# Regularity of Dirac-harmonic maps with \(\lambda \)-curvature term in higher dimensions

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## Abstract

In this paper, we will study the partial regularity for stationary Dirac-harmonic maps with \(\lambda \)-curvature term. For a weakly stationary Dirac-harmonic map with \(\lambda \)-curvature term \((\phi ,\psi )\) from a smooth bounded open domain \(\Omega \subset {\mathbb {R}}^m\) with \(m\ge 2\) to a compact Riemannian manifold *N*, if \(\psi \in W^{1,p}(\Omega )\) for some \(p>\frac{2m}{3}\), we prove that \((\phi , \psi )\) is smooth outside a closed singular set whose \((m-2)\)-dimensional Hausdorff measure is zero. Furthermore, if the target manifold *N* does not admit any harmonic sphere \(S^l\), \(l=2,\ldots , m-1\), then \((\phi ,\psi )\) is smooth.

## Mathematics Subject Classification

53C43 58E20## 1 Introduction

Variational problems from fields of theoretical physics, like quantum field or string theory, usually come in some particular dimension, with some finite dimensional, but non-compact symmetry group. These include harmonic maps coming from the nonlinear sigma model in dimension 2 or Yang–Mills fields in dimension 4. Typically, they then represent borderline cases of the Palais–Smale condition, and therefore, standard PDE methods for proving the regularity of solutions may not apply. In those dimensions, geometric analysis can usually identify a particular blow-up behavior, that is, a special scheme for the emergence and the control of singularities. That is, minimizing sequences can develop singularities, but in the limit, these singularities can be described as regular solutions on some blown-up domain.

The mathematical aspects, however, are also of much interest and subtlety in higher dimensions. In those dimensions, solutions can really become singular. Again, this has been widely explored in geometric analysis. For instance, the equations for minimal submanifolds in Euclidean or Riemannian spaces loose the conformal invariance, and completely new phenomena emerge, in particular around the Bernstein problem, and this has been a key trigger for the development of geometric measure theory. For harmonic mappings, see [16, 17, 19, 37], and for Yang–Mills, Rivière has carried out the systematic investigation in dimensions larger than 4, see [34] and the references therein. In those cases, the best analytical results that can be obtained are usually partial regularity results, that is, one can control the Hausdorff dimension of the singular set and often also the structure of the singularities.

Here, we engage in such an investigation for Dirac-harmonic maps, a variational problem motivated by the supersymmetric non-linear sigma model of quantum field theory. They arise again naturally in dimension 2, where we again find conformal invariance and can perform a—rather subtle—blow-up analysis. Dirac-harmonic maps were first introduced and studied in [8, 9] in dimension 2. In light of the above, it seems worthwhile to also investigate them in higher dimensions, and again, we expect that the analytical behavior will be rather different. Such an analysis has been started by Wang and Xu [42]. In particular, they derived a monotonicity formula and controlled the singular set as for harmonic maps. In fact, since Dirac-harmonic maps generalize harmonic maps in the sense that they couple a harmonic map type field with a nonlinear Dirac field, one should naturally expect that the structure of harmonic map regularity theory can serve as a guideline. Nevertheless, as it turns out already in dimension 2, while the results are indeed roughly similar to those known for harmonic maps, their proofs can become considerably more difficult. This forces the development of new techniques, some of which then in turn also lead to deeper insights for harmonic maps. Here, we take a step further by implementing the important analysis of Lin [27] who could show regularity in the absence of obstructions, represented by harmonic spheres in a certain range of dimensions. Also, we consider a model that is more general than that in [42], but which is important from the original perspective of quantum field theory, that of Dirac-harmonic maps with curvature term. While the curvature term usually only comes with a constant factor in the literature, we find that we can also admit a field-dependent, variable factor, without impeding the analysis.

We now recall the technical details of the models, and then state our main results at the end of this introduction. Let (*M*, *g*) be an *m*-dimensional compact spin Riemannian manifold, \(\Sigma M\) the spinor bundle over *M* and \(\langle \cdot ,\cdot \rangle _{\Sigma M}\) the metric on \(\Sigma M\). Choosing a local orthonormal basis \({e_\alpha ,\alpha =1, \ldots ,m}\) on *M*, the usual Dirac operator is defined as Open image in new window, where \(\nabla \) is the spin connection on \(\Sigma M\) and \(\cdot \) is the Clifford multiplication. For more details on spin geometry and Dirac operators, one can refer to [26].

*M*to another compact Riemannian manifold (

*N*,

*h*) with dimension \(n\ge 2\). If \(\phi ^{*}TN\) is the pull-back bundle of

*TN*by \(\phi \), we get the twisted bundle \(\Sigma M\otimes \phi ^{*}TN\). Naturally, there is a metric \(\langle \cdot ,\cdot \rangle _{\Sigma M\otimes \phi ^{*}TN}\) on \(\Sigma M\otimes \phi ^{*}TN\) which is induced from the metrics on \(\Sigma M\) and \(\phi ^{*}TN\). Also we have a natural connection \(\widetilde{\nabla }\) on \(\Sigma M\otimes \phi ^{*}TN\) which is induced from the connections on \(\Sigma M\) and \(\phi ^{*}TN\). Let \(\psi \) be a section of the bundle \(\Sigma M\otimes \phi ^{*}TN\). In local coordinates \(\{y^i\}\), it can be written as

*N*. Then \(\widetilde{\nabla }\) becomes

*N*. The Dirac operator along the map \(\phi \) is defined byNow, consider the action functional introduced in [8, 9]

*L*are called

*Dirac-harmonic maps*from

*M*to

*N*.

*L*are given as follows

*Re*(

*z*) denotes the real part of \(z\in \mathbb {C}\) and \(R^m_{lij}\) stands for the Riemann curvature tensor of the target manifold (

*N*,

*g*). See [8, 9] for details.

Dirac-harmonic maps are motivated from the supersymmetric nonlinear sigma model from quantum field theory [12, 20]. They have been investigated extensively in recent years. This subject generalizes the theory of harmonic maps and harmonic spinors. The regularity problem for harmonic maps has been extensively studied in the literature, see e.g. [3, 6, 13, 18, 29, 37] for the classical regularity theory of minimizing harmonic maps and stationary harmonic maps. Based on the geometric analysis techniques developed for harmonic maps and more generally critical elliptic systems with an antisymmetric structure [33, 35], regularity issues for Dirac-harmonic maps in dimension two were systematically studied in [8, 11, 39, 42, 45]. In higher dimensions, Wang-Xu [42] introduced the notion of stationary Dirac-harmonic maps and derived a monotonicity formula for stationary Dirac-harmonic maps, based on which some partial regularity results were obtained. They proved the singular set has Hausdorff dimension at most \(m-2\). In this paper, we give conditions on the target manifold under which the dimension can be reduced further. Moreover, we prove these properties hold for a general case, i.e. Dirac-harmonic maps with \(\lambda \)-curvature term. The blow-up analysis for Dirac-harmonic maps has been investigated in [8, 28, 44, 46]. To study the existence problem, a heat flow approach was introduced in [10] and further explored in [22, 23, 24].

*Dirac-harmonic maps with curvature term*from

*M*to

*N*. They were first proposed and studied by Chen–Jost–Wang [7], where a type of Liouville theorem was proved. The regularity for weak solutions in dimension two was considered in [4]. The blow-up theory, including the energy identity and bubble tree convergence, for a sequence of Dirac-harmonic maps with curvature term from a closed Riemann surface with uniformly bounded energy has been systematically investigated in [21]. For the regularity problem of a similar model with a different type of curvature term, i.e., Dirac-harmonic maps with Ricci type spinor potential, we refer to Xu–Chen [43].

*N*. Since (

*N*,

*h*) ia a compact Riemannian manifold, we define a nonnegative constant:

^{1}

*Dirac-harmonic maps with*\(\lambda \)-

*curvature term*from

*M*to

*N*. Thus, \((\phi ,\psi )\) is a Dirac-harmonic map iff \(\lambda \equiv 0\) and it is a Dirac-harmonic map with curvature term iff \(\lambda \equiv 1\).

*N*isometrically into \(\mathbb {R}^K\). Following Wang-Xu’s analysis set up for Dirac-harmonic maps in higher dimensions in [42], we denote

*K*-tuple of spinors \((\psi ^1, \ldots ,\psi ^K)\) satisfying

In the sequel, for simplicity, we shall consider the case that \(M=\Omega \) is a bounded open domain of \({\mathbb {R}}^m\) with smooth boundary and equipped with the Euclidean metric. Then, the spinor bundle \(\Sigma M\) over *M* can be identified with \(\Sigma =\Omega \times \mathbb {C}^L\), \(L=\mathrm {rank}_{\mathbb {C}}\Sigma \). See [26].

### Definition 1.1

We call \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}(\mathbb {C}^L\otimes \phi ^{*}TN)\) a weakly Dirac-harmonic map with \(\lambda \)-curvature term if it is a critical point of \(L_\lambda \) over the Sobolev space \(W^{1,2}(\Omega ,N)\times S^{1,4/3}(\mathbb {C}^L\otimes \phi ^{*}TN)\).

Our first main result is the following small regularity theorem.

### Theorem 1.2

When \(\lambda =0\), the conclusion in the above theorem has been proven in [42]. When \(m=2\) and \(\lambda =1\), one can refer to [4].

Similarly to the classical regularity theory of harmonic maps, in order to study the partial regularity in higher dimensions, we need to introduce the notion of stationary solutions.

### Definition 1.3

*stationary*if it is also a critical point of \(L_{\lambda }\) with respect to the domain variations,

*i*.

*e*. for any \(Y\in C^\infty _0(\Omega ,{\mathbb {R}}^n)\), it holdswhere \(\phi _t(x)=\phi (x+tY(x))\) and \(\psi _t(x)=\psi (x+tY(x))\).

In this paper, we shall impose the same extra condition for the spinor as in Wang-Xu [42] and get the following partial regularity theorem for stationary Dirac-harmonic maps with \(\lambda \)-curvature term. For similar results for stationary harmonic maps and stationary Dirac-harmonic maps, we refer to [3, 13, 42].

### Theorem 1.4

For \(m\ge 2\), let \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}( \mathbb {C}^L\otimes \phi ^{*}TN)\) be a weakly stationary Dirac-harmonic map with \(\lambda \)-curvature term. Suppose \(\Vert \psi \Vert _{ W^{1,p}(\Omega )}<\infty \) for some \(p>\frac{2m}{3}\), then there exists a closed subset \(S(\phi )\subset \Omega \), with \(H^{m-2}(S(\phi ))=0\), such that \((\phi ,\psi )\in C^\infty (\Omega {\setminus } S(\phi ))\).

Furthermore, we have

### Theorem 1.5

Under the same assumption as in the above theorem, if *N* does not admit harmonic spheres \(S^2\), then the Hausdorff dimension of the singular set \(S(\phi )\) is \(d_h\le m-3\). Moreover, if *N* does not admit harmonic spheres \(S^l\), \(l=2, \ldots ,m-1\), then \((\phi ,\psi )\) is smooth.

Similarly to the (approximate) harmonic maps (see [30, 31]), one can also consider detailed properties of the stratification of the singular set. This will be addressed in future works.

To prove Theorem 1.2, we firstly use the idea of Wang in [41] to improve the regularity of the spinor \(\psi \) and then apply regularity results for elliptic system with an antisymmetric structure (see e.g. Theorem 5.2 in Appendix) to handle the map \(\phi \). For Theorem 1.4, since \(\nabla \psi \in L^p\) for some \(p>\frac{2m}{3}\), it follows from using Theorem 1.2, the monotonicity formula and applying similar arguments as in Wang-Xu [42]. As for our last Theorem 1.5, thanks to the observation in Proposition 4.5 for some formulas of the spinors, following Lin’s scheme in [27], we consider the concentration set of a blow-up sequence of Dirac-harmonic maps with \(\lambda \)-curvature term. The proof is based on the analysis of defect measures by geometric measure theory.

The rest of the paper is organized as follows. In Sect. 2, we first derive the Euler–Lagrange equation for stationary Dirac-harmonic maps with \(\lambda \)-curvature term. Secondly, we establish the monotonicity formula crucial to prove Theorems 1.4 and 1.5. In Sect. 3, we prove the small regularity Theorem 1.2 and then Theorem 1.4 follows immediately by applying some monotonicity formula argument. In Sect. 4, we use the blow-up analysis to prove Theorem 1.4. For the reader’s convenience, we will state some well-known regularity results and estimates for some first and second order elliptic systems in Sect. 5.

## 2 Euler–Lagrange equations and monotonicity formula

In this section, we will derive the Euler–Lagrange equation and the monotonicity formula for Dirac-harmonic maps with \(\lambda \)-curvature term.

First, similarly to the cases \(\lambda =0, 1\) considered in [7, 9], respectively, the Euler–Lagrange equations of the functional \(L_\lambda \) can be derived in terms of local coordinates as follows:

### Lemma 2.1

*M*to

*N*. Then, in local coordinates, \((\phi ,\psi )\) satisfies

*N*.

### Proof

*N*isometrically into \(\mathbb {R}^N\), denoted by \(f:N\rightarrow \mathbb {R}^K\). Since \(\lambda \in C^\infty (N)\), there exists an extended function \(\lambda \in C_0^\infty ({\mathbb {R}}^K)\) (for simplicity, we still denote it by \(\lambda \)), such that

### Lemma 2.2

*B*along the map \(\phi \), \(P(\cdot ;\cdot )\) is the shape operator, i.e.

*A*is the second fundamental form of

*N*in \({\mathbb {R}}^K\) and

### Proof

The proof here is almost the same as the computations in the case of \(\lambda =1\) (see Section 3 in [21] where the inner product for the spinors was taken to be Hermitian as in this paper and hence one needs to take the real parts for certain terms. See also Lemma 3.5 in [4]). We omit the details here. \(\square \)

Secondly, we will derive some useful formulae (i.e. Lemma 2.3 and Lemma 2.4) for stationary Dirac-harmonic maps with \(\lambda \)-curvature term which are just Lemma 4.2 and Lemma 4.4 in [42] for \(\lambda =0\) and Proposition 5.3 and Proposition 5.5 in [5] for \(\lambda =1\).

### Lemma 2.3

### Proof

Now, we can derive the monotonicity formula for weakly stationary Dirac-harmonic maps with \(\lambda \)-curvature term (see [5, 42] for the cases of \(\lambda =0, 1\))

### Lemma 2.4

### Proof

*r*from \(r_1\) to \(r_2\). \(\square \)

The following corollary is a small extension of the case of \(\lambda =0\) considered in [42]:

### Corollary 2.5

### Proof

Then the conclusion of the corollary follows immediately from (2.8).

## 3 Proof of Theorem 1.2 and 1.4

In this section, we will prove our main results: Theorem 1.2 and 1.4.

### Lemma 3.1

The idea of proving this lemma is similar to Lemma 2.2 in [41] which has been applied to some other Dirac type equation in dimension 2 in [4, 39]. Recently, [25] (Lemma 6.1) proved a similar lemma for a more general equation in higher dimensions which can be used in our case.

### Proof

Combining Lemma 3.1 with Theorem 5.2, we can now prove Theorem 1.2.

### Proof of Theorem 1.2

*f*satisfying

Now, we prove our main Theorem 1.4.

### Proof of Theorem 1.4

## 4 Proof of Theorem 1.5

In this section, we consider a weakly converging sequence of stationary Dirac-harmonic maps with \(\lambda \)-curvature term.

Without loss of generality, we assume \(B_1(0)\subseteq \Omega \). Then, we have

### Lemma 4.1

*i*.

*e*. for any \(x_0\in \mathrm {sing}(\phi )\), \(\phi \) is not smooth at \(x_0\).

### Proof

*n*, there holds

It is standard to get \(H^{m-2}(\Sigma )\le C(\epsilon _0,m,\Lambda )\) by a covering lemma (cf. [27]).

*n*is sufficient large, we have

### Lemma 4.2

### Proof

By modifying Lin’s method in [27] or applying Preiss’s result [32], we have

### Corollary 4.3

The set of energy concentration points \(\Sigma \) is \((m-2)\)-rectifiable.

*y*. (See [14, 40]).

### Lemma 4.4

Suppose \(H^{m-2}(\Sigma )>0\), then there exists a nonconstant harmonic sphere \(S^2\) into *N*.

Before we prove this lemma, let us state a basic proposition for the Dirac operator.

### Proposition 4.5

*TM*and \(X\in \Gamma (TM)\) a section satisfying

### Proof

The proof is similar to the Proposition 2.2 in [23] (see also the computations of Proposition 3.3 in [9]), where the case of a two dimensional domain was considered. \(\square \)

### Proof of Lemma 4.4

*n*is big enough, we must have

*n*is big enough.

*n*is big enough, we have

*n*is large enough.

*u*,

*v*) in \(C^1_{loc}(B^{m-2}_{3/2}(0)\times {\mathbb {R}}^2)\). Moreover, by (4.18)–(4.20), for any \(R>0\), we have

*u*can be extended to a nonconstant harmonic sphere.

## Footnotes

- 1.
Here, the constant is the \(C^1\)-norm of \(\lambda \) which can be replaced by the \(W^{1,q}\)-norm for some big constant \(q>1\) in the whole paper, we leave it to interested readers.

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society.

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