Regularity of Solutions of the Nonlinear Sigma Model with Gravitino

We propose a geometric setup to study analytic aspects of a variant of the super symmetric two-dimensional nonlinear sigma model. This functional extends the functional of Dirac-harmonic maps by gravitino fields. The system of Euler–Lagrange equations of the two-dimensional nonlinear sigma model with gravitino is calculated explicitly. The gravitino terms pose additional analytic difficulties to show smoothness of its weak solutions which are overcome using Rivière’s regularity theory and Riesz potential theory.


Introduction
The various versions of the two-dimensional sigma models are among the most important and best studied models of quantum field theory. On one hand, such models possess important symmetries, in particular conformal invariance. On the other hand, they can be analyzed in detail with difficult, but currently available mathematical methods. Here, we shall investigate its probably most general and physically and mathematically richest version, the two-dimensional supersymmetric nonlinear sigma-model, introduced in [7,13]. This model possesses a subtle mathematical structure, see [12,19]. The physical and mathematical structure of the model depends on the symmetries it possesses. These include generalized conformal invariance, super Weyl symmetry, and supersymmetry, hence the name of the model.
While supersymmetry requires anti-commuting variables, a version of this model with all fields commuting has been intensively studied by mathematicians in the last decade. The mathematical analysis started with various reduced forms of this model. The simplest instance is harmonic functions, which correspond to the linear sigma model, and they have played an important role in analysis and geometry for a long time. The nonlinear version leads to harmonic maps instead of functions, and these are likewise well studied objects with many applications in geometric analysis. In the super version, the map gets coupled with a super partner, a vector spinor. Chen-Jost-Li-Wang [8,9] initiated the analysis of such coupled fields, which they called Dirac-harmonic maps. The full physical model contains still more additional terms, some of which were considered in [4][5][6]10,21]. Based on those works, we are now in a position to address the full model, including the gravitino terms. The supersymmetric action functional has been mathematically studied from an algebraic and geometric perspective in a systematic way in [20]. Here we shall start to explore the analytic aspects.
Let (M, g) be a closed, oriented surface and (N , h) a closed Riemannian manifold. We will study the super action functional A defined on the space where by Γ 1,4/3 (S ⊗ φ * T N) we mean the space of W 1,4/3 sections of the twisted spinor bundle S ⊗ φ * T N. Furthermore, in this paper the Riemannian metric g and the gravitino χ are considered parameters of the functional. Even though an L 4 -integrability condition suffices for the finiteness of A, we will always assume the gravitino χ is a smooth section of S ⊗ T M. The action functional is Here Q is a projection operator mapping to a subspace of S ⊗ T M, 1 ⊗ φ * : S ⊗ T M → S ⊗ φ * T N and R N (ψ) is a contraction of the pullback of the curvature of N along φ with the field ψ to the fourth order. While the precise geometric setup will be explained in Section 2, we already give local expressions for the third and fifth summand. Let {e α } be a local orthonormal frame of T M and {y i } local coordinates on N . Writing χ = χ α ⊗ e α and ψ = ψ i ⊗ φ * ∂ y i it holds −4 (1 ⊗ φ * )(Qχ), ψ S⊗φ * T N = 2 e α · e β · χ α ⊗ φ * e β , ψ S⊗φ * T N , Since the action functional is somewhat involved and contains many different fields and at the same time possesses rich symmetries, the derivation of the associated Euler-Lagrange equations requires substantial computations. This will be the first achievement of this paper. The result is: Theorem 1. The Euler-Lagrange equations for the super action functional A are given by − ( ∇ S e β (e α · e β · χ α ), ψ S + e α · e β · χ α , ∇ S⊗φ * T N e β ψ S ), These equations already make the growth order transparent with which the various fields enter. S R(ψ) stands for a term involving the curvature of the target N that is cubic in ψ, see (6) and S∇ R(ψ) involves derivatives of that curvature and is quartic in ψ, see (7). We shall then turn to the properties of their solutions. More precisely, we want to show the regularity of weak solutions, that is, those that satisfy the Euler-Lagrange equations in the sense of distributions.
The basic issues in geometric analysis are the existence, uniqueness and smoothness of nontrivial critical points. That is, one wishes to show the existence of weak solutions and then their uniqueness and regularity. In this paper, we settle the smoothness. The Euler-Lagrange equations (2) of this action functional turn out to be critical for the Sobolev framework, in the sense that, with initial data assumed to lie in some Sobolev spaces, the classical bootstrap arguments are not strong enough to improve the regularity. That is, the powerful scheme of elliptic regularity theory does not directly apply, and we need to utilize the structure of the equations, and in particular their symmetries, in a subtler way. Our analytical tools are the Morrey spaces, which can be viewed as finer subspaces of the Lebesgue spaces. With estimates on Riesz potentials, we can then iteratively improve the regularity, and get the system away from the critical case. Related methods have been used in [4,27,29]. Then the Rivière regularity theory (see e.g. [24][25][26]) can be applied to the map component of the critical pairs. Finally, we can show that

Theorem 2. The critical points of the super action functional
are smooth, provided g and χ are smooth.
This result should also help in finding solutions of its associated Euler-Lagrange equations. Moreover, our method is of interest in its own right, as we shall explain later. Further geometric and analytic aspects of this model will be addressed in subsequent work. As in the aforementioned works, we shall work with the version of the model that only has commuting fields. As explained in [11], this depends on an appropriate representation of the Clifford algebra involved. Thus, in contrast to [20], we shall not have to work in the category of supermanifolds, but can confine ourselves to the setting of Riemannian geometry. Yet, in the framework of supermanifolds, the action functional (1) and its symmetries obtain a natural geometric interpretation. In [20] it was shown that the fields g and χ determine a super Riemann surface, a super geometric generalization of a Riemann surface. Recall that Teichmüller theory can be developed with the help of the harmonic action functional. The functional A can be seen as a super analogue of the harmonic action functional. Hence it is expected that an understanding of the solution space of the Euler-Lagrange equations (2) helps to study geometric properties of the moduli space of super Riemann surfaces.
Concerning the organization of this paper, we shall first set up the geometric background for the model and introduce the action functional as well as its basic properties. Then we shall derive its Euler-Lagrange equations. For our regularity scheme, we need to bring the equations into a suitable form. This treatment of the Euler-Lagrange equations, which builds upon [4,11,27,30,31], is crucial for our paper, and we hope that it will also be useful for the further mathematical investigation of the model. We can then finally show the regularity of weak solutions of the Euler-Lagrange equations. The main lemma in improving the regularity appears in the last section in a somewhat more general form than needed for our present purposes.

Preliminaries
In this section we summarize the geometrical background and thereby also fix the notation used in what follows in the subsequent sections. The main purpose of this section is to provide a geometrical setup such that the action functional (1) can be seen as a realvalued action functional with non-vanishing Dirac-action. Those two requirements will be satisfied using a real four-dimensional spinor representation. In contrast, in the description of non-linear sigma models on two-dimensional manifolds, two-dimensional real or complex spinor representations are usually taken into account, see for example [9,20]. For the convenience of the reader we add some comments on how these different geometrical settings are related.

2.1.
Let (M, g) be a closed, oriented, two-dimensional Riemannian spin manifold with fixed spin structure. The corresponding Spin(2) principal bundle is denoted by P Spin (2) . For any bilinear form b on T M we denote by Cl(M, b) the corresponding Clifford algebra bundle, which is isomorphic to the quotient of the tensor algebra by the two-sided ideal generated by where X, Y ∈ Γ (T M). In the following we will only use b = ±g. The typical fiber of Cl (M, g), denoted by Cl 2,0 , is a simple algebra and isomorphic to gl(2, R). We denote this isomorphism by γ + : Cl 2,0 → gl(2, R). Hence, the spinor bundle of Cl (M, g) is given by Σ = P Spin(2) × γ + R 2 where Spin(2) ⊂ gl(2, R) acts by left-multiplication on R 2 . We denote the Clifford multiplication of a tangent vector X with s ∈ Γ (Σ) by γ + (X )s or simply by X · s if no confusion arises. By its construction as an associated bundle to P Spin(2) , the bundle Σ possesses a natural fiber metric g Σ such that the Clifford action by tangent vectors is symmetric. The Levi-Civita connection on T M lifts to the spin connection ∇ Σ on Σ.
The spin Dirac operator is defined with respect to a local g-orthonormal frame e α by / ∂ Σ s = e α · ∇ Σ e α s for a section s of Σ. It is easy to see that / ∂ Σ is antisymmetric and hence for any spinor s the Dirac action vanishes, that is, In order to avoid the vanishing of the Dirac action one may work with anti-commuting spinors, see for example [20] and references therein. Another possibility to obtain a non-vanishing Dirac action is to consider the complexification Σ C = Σ ⊗ C and the resulting Hermitian form h Σ C . Then the operator i / does not vanish identically and is real valued. An equivalent reformulation of this approach was introduced in [9]. Notice, however, that the third summand of (1) involves a scalar product of two different spinors. If this scalar product were to be implemented by h Σ C , the action functional (1) would not be guaranteed to be real-valued. Whence we replace the two-dimensional complex spinor representation of the approach presented in [9] by a four-dimensional real one. This step will be explained next.

2.2.
The typical fiber of the Clifford algebra bundle Cl (M, −g) is the Clifford algebra Cl 0,2 . As a real associative algebra with unit the Clifford algebra Cl 0,2 is isomorphic to the quaternions H. Consequently, the left-regular representation of Cl 0,2 on itself is irreducible. Hence, we may regard the vector bundle S = P Spin(2) × Spin(2) Cl 0,2 as a spinor bundle, where Spin(2) ⊂ Cl 0,2 acts via the left-regular representation of Cl 0,2 . The spinor bundle S is a real rank-four vector bundle. Notice that Cl 0,2 is a Z 2 -graded module over the Z 2 -graded algebra Cl 0,2 . As a consequence also the spinor bundle S = S 0 ⊕ S 1 is a Z 2 -graded module over the Z 2 -graded algebra bundle Cl (M, −g). Here, both the even and the odd part of S are isomorphic to Σ as associated bundles to P Spin (2) . The Clifford action γ (X ) of a tangent vector X on S must be of the form because it is odd with respect to the Z 2 -grading. Recall that γ + (X ) denotes the Clifford multiplication of X on Σ, where X is considered as an element of Cl (M, g).
The induced metric and spin connection on S are denoted, respectively, by In particular, the Dirac action s, / ∂s L 2 (S) is non-trivial, as opposed to its Cl 2,0 counterpart (3). Furthermore, / ∂ is essentially self-adjoint, see [23, Chapter II, Theorem 5.7]. 1

2.3.
We now explain the different complex structures on the spinor bundles Σ and S. This will be needed later on and help to clarify the relation to the geometrical setup introduced in [9]. Recall that the Riemann surface M possesses an integrable almost complex structure J M that is defined by for all tangent vectors X and Y . Consequently, the tangent bundle T M is a holomorphic line bundle. When seen as T M ⊂ Cl (M, g), the almost complex structure J M can be realized as right-multiplication by the volume form ω. With respect to a local oriented g-orthonormal frame e α the volume form is given by ω = e 1 · e 2 . Similarly, left-multiplication by ω induces an almost complex structure on Σ, which we denote by J Σ . The bundle Σ C = Σ ⊗ C decomposes in eigen bundles of i J C Σ , where J C Σ denotes the complex linear extension of J Σ . The complex line bundles W = (Σ, J Σ ) of eigenvalue −1 and W = (Σ, −J Σ ) of eigenvalue +1 are, respectively, the so-called bundles of "left-and righthanded" Weyl spinors. On W = (Σ, J Σ ) there is a bilinear form with values in T * M given by where e α is the dual basis to the g-orthonormal frame e α . The compatibility of Clifford multiplication and almost complex structures, In particular W is a holomorphic vector bundle. In other words, holomorphic cotangent vector fields on a Riemann surface with fixed spin structure have a "square root". Conversely, on a Riemann surface (M, J M ) every square root of T * M gives rise to a spin structure on M.
Obviously, the complex vector bundle (S, In addition, the spinor bundle S possesses three almost complex structures I S , J S , K S ∈ End(S) that commute with the Clifford multiplication and satisfy the quaternionic relations: Hence, S may alternatively be viewed as a quaternionic line bundle. This may not come as a big surprise for Cl 0,2 H = R ⊕ R 3 . When viewed as complex vector bundles of rank two, the three complex spinor bundles (S, I S ), (S, J S ) and (S, K S ) are isomorphic and may be identified with Let us take a closer look at the identification of (S, the Clifford multiplication by X on S differs from the Clifford multiplication by X on Σ by a factor of i. In this way any representation of Cl (M, g) on Σ yields a purely imaginary representation of Cl (M, −g) on Σ C . Furthermore, we obtain the following identifications of Dirac-operators: We now derive a convenient local expression for the Dirac operator. Let us first assume that (M, g) is the Euclidean space with standard coordinates x and y. The holomorphic tangent bundle of M is then spanned by ∂ z = 1 2 ∂ x − i∂ y . The spinor bundle (S, I S ) = W ⊕ W possesses a complex base s, s such that s ∈ W , the spinor s is the complex conjugate of s and s ⊗ s = dz. With respect to this basis the Clifford multiplication of Cl (M, −g) on (S, I S ) is represented by Hence the Euclidean Dirac-operator is given by that is, by the standard Cauchy-Riemann operators. The general, non-Euclidean Diracoperator differs from the Euclidean one by a rescaling and zero-order terms. In particular, this means that the regularity theory developed for Cauchy-Riemann equations applies.

2.4.
In this paragraph we introduce the "super partner" of the metric, called gravitino.

Definition 1.
A gravitino is a smooth section of the bundle S ⊗ T M.
Remark 1. Sometimes in the literature, e.g. [20], a gravitino is defined as a section of the bundle S ⊗ T * M, but here we use the Riemannian metric g to identify T * M with T M, for later convenience.
The Clifford multiplication gives a surjective map and has a canonical right-inverse that is given with respect to a local g-orthonormal Consequently the bundle S⊗T M has an orthogonal direct sum decomposition S⊗T M ∼ = S ⊕ ker γ and the maps P = σ • γ and Q = 1 − P are projection operators on S and ker γ respectively. With respect to the g-orthonormal frame {e α } the gravitino χ can locally be expressed as χ = χ α ⊗ e α with χ α ∈ Γ loc (S). The projection operators P and Q are given by Later we will mostly be concerned with the sections of ker γ , because only Qχ appears in the action functional. Notice that ker γ can be identified with (S, Using the almost complex structure J Σ ⊕ J Σ on S and T * M = W ⊗ C W we obtain the following decomposition This is the decomposition of S ⊗ T M into irreducible representations of Spin(2). Up to a metric identification, the bundle S ⊗ T M decomposes into two representations of type 1 2 and two of type 3 2 . The operator Q projects onto the 3 2 -parts.

2.5.
We recall the definition of the field φ and its super partner ψ, see [9].
Equip the tensor product bundle S ⊗ φ * T N with the induced metric and connection. More precisely, let {y i } be local coordinates on N, so that {φ * ( ∂ ∂ y i )} is a local frame of φ * T N. Then the local sections, which will be referred to as "(local) vector spinors", can be written as The induced metric and connection can be expressed by , for any X ∈ T M. The twisted spin Dirac operator / D on S ⊗ φ * T N is defined as follows: In a local g-orthonormal frame e α as above, Similarly to the spin Dirac operator / ∂ the twisted spin Dirac operator / D is essentially self-adjoint with respect to the scalar product in L 2 (S ⊗ φ * T N).

The Action Functional
We want to consider the following action functional: where the last curvature term is locally defined by Notice that we use the following conventions for the curvature tensor: We will abbreviate R φ * T N as R N . Hence, the curvature term can be written as then Note that since P and Q give an orthogonal decomposition, |Qχ | 2 S⊗T M = χ, Qχ . This formula is convenient when expressing the terms locally.

Remark 2.
In order to obtain a real-valued action functional we work here with the real spinor bundle S and the real scalar product g S = ·, · S . Alternatively we might also work with the complex spinor bundle Σ C and the hermitian form h Σ C . We recall that the hermitian form h S on (S, I S ) induced by g S can be written as and coincides with h Σ C under the complex linear isomorphism Σ C (S, I S ). All summands in (5) except the third one are symmetric in the spinors and would consequently be real. For those terms the approach here and in [9] coincide. For the third term one could use equally the real part of We will refrain from using that expression later on.
The functional A(φ, ψ; g, χ) has rich symmetries. It is invariant under generalized conformal transformations of the metric in the sense that where u ∈ C ∞ (M). To verify the conformal invariance we use the rescaling of the spinor metric g S by e u g S and that / D e 2u g e −u ψ = e −2u / D g ψ, see also [16, Proposition 1.3.10]. Here / D g denotes the Dirac operator defined with respect to the metric g. Moreover, the functional stays invariant under super Weyl transformations: with Qχ = 0. This follows directly from the fact that the action functional only involves Qχ and not Pχ . A is also Spin(2)-gauge-invariant, in particular under the following Z 2 -action on the spinor bundle S: These symmetries will be naturally inherited by its critical points. They are useful when dealing with the solution space of the Euler-Lagrange equations. A detailed discussion of the symmetries of A and the corresponding conservation laws can be found in [22].
As already mentioned in the introduction the functional (5) is essentially the action functional of the two-dimensional nonlinear supersymmetric sigma model, see [7,13,20]. In contrast to what is discussed there, we deal with commuting spinors. For that matter the action functional (5) does in general not possess supersymmetry, except in special cases, see [22]. Furthermore, a term which vanishes identically at critical points is omitted here. A. Fix (g, χ) and vary (φ, ψ) via (Φ, Ψ ) with variational fields (ξ, η). Here

Now we derive the Euler-Lagrange equations for
.
At a critical point, we have Here we denote by the roman numerals I, . . . , V the summands under the integral in the action functional A. We calculate them term by term.
1. As for harmonic maps,

With
3. Under a local orthonormal frame {e α }, where the first integrand can be rewritten as The first summand vanishes after integration on the closed manifold M since it is a divergence of some vector field. Therefore Here, by abuse of notation we denote by ∇ S e β (e α · e β · χ α ), ψ , the section of φ * T N that arises by metric contraction of ψ by ∇ S e β (e α · e β · χ α ).

Likewise we have
Thus, 5. In local coordinates, we compute

The integrand reads
We define S∇ R analogously to S R, that is, Using the metric to identify it with the corresponding vector field, we get Then, From the preceding computations, we obtain We can thus verify Theorem 1 which we restate here:

4.2.
We rewrite the Euler-Lagrange equations (2) in terms of local coordinates on N .
Let {y i } be a local coordinate system on N . Then {φ * ( ∂ ∂ y i )} is a local frame for the vector bundle φ * T N. Then (2) can be written as Since the curvature of M does not appear in those formulas, we may omit the upper index N for the curvature terms, and we will label it again whenever needed. We may introduce local coordinates on M such that a conformal transformation brings the metric into the following form and then {e α ≡ ∂ ∂ x α } is a local orthonormal frame. We define the vector fields V j on M, j = 1, . . . , n, via for any vector field W on M. Thus, In particular, noting that ∇ e α e β = 0, we have div V j = e β (V j,β ) = e β e α · e β · χ α , ψ j . and e α · e β · χ α , ψ k e β (φ j ) Thus, in those local coordinates the Euler-Lagrange equations become for 1 ≤ i ≤ n. One sees that the right hand side of the first equation lies in L 1 while that of the second equation lies in L 4/3 . This shows that the Euler-Lagrange equations are critical for the Sobolev elliptic theory. Thus, the regularity of weak solutions is a subtle issue.

4.3.
To get the regularity of weak solutions, we embed N isometrically into some Euclidean space. In order to see what happens to the various fields involved, we start with a general consideration. Let (N , h ) be another Riemannian manifold and f : N → N a smooth immersion. We get a composition and induced maps of vector bundles which fit into the following commutative diagram Then the tension fields of φ and φ are related by Now let (N , h ) = (R K , δ) be a Euclidean space with standard global coordinate functions (u a ) a=1,...,K , and let f : (N , h) → (R K , δ) be an isometric embedding. Then the second fundamental form A is perpendicular to N in the sense that, for any X, Y ∈ Γ (T N), extended locally to R K and still denoted by X , Y respectively, the following orthogonal decomposition holds: where ∇ e denotes the flat connection on Euclidean space. See [2,18]. Moreover, for any normal vector field ξ ∈ Γ (T ⊥ N ), where P(ξ ; X ) = −(∇ e X ξ) is the shape operator of N .
As in [31] and [11], we take a local orthonormal frame {ν l |l = n + 1, . . . , K } of T ⊥ N . (These can be smoothly extended to a tubular neighborhood of N , and thus be defined in an open subset of R K ). Then In terms of the global frame { ∂ ∂u a } we write the vector fields X, Y, Z tangent to the submanifold N as Since A is symmetric: We recall here the Gauss equation for X, Y, Z , W ∈ Γ (T N): Since this holds for all W ∈ Γ (T N), we have We will denote the induced map on the tensor product bundles by Then ψ ≡ f # (ψ) is a section of the latter bundle, i.e., a spinor field along the map φ . In local coordinates, Moreover, the Dirac terms corresponding to φ and φ are related via (see [11]) where A(φ * e α , e α · ψ) ≡ e α · ψ i ⊗ φ * A(T φ(e α ), ∂ ∂ y i ) .

4.4.
We are now ready to write the Euler-Lagrange equations in terms of (φ , ψ ).
Apply f # to / Dψ and use (12): We compute the following terms: -Note that Using (11) and the expression for A, we have -Recalling (6) and (10), -For the last term: We thus obtain the equation for ψ : In components, for each a, Here / ∂ is the Dirac operator / ∂ on S and each ψ a is a local pure spinor field. Next we applyφ * ( f * ) to τ (φ) to get Since R K is flat, We deal with the terms on the right hand side as follows: -Using (10) we get -To push S∇ R forward, we note that we can Moreover, using Gauss equation again, one has where we have written A i j ≡ A( ∂ ∂ y i , ∂ ∂ y j ). See for example [4,21]. Hence, -In the same way as we have defined the vector fields V j , j = 1, . . . , n, we can define vector fields V a , a = 1 . . . , K , on M by V a , W T M = e α · W · χ α , ψ a S , ∀W ∈ Γ (T M). Thenφ Therefore the equation for φ is In components, for each a, Remark 3. Actually, for our proof of the local regularity of weak solutions, we don't need to write the second term and the last term into such an antisymmetric structure, see [4] for a similar treatment for a simpler model. Former regularity proofs, see e.g. [11,30,31], however, did need that structure. But it is also convenient to have such a structure.
Therefore, the equations for φ appear in the elegant form: for a = 1, . . . , K , where the coefficients of first derivative of φ are antisymmetric.

Regularity of Weak Solutions
We now come to the crucial contribution of our paper, the regularity of weak solutions. In order to make the action functional A well-defined and finite-valued, we need to assume The issue then is higher regularity of such weak solutions. More precisely, we shall show that (φ, ψ) are smooth when they satisfy (2) in the weak sense. By the Sobolev embedding theorem, φ ∈ L p (M, N ) for any p ∈ [1, ∞) and ψ ∈ L 4 (Γ (S ⊗ φ * T N)).
Since f : N → R K is a smooth embedding, (φ , ψ ) have the same regularity as (φ, ψ), and so it suffices to show smoothness of the former. As the regularity is a local issue, we can take φ : B 1 → R K defined in the euclidean unit disc B 1 ⊂ R 2 ∼ = C 1 . Over B 1 the bundle S ⊗ φ * T N is trivial with typical fiber C 2 ⊗ R K . Hence ψ : B 1 → C 2 ⊗ R K is a vector valued function.

5.1.
As we have seen, ψ satisfies (13) or equivalently (14). By the following lemma, which will be proved in Sect. 6, all powers of ψ are integrable. Lemma 1. Let p ∈ (4, ∞) and ϕ ∈ L 4 (B 1 , C 2 ⊗R K ) be a weak solution of the nonlinear system It follows from Lemma 1 that ψ ∈ L p loc (B 1 ) for any p ∈ [1, ∞). Since locally the Dirac operator is given by the classical Cauchy-Riemann operators ∂ z and ∂ z , it follows from the elliptic theory that ψ ∈ W 1,q (B 1/2 ) for any q ∈ [1, 2).

5.2.
We use the aforementioned Rivière's regularity theory to deal with φ . More precisely, we use the following result which is an extension of [24] to improve the regularity of φ .
In the previous section we have written the equation for φ into such a form, see (17). Since we have seen ψ ∈ L p loc (B 1 ), 1 ≤ p < ∞, the hypotheses of Theorem 3 are satisfied. Thus we can conclude that φ ∈ W 2, p loc (B 1 ) for any p ∈ [1,2). It follows from the Sobolev embedding theorems that φ ∈ W 1,q (B 1/2 ) for any q ∈ [1, ∞).

5.3.
We can now apply the standard elliptic theory for a bootstrap argument, see e.g. [3,15], and hence conclude that (φ , ψ ) are smooth. The smoothness of φ then follows directly. For ψ, one can use (8) and the elliptic theory for Cauchy-Riemann operators (e.g. [3]) to conclude that ψ is also smooth. Therefore the full regularity of weak solutions is obtained, completing the proof of Theorem 2.

Proof of Lemma 1
In this section, we provide the proof of Lemma 1. We shall use the Dirac type equation to improve the integrability of the spinor. Results of this type were first obtained by [29] and further developed in [4,27]. Actually a stronger result holds in general. Before stating the general result, we recall some basic facts on Morrey spaces, see for example [14].
Let U be a domain in R n . For 0 ≤ λ ≤ n and 1 ≤ p < ∞, the Morrey space on U is defined as Here the ( p, λ)-Morrey norm of u is defined by Note that on a bounded domain U ⊂ R n , for 1 ≤ p < ∞ and 0 ≤ λ ≤ n, it holds that In this section we consider a map ϕ : B 1 → C L ⊗ R K satisfying a first order elliptic system, where B 1 ⊂ R n is the euclidean unit ball and C L ⊗ R K is supposed to be the typical fiber of a twisted complex spinor bundle over B 1 .
The proof is motivated from that in [29] and is adapted to this system with minor changes. The idea is to use the fundamental solution of the Euclidean Dirac operator and apply Riesz potential estimates. Thanks to the Bochner-Lichnerowicz-Weitzenböck type formulas, e.g. see [23,Theorem II.8.17], [28,Lemma 4.1], [18,Theorem 4.4.2], the fundamental solution of the Euclidean Dirac operator can be derived from that of the Euclidean Laplacian. We remark that the M 2,2 -assumption on B here fits quite well to the proof.
Finally note that in the 2-dimensional case, So Lemma 1 follows from Lemma 2.