Energy Estimates for the Supersymmetric Nonlinear Sigma Model and Applications

We derive gradient and energy estimates for critical points of the full supersymmetric sigma model and discuss several applications.


Introduction and Results
The full nonlinear supersymmetric σ -model is an important model in modern quantum field theory. In the physical literature [7,18] it is usually formulated in terms of supergeometry, which includes the use of Grassmann-valued spinors. However, taking ordinary instead of Grassmann-valued spinors one can investigate the full nonlinear supersymmetric σ -model as a geometric variational problem. This study was initiated in [10], where the notion of Dirac-harmonic maps was introduced. These form a pair of a map between Riemannian manifolds and a vector spinor. More precisely, the equations for Dirac-harmonic maps couple the harmonic map equation to spinor fields. As limiting cases both harmonic maps and harmonic spinors can be obtained. In the case of a two-dimensional domain Diracharmonic maps belong to the class of conformally invariant variational problems yielding a rich structure.
However, to analyze the full nonlinear supersymmetric σ -model one has to go beyond the notion of Dirac-harmonic maps. Considering an additional two-form in the action functional one is led to magnetic Dirac-harmonic maps introduced in [5]. Dirac-harmonic maps to target spaces with torsion are analyzed in [4]. Finally, taking into account a curvature term in the action functional one is led to Dirac-harmonic maps with curvature term, which were introduced in [8].
In this note we study general properties of the system of partial differential equations that arises as critical points of the full nonlinear supersymmetric σ -model. This article is organized as follows. In Section 2 we recall the mathematical background that we are using to perform our analysis. In Section 3 we present an ε-regularity theorem for the domain being a closed surface and as an application, we prove the removable singularity theorem for Dirac-harmonic maps with curvature term. In Section 4 we derive gradient estimates and point out several applications.

The Full Supersymmetric Nonlinear Sigma Model
Throughout this article, we assume that (M, h) is a Riemannian spin manifold with spinor bundle M, for more details about spin geometry see the book [20]. Moreover, let (N, g) be another Riemannian manifold and let φ : M → N be map. Together with the pullback bundle φ −1 T N we can consider the twisted bundle M ⊗φ −1 T N. The induced connection on this bundle will be denoted by∇. Sections ψ ∈ ( M ⊗ φ −1 T N) in this bundle are called vector spinors and the natural operator acting on them is the twisted Dirac operator, denoted by / D. This is an elliptic, first order operator, which is self-adjoint with respect to the L 2 -norm. More precisely, the twisted Dirac operator is given by / D = e α ·∇ e α , where {e α } is an orthonormal basis of T M and · denotes Clifford multiplication. We are using the Einstein summation convention, that is we sum over repeated indices. Clifford multiplication is skew-symmetric, namely χ, X · ξ M = − X · χ, ξ M for all χ, ξ ∈ ( M) and all X ∈ T M. Moreover, the twisted Dirac-operator / D satisfies the following Weitzenböck formula Here,˜ denotes the connection Laplacian on M ⊗φ −1 T N, R denotes the scalar curvature on M and R N is the curvature tensor on N . This formula can be deduced from the general Weitzenböck formula for twisted Dirac operators, see [20], p. 164, Theorem 8.17. We do not present the full energy functional here but rather focus on its critical points. These satisfy a coupled system of the following form Here, τ (φ) ∈ (φ −1 T N) denotes the tension field of the map φ and the other terms represent the analytical structure of the right hand side. We will always assume that the endomorphisms A, B, C, E and F are bounded.
At some points we will assume that the target manifold N is isometrically embedded in some R q by the Nash embedding theorem. Then, we have that φ : M → R q with φ(x) ∈ N . The vector spinor ψ becomes a vector of usual spinors ψ 1 , ψ 2 , . . . , ψ q , more precisely ψ ∈ ( M ⊗ T R q ). The condition that ψ is along the map φ is then encoded as q i=1 ν i ψ i = 0 for any normal vector ν at φ(x).
The quantities A, B, C, E and F can be extended to the ambient space (denoted by a tilde) and depend only on geometric data. However, this does not alter the analytic structure of the right hand side of Eqs. 2.2, 2.3.

Remark 2.1
The regularity of the system (2.4), (2.5) is already fully understood. By now, there are powerful tools available to ensure the smoothness of a system like (2.4), (2.5), see [22,23] and [6]. However, it should be noted that in order to apply the main result from [22] we need a certain antisymmetry of the endomorphism A. It is quite remarkable that the actual A from the nonlinear supersymmetric sigma model has the necessary antisymmetry.

Remark 2.2
In the physical literature the energy functional for the full supersymmetric nonlinear sigma model is fixed by the requirements of superconformal invariance (conformal invariance + supersymmetry) and invariance under diffeomorphisms on the domain.

Energy Estimates and Applications
Throughout this section we assume that the domain M is a closed Riemannian spin surface.

Epsilon Regularity Theorem
We derive an ε-regularity Theorem for smooth solutions of the system (2.4), (2.5). To this end, we combine the methods for Dirac-harmonic maps from [10], Theorem 3.2 and nonlinear Dirac equations from [13], Theorem 2.1. To establish the ε-regularity theorem we make use of the invariance under scaling of the system (2.4), (2.5).
However, we should not assume that the energy is small globally. Proof See the proof of Lemma 4.8 in [6].
We define the following local energy: We define the energy of the pair (φ, ψ) on U by Similar as in the case of Dirac-harmonic maps [10] we prove the following Then the following estimate holds for allD ⊂ D, p > 1, where C(D, p) is a positive constant depending only onD and p.
We divide the proof into several steps, we will assume thatD ⊂ D 3 ⊂ D 2 ⊂ D 1 ⊂ D.
As a first step, we derive an estimate for the spinor ψ, similar to Lemma 3.4 in [10].

Lemma 3.4 Assume that the pair (φ, ψ) is a smooth solution of
where C(D 1 ) is a constant depending only on D 1 .
Consider the spinor ξ := ηψ and moreover, since the unit disc D is flat, we have / ∂ 2 = − . Using Eq. 2.5, we calculate Hence, employing elliptic estimates we get ). By Hölder's inequality we can estimate with the conjugate Sobolev index q * = 2q 2−q . By the Sobolev embedding theorem we may then follow Thus, if the energy E(φ, ψ) is small enough, we have At this point for any p > 1 one can always find some q < 2 such that p = q * and this yields the first claim.

Lemma 3.5 Assume that the pair (φ, ψ) is a smooth solution of
where the constant C depends only on D 2 .
Hence, for any p > 1 we have Choosing p = 4 3 on the disc D, we find .
Without loss of generality we assume D φ = 0 such that |φ| W 1,p (D) ≤ C|dφ| L p (D) for any p > 0. Moreover, by Hölder's inequality we have such that we may conclude .
By the Sobolev embedding theorem we find |ηφ| and we may follow .
(3.10) Regarding the last two terms in Eq. 3.10 we note that using Eq. 3.6 Applying these estimates and choosing ε small enough, Eq. 3.10 gives which can be rearranged as Finally, by the properties of η we have that for some ε > 0 Proof We choose a cut-off function η satisfying 0 ≤ η ≤ 1 with η| D 2 = 1 and supp η ⊂ D.
Again, consider the spinor ξ := ηψ and using Eq. 3.7 we estimate which proves the claim.
where C is a constant depending only on D 2 .
Proof Choose a cut-off function η : 0 ≤ η ≤ 1 with η| D 2 = 1 and supp η ⊂ D. By Eq. 3.10 we have . Using we obtain the result. 14) where the constant C depends only on D 3 .
Proof Choose a cut-off function η : 0 ≤ η ≤ 1 with η| D 3 = 1 and supp η ⊂ D 2 . By Eq. 3.9 we have By the Sobolev embedding theorem we get Moreover, applying we find Hence, we may conclude Again, by the Sobolev embedding theorem we may thus follow Having gained control over the W 2,2 norm of φ we now may control the W 2,p norm of φ for p > 2. Again, suppose thatD ⊂ D 3 and choose a cut-off function η : 0 ≤ η ≤ 1 with η|D = 1 and supp η ⊂D. By Eq. 3.9 we have for any p > 1 By application of Eq. 3.15 we find . Finally, we conclude by Eq. 3.13 that which proves the assertion.
After having gained control over φ we may now control the spinor ψ.  (3.17) where the constants depend only on D 2 .
Now for D 2 ⊂ D 1 choose a cut-off function η : 0 ≤ η ≤ 1 with η| D 2 = 1 and supp η ⊂ D 1 . For any p > 1 we then have Setting p = 4 3 and making using of Hölder's inequality we obtain By application of Eqs. 3.6, 3.12, 3.13 and 3.14 we get By the Sobolev embedding theorem this yields ψ| W 1,4 (D 2 ) ≤ C|ψ| L 4 (D) (3.19) and also This proves the first estimate for the spinor.
Using the same method as before, we now get an estimate on |∇ψ|. Thus, for D 3 ⊂ D 2 choose a cut-off function η : 0 ≤ η ≤ 1 with η| D 3 = 1 and supp η ⊂ D 2 . Setting p = 2 in Eq. 3.18 we obtain By the Sobolev embedding theorem we may then follow (3.20) At this point forD ⊂ D 3 we again use Eq. 3.18 with a cut-off function η : 0 ≤ η ≤ 1 with η|D = 1 and supp ⊂ D 3 . Using Eqs. 3.20, 3.6, 3.13 and 3.14 we can follow Thus and, finally, we obtain

It is easy to see that (φ,ψ) is a smooth solution of Eqs. 2.4 and 2.5 on D with E(φ,ψ, D) < ε. By application of Theorem 3.3, we have
and scaling back yields the assertion.

Application: Removable Singularity Theorem for Dirac-harmonic Maps with Curvature Term
Using the previous estimates we sketch how to prove the removable singularity theorem for Dirac-harmonic maps with curvature term.
Dirac-harmonic maps with curvature term are critical points of the functional with the indices contracted as The critical points of the energy functional (3.23) are given by (see [6], Proposition 2.1) where τ (φ) is the tension field of the map φ, R N denotes the curvature tensor on N and : φ −1 T * N → φ −1 T N represents the musical isomorphism. By embedding N into R q isometrically the Eqs. 3.24 and 3.25 acquire the form (2.4) and (2.5). For more details see Lemma 3.5 in [6]. (3.26) where (r, θ ) are polar coordinates on the disc D around the origin, φ r denotes differentiation of φ with respect to r and φ θ denotes differentiation of φ with respect to θ .

Lemma 3.11 Let (φ, ψ) be a smooth Dirac-harmonic map with curvature term on
Proof On a small domainM of M we choose a local isothermal parameter z = x + iy and set with ∂ x = ∂ ∂x and ∂ y = ∂ ∂y . It was shown in [6], Proposition 3.3, that the quadratic differential (3.27) is holomorphic. By Corollary 3.10 we know that which, altogether gives |T (z)| ≤ Cz −2 . Moreover, it is easy to see that D |T (z)| < ∞.
Hence, we may follow that zT (z) is holomorphic on the disc D and by Cauchy's integral theorem we deduce 0 = Im |z|=r zT (z)dz = 2π 0 Re(z 2 T (z))dθ. (3.28) By a direct calculation we find Using the equation for ψ in polar coordinates we find that the term is both purely real and purely imaginary and thus vanishes. Thus, we obtain which together with Eq. 3.29 proves the result. Proof We do not give a full proof here. Using the ε-regularity Theorem 3.3 and Lemma 3.11 the removable singularity theorem can be proven the same way as for Dirac-harmonic maps, see the proof of Theorem 4.6 in [11] and the proof of Theorem 3.1 in [13].

Gradient Estimates and Applications
In this section we derive gradient estimates for solutions (φ, ψ) of the coupled system (2.2), (2.3). To achieve this we extend the techniques from [12] and [17], see also [8].  ∇ψ, ψ, ψ, ψ), dφ and thus we may estimate where δ i , i = 2, 3, 4, 6 are positive constants to the determined later. As a next step we derive an estimate for |ψ| 4 . By a direct calculation we obtain (with R being the scalar curvature on M) 1 2 |ψ| 4 = 2|ψ| 2 |∇ψ| 2 + d|ψ| 2 + R 2 |ψ| 4 + |ψ| 2 e α · e β · R N (dφ(e α ), dφ(e β ))ψ, ψ where we applied (2.1). To estimate the last term, we use the equation for ψ, (2.3), and find Due to the skew-symmetry of the Clifford multiplication the first terms on the right hand side are both purely imaginary and purely real and thus vanish. Moreover, we have the estimate Again, we may rearrange This allows us to derive a first (similar to [21] for harmonic maps and [12] for Diracharmonic maps)  For the sake of completeness we give the following Lemma 4. 3 We have the following inequality: where the value of the positive constants c 13 and c 14 is determined along the proof.
Proof We choose δ j , j = 2, 4, 6, 7, 8 such that and 1 − t > 0. Using Eq. 4.6 we find  in Eq. 4.8 then we would get an inequality of the form This energy inequality has the same analytic structure as in the case of harmonic maps.
To obtain a gradient estimate from Eq. 4.7 for non-compact M and N we need the following tools: Let ρ be the Riemannian distance function from the point y 0 in the target manifold N . We define ξ := d 1 cos( d 1 ρ) (4.9) for some positive number √ d 1 to be fixed later, where B R (y 0 ) denotes the geodesic ball of radius R around the point y 0 . We will assume that R < π/ ( on the geodesic ball B r (x 0 ) in M with some positive number p. Clearly, the function F vanishes on the boundary B a (x 0 ), hence F attains its maximum at an interior point x max . Moreover, we can assume that the distance function r is smooth near the point x max , see [16], Section 2.
Proof Differentiating log F at its maximum x max we obtain and also 0 ≥ − r 2 In the following, we apply the Laplacian comparison Theorem, see [19], p.20, that is with some positive constant C L . Moreover, we make use of the Gauss Lemma, that is |dr| 2 = 1.