={Nonlinear Dirac equations on Riemann surfaces}, journal={Ann. Global Anal. Geom.}, volume={33}, date={2008}, number={3}, pages={253--270}, issn={0232-704X}, review={\MR {2390834}}, doi={10.1007/s10455-007-9084-6}, }
Reference [12]
Ya-Zhe Chen and Lan-Cheng Wu, Second order elliptic equations and elliptic systems, translated from the 1991 Chinese original by Bei Hu, Translations of Mathematical Monographs, vol. 174, American Mathematical Society, Providence, RI, 1998. MR1616087,
Show rawAMSref \bib{chen1998elliptic}{book}{ author={Chen, Ya-Zhe}, author={Wu, Lan-Cheng}, title={Second order elliptic equations and elliptic systems}, series={Translations of Mathematical Monographs}, volume={174}, contribution={translated from the 1991 Chinese original by Bei Hu}, publisher={American Mathematical Society, Providence, RI}, date={1998}, pages={xiv+246}, isbn={0-8218-0970-9}, review={\MR {1616087}}, }
Reference [13]
Pierre Deligne, Super space descriptions of super gravity, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, pp. 1013–1016. MR1701612,
Show rawAMSref \bib{deligne1999quantum}{article}{ author={Deligne, Pierre}, title={Super space descriptions of super gravity}, conference={ title={Quantum fields and strings: a course for mathematicians, Vol. 1, 2}, address={Princeton, NJ}, date={1996/1997}, }, book={ publisher={Amer. Math. Soc., Providence, RI}, }, date={1999}, pages={1013--1016}, review={\MR {1701612}}, }
Reference [14]
S. Deser and B. Zumino, A complete action for the spinning string, Phys. Lett. B 65 (1976), no. 4, 369–373, DOI 10.1016/0370-2693(76)90245-8. MR0452210,
Show rawAMSref \bib{deser1976complete}{article}{ author={Deser, S.}, author={Zumino, B.}, title={A complete action for the spinning string}, journal={Phys. Lett. B}, volume={65}, date={1976}, number={4}, pages={369--373}, issn={0370-2693}, review={\MR {0452210}}, doi={10.1016/0370-2693(76)90245-8}, }
Reference [15]
Weiyue Ding and Gang Tian, Energy identity for a class of approximate harmonic maps from surfaces, Comm. Anal. Geom. 3 (1995), no. 3-4, 543–554, DOI 10.4310/CAG.1995.v3.n4.a1. MR1371209,
Show rawAMSref \bib{ding1995energy}{article}{ author={Ding, Weiyue}, author={Tian, Gang}, title={Energy identity for a class of approximate harmonic maps from surfaces}, journal={Comm. Anal. Geom.}, volume={3}, date={1995}, number={3-4}, pages={543--554}, issn={1019-8385}, review={\MR {1371209}}, doi={10.4310/CAG.1995.v3.n4.a1}, }
Reference [16]
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. MR1814364,
Show rawAMSref \bib{gilbarg2001elliptic}{book}{ author={Gilbarg, David}, author={Trudinger, Neil S.}, title={Elliptic partial differential equations of second order}, series={Classics in Mathematics}, contribution={reprint of the 1998 edition}, publisher={Springer-Verlag, Berlin}, date={2001}, pages={xiv+517}, isbn={3-540-41160-7}, review={\MR {1814364}}, }
Reference [17]
Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55, DOI 10.1016/0001-8708(74)90021-8. MR0358873,
Show rawAMSref \bib{hitchin1974harmonic}{article}{ author={Hitchin, Nigel}, title={Harmonic spinors}, journal={Advances in Math.}, volume={14}, date={1974}, pages={1--55}, issn={0001-8708}, review={\MR {0358873}}, doi={10.1016/0001-8708(74)90021-8}, }
Reference [18]
Jürgen Jost, Geometry and physics, Springer-Verlag, Berlin, 2009, DOI 10.1007/978-3-642-00541-1. MR2546999,
Show rawAMSref \bib{jost2009geometry}{book}{ author={Jost, J\"{u}rgen}, title={Geometry and physics}, publisher={Springer-Verlag, Berlin}, date={2009}, pages={xiv+217}, isbn={978-3-642-00540-4}, review={\MR {2546999}}, doi={10.1007/978-3-642-00541-1}, }
Reference [19]
Jürgen Jost, Riemannian geometry and geometric analysis, 6th ed., Universitext, Springer, Heidelberg, 2011, DOI 10.1007/978-3-642-21298-7. MR2829653,
Show rawAMSref \bib{jost2011riemannian}{book}{ author={Jost, J\"{u}rgen}, title={Riemannian geometry and geometric analysis}, series={Universitext}, edition={6}, publisher={Springer, Heidelberg}, date={2011}, pages={xiv+611}, isbn={978-3-642-21297-0}, review={\MR {2829653}}, doi={10.1007/978-3-642-21298-7}, }
Reference [20]
Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. MR1100926,
Show rawAMSref \bib{jost1991two}{book}{ author={Jost, J\"{u}rgen}, title={Two-dimensional geometric variational problems}, series={Pure and Applied Mathematics (New York)}, publisher={John Wiley \& Sons, Ltd., Chichester}, date={1991}, pages={x+236}, isbn={0-471-92839-9}, review={\MR {1100926}}, }
[21]
Jürgen Jost, Enno Kessler, and Jürgen Tolksdorf, Super Riemann surfaces, metrics and gravitinos, Adv. Theor. Math. Phys. 21 (2017), no. 5, 1161–1187, DOI 10.4310/ATMP.2017.v21.n5.a2. MR3780267,
Show rawAMSref \bib{jost2014super}{article}{ author={Jost, J\"{u}rgen}, author={Kessler, Enno}, author={Tolksdorf, J\"{u}rgen}, title={Super Riemann surfaces, metrics and gravitinos}, journal={Adv. Theor. Math. Phys.}, volume={21}, date={2017}, number={5}, pages={1161--1187}, issn={1095-0761}, review={\MR {3780267}}, doi={10.4310/ATMP.2017.v21.n5.a2}, }
Reference [22]
Jürgen Jost, Enno Keßler, Jürgen Tolksdorf, Ruijun Wu, and Miaomiao Zhu, Regularity of solutions of the nonlinear sigma model with gravitino, Comm. Math. Phys. 358 (2018), no. 1, 171–197, DOI 10.1007/s00220-017-3001-z. MR3772035,
Show rawAMSref \bib{jost2016regularity}{article}{ author={Jost, J\"{u}rgen}, author={Ke\ss ler, Enno}, author={Tolksdorf, J\"{u}rgen}, author={Wu, Ruijun}, author={Zhu, Miaomiao}, title={Regularity of solutions of the nonlinear sigma model with gravitino}, journal={Comm. Math. Phys.}, volume={358}, date={2018}, number={1}, pages={171--197}, issn={0010-3616}, review={\MR {3772035}}, doi={10.1007/s00220-017-3001-z}, }
Reference [23]
Jürgen Jost, Enno Keßler, Jürgen Tolksdorf, Ruijun Wu, and Miaomiao Zhu, Symmetries and conservation laws of a nonlinear sigma model with gravitino, J. Geom. Phys. 128 (2018), 185–198, DOI 10.1016/j.geomphys.2018.01.019. MR3786194,
Show rawAMSref \bib{jost2016note}{article}{ author={Jost, J\"{u}rgen}, author={Ke\ss ler, Enno}, author={Tolksdorf, J\"{u}rgen}, author={Wu, Ruijun}, author={Zhu, Miaomiao}, title={Symmetries and conservation laws of a nonlinear sigma model with gravitino}, journal={J. Geom. Phys.}, volume={128}, date={2018}, pages={185--198}, issn={0393-0440}, review={\MR {3786194}}, doi={10.1016/j.geomphys.2018.01.019}, }
Reference [24]
Jürgen Jost, Lei Liu, Miaomiao Zhu, Geometric analysis of the action functional of the nonlinear supersymmetric sigma model, MPI MIS preprint 77/2015, 2015.
Reference [25]
Enno Keßler and Jürgen Tolksdorf, The functional of super Riemann surfaces—a “semi-classical” survey, Vietnam J. Math. 44 (2016), no. 1, 215–229, DOI 10.1007/s10013-016-0183-1. MR3470759,
Show rawAMSref \bib{kessler2014functional}{article}{ author={Ke\ss ler, Enno}, author={Tolksdorf, J\"{u}rgen}, title={The functional of super Riemann surfaces---a ``semi-classical'' survey}, journal={Vietnam J. Math.}, volume={44}, date={2016}, number={1}, pages={215--229}, issn={2305-221X}, review={\MR {3470759}}, doi={10.1007/s10013-016-0183-1}, }
Reference [26]
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR1031992,
Show rawAMSref \bib{lawson1989spin}{book}{ author={Lawson, H. Blaine, Jr.}, author={Michelsohn, Marie-Louise}, title={Spin geometry}, series={Princeton Mathematical Series}, volume={38}, publisher={Princeton University Press, Princeton, NJ}, date={1989}, pages={xii+427}, isbn={0-691-08542-0}, review={\MR {1031992}}, }
Reference [27]
Lei Liu, No neck for Dirac-harmonic maps, Calc. Var. Partial Differential Equations 52 (2015), no. 1-2, 1–15, DOI 10.1007/s00526-013-0702-9. MR3299172,
Show rawAMSref \bib{liu}{article}{ author={Liu, Lei}, title={No neck for Dirac-harmonic maps}, journal={Calc. Var. Partial Differential Equations}, volume={52}, date={2015}, number={1-2}, pages={1--15}, issn={0944-2669}, review={\MR {3299172}}, doi={10.1007/s00526-013-0702-9}, }
[28]
Dusa McDuff and Dietmar Salamon, -holomorphic curves and quantum cohomology, University Lecture Series, vol. 6, American Mathematical Society, Providence, RI, 1994, DOI 10.1090/ulect/006. MR1286255,
Show rawAMSref \bib{mcduff1994jholomorphic}{book}{ author={McDuff, Dusa}, author={Salamon, Dietmar}, title={$J$-holomorphic curves and quantum cohomology}, series={University Lecture Series}, volume={6}, publisher={American Mathematical Society, Providence, RI}, date={1994}, pages={viii+207}, isbn={0-8218-0332-8}, review={\MR {1286255}}, doi={10.1090/ulect/006}, }
Reference [29]
Thomas H. Parker, Bubble tree convergence for harmonic maps, J. Differential Geom. 44 (1996), no. 3, 595–633. MR1431008,
Show rawAMSref \bib{parker1996bubble}{article}{ author={Parker, Thomas H.}, title={Bubble tree convergence for harmonic maps}, journal={J. Differential Geom.}, volume={44}, date={1996}, number={3}, pages={595--633}, issn={0022-040X}, review={\MR {1431008}}, }
Reference [30]
J. Sacks and K. Uhlenbeck, The existence of minimal immersions of -spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24, DOI 10.2307/1971131. MR604040,
Show rawAMSref \bib{sacks1981existence}{article}{ author={Sacks, J.}, author={Uhlenbeck, K.}, title={The existence of minimal immersions of $2$-spheres}, journal={Ann. of Math. (2)}, volume={113}, date={1981}, number={1}, pages={1--24}, issn={0003-486X}, review={\MR {604040}}, doi={10.2307/1971131}, }
Reference [31]
Ben Sharp and Miaomiao Zhu, Regularity at the free boundary for Dirac-harmonic maps from surfaces, Calc. Var. Partial Differential Equations 55 (2016), no. 2, Art. 27, 30, DOI 10.1007/s00526-016-0960-4. MR3465443,
Show rawAMSref \bib{sharp2016regularity}{article}{ author={Sharp, Ben}, author={Zhu, Miaomiao}, title={Regularity at the free boundary for Dirac-harmonic maps from surfaces}, journal={Calc. Var. Partial Differential Equations}, volume={55}, date={2016}, number={2}, pages={Art. 27, 30}, issn={0944-2669}, review={\MR {3465443}}, doi={10.1007/s00526-016-0960-4}, }
Reference [32]
Anthony J. Tromba, Teichmüller theory in Riemannian geometry, lecture notes prepared by Jochen Denzler, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992, DOI 10.1007/978-3-0348-8613-0. MR1164870,
Show rawAMSref \bib{tromba2012teichmuller}{book}{ author={Tromba, Anthony J.}, title={Teichm\"{u}ller theory in Riemannian geometry}, series={Lectures in Mathematics ETH Z\"{u}rich}, contribution={lecture notes prepared by Jochen Denzler}, publisher={Birkh\"{a}user Verlag, Basel}, date={1992}, pages={220}, isbn={3-7643-2735-9}, review={\MR {1164870}}, doi={10.1007/978-3-0348-8613-0}, }
Reference [33]
Changyou Wang and Deliang Xu, Regularity of Dirac-harmonic maps, Int. Math. Res. Not. IMRN 20 (2009), 3759–3792, DOI 10.1093/imrn/rnp064. MR2544729,
Show rawAMSref \bib{wang2009regularity}{article}{ author={Wang, Changyou}, author={Xu, Deliang}, title={Regularity of Dirac-harmonic maps}, journal={Int. Math. Res. Not. IMRN}, date={2009}, number={20}, pages={3759--3792}, issn={1073-7928}, review={\MR {2544729}}, doi={10.1093/imrn/rnp064}, }
Reference [34]
Rugang Ye, Gromov’s compactness theorem for pseudo holomorphic curves, Trans. Amer. Math. Soc. 342 (1994), no. 2, 671–694, DOI 10.2307/2154647. MR1176088,
Show rawAMSref \bib{ye1994gromov}{article}{ author={Ye, Rugang}, title={Gromov's compactness theorem for pseudo holomorphic curves}, journal={Trans. Amer. Math. Soc.}, volume={342}, date={1994}, number={2}, pages={671--694}, issn={0002-9947}, review={\MR {1176088}}, doi={10.2307/2154647}, }
Reference [35]
Liang Zhao, Energy identities for Dirac-harmonic maps, Calc. Var. Partial Differential Equations 28 (2007), no. 1, 121–138, DOI 10.1007/s00526-006-0035-z. MR2267756,
Show rawAMSref \bib{zhao2006energy}{article}{ author={Zhao, Liang}, title={Energy identities for Dirac-harmonic maps}, journal={Calc. Var. Partial Differential Equations}, volume={28}, date={2007}, number={1}, pages={121--138}, issn={0944-2669}, review={\MR {2267756}}, doi={10.1007/s00526-006-0035-z}, }
Reference [36]
Miaomiao Zhu, Dirac-harmonic maps from degenerating spin surfaces. I. The Neveu-Schwarz case, Calc. Var. Partial Differential Equations 35 (2009), no. 2, 169–189, DOI 10.1007/s00526-008-0201-6. MR2481821,
Show rawAMSref \bib{zhu1}{article}{ author={Zhu, Miaomiao}, title={Dirac-harmonic maps from degenerating spin surfaces. I. The Neveu-Schwarz case}, journal={Calc. Var. Partial Differential Equations}, volume={35}, date={2009}, number={2}, pages={169--189}, issn={0944-2669}, review={\MR {2481821}}, doi={10.1007/s00526-008-0201-6}, }
Reference [37]
Miaomiao Zhu, Regularity for weakly Dirac-harmonic maps to hypersurfaces, Ann. Global Anal. Geom. 35 (2009), no. 4, 405–412, DOI 10.1007/s10455-008-9142-8. MR2506243,
Show rawAMSref \bib{zhu2}{article}{ author={Zhu, Miaomiao}, title={Regularity for weakly Dirac-harmonic maps to hypersurfaces}, journal={Ann. Global Anal. Geom.}, volume={35}, date={2009}, number={4}, pages={405--412}, issn={0232-704X}, review={\MR {2506243}}, doi={10.1007/s10455-008-9142-8}, }

Article Information

MSC 2010
Primary: 53C43 (Differential geometric aspects of harmonic maps), 58E20 (Harmonic maps)
Keywords
  • Nonlinear sigma-model
  • Dirac-harmonic map
  • gravitino
  • supercurrent
  • Pohozaev identity
  • energy identity
Author Information
Jürgen Jost
Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22–26, D-04103 Leipzig, Germany
jjost@mis.mpg.de
Ruijun Wu
Ennio De Giorgi Mathematical Research Center, Collegio Puteano, Scuola Normale Superiore, Piazza dei Cavalieri, 3, I-56100 PISA, SNS, Pisa, Italy
ruijun.wu@sns.it
ORCID
MathSciNet
Miaomiao Zhu
School of Mathematical Sciences, Shanghai Jiao Tong University, Dongchuan Road 800, 200240 Shanghai, People’s Republic of China
mizhu@sjtu.edu.cn
MathSciNet
Additional Notes

The third author was supported in part by National Natural Science Foundation of China (No. 11601325).

The third author is the corresponding author.

Journal Information
Transactions of the American Mathematical Society, Series B, Volume 6, Issue 7, ISSN 2330-0000, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , and published on .
Copyright Information
Copyright 2019 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
Article References
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  • DOI 10.1090/btran/36
  • MathSciNet Review: 3960707
  • Show rawAMSref \bib{3960707}{article}{ author={Jost, J\"urgen}, author={Wu, Ruijun}, author={Zhu, Miaomiao}, title={Energy quantization for a nonlinear sigma model with critical gravitinos}, journal={Trans. Amer. Math. Soc. Ser. B}, volume={6}, number={7}, date={2019}, pages={215-244}, issn={2330-0000}, review={3960707}, doi={10.1090/btran/36}, }

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