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 [18], [7] it is usually formulated in terms of supergeometry, which includes the use of Grassmann-valued spinors. However, taking ordinary instead of Grassmannvalued spinors one can investigate the full nonlinear supersymmetric σ-model as a geometric variational problem. This study was initiated in [9], 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 Dirac-harmonic maps belong to the class of conformally invariant variational problems yielding a rich structure. Many important results for Dirac-harmonic maps have already been established. This includes the regularity of weak solutions [24] and an existence result for uncoupled solutions [1]. The boundary value problem for Dirac-harmonic maps is studied in [14], [13]. The heat-flow for Dirac-harmonic maps was studied recently in [2], [3] and [15]. 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 ΣM , 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 ΣM ⊗ φ −1 T N . The induced connection on this bundle will be denoted by∇. Sections ψ ∈ Γ(ΣM ⊗ 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 ·∇ eα , 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 ΣM = − X · χ, ξ ΣM for all χ, ξ ∈ Γ(ΣM ) and all X ∈ T M . Moreover, the twisted Dirac-operator / D satisfies the following Weitzenböck formula Here,∆ denotes the connection Laplacian on ΣM ⊗ 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) ∈ Γ(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 ∈ Γ(ΣM ⊗ 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 system (2.2), (2.3) then acquires the form Here / ∂ := e α · ∇ ΣM eα denotes the usual Dirac-operator acting on sections in ψ ∈ Γ(ΣM ⊗ T R q ). 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 (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.
3.1. 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 [9], Theorem 3.2 and nonlinear Dirac equations from [12], 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.
Lemma 3.1. Assume that the pair (φ, ψ) is a smooth solution of (2.4) and (2.5) satisfying with ε 0 small enough. Moreover, assume that there are no harmonic spinors on M . Then both φ and ψ are trivial.
We define the following local energy: Let U be a domain on M . We define the energy of the pair (φ, ψ) on U by Similar as in the case of Dirac-harmonic maps [9] we prove the following 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 [9].
Lemma 3.4. Assume that the pair (φ, ψ) is a smooth solution of (2.4) and (2.5) satisfying (3.3). Then the following estimate holds where C(D 1 ) is a constant depending only on D 1 .
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 (2.4) and (2.5) satisfying (3.3). Then the following estimate holds where the constant C depends only on D 2 .
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| W 1,4 (D) ≤ c|ηφ| and we may follow ). (3.10) Regarding the last two terms in (3.10) we note that using (3.6) Applying these estimates and choosing ε small enough, (3.10) gives ), which can be rearranged as Finally, by the properties of η we have that for some ε > 0 holds.

12)
where C(D 2 ) is a constant depending only on D 2 .
Again, consider the spinor ξ := ηψ and using (3.7) we estimate which proves the claim.
Lemma 3.7. Assume that the pair (φ, ψ) is a smooth solution of (2.4) and (2.5) satisfying (3.3). Then the following estimate holds where C is a constant depending only on D 2 .
Lemma 3.8. Assume that the pair (φ, ψ) is a smooth solution of (2.4) and (2.5) satisfying (3.3). Then the following estimate holds 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 (3.9) we have ). By the Sobolev embedding theorem we get Moreover, applying ). 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 (3.9) we have for any p > 1 ). By application of (3.15) we find ). Finally, we conclude by (3.13) that ), which proves the assertion.
After having gained control over φ we may now control the spinor ψ. Lemma 3.9. Assume that the pair (φ, ψ) is a smooth solution of (2.4) and (2.5) satisfying (3.3). Then the following estimates hold: where the constants depend only on D 2 .
This proves the first estimate for the spinor.

3.2.
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], Prop. 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 equations (3.24) and (3.25) acquire the form (2.4) and (2.5). For more details see Lemma 3.5 in [6].
Lemma 3.11. Let (φ, ψ) be a smooth Dirac-harmonic map with curvature term on D \ {0} satisfying E(φ, ψ, D) < ε. Then we have 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.
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], Prop. 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 Re(z 2 T (z)) =r 2 |φ r | 2 − |φ θ | 2 + r 2 cos 2 θ ψ, ∂ r ·∇ ∂r ψ − sin 2 θ ψ, which together with (3.29) proves the result.
Theorem 3.12 (Removable Singularity Theorem). Let (φ, ψ) be a Dirac-harmonic map with curvature term which is smooth on U \ {p} for some p ∈ U ⊂ M . If the pair (φ, ψ) has finite energy, then (φ, ψ) extends to a smooth solution on U .
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 [10] and the proof of Theorem 3.1 in [12].

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 [11] and [17], see also [8].
Remark 4.1. In this section we do not necessarily have to assume that the domain M is compact. Moreover, we do not have to restrict to a two-dimensional domain M . However, in the case of the nonlinear supersymmetric sigma model the term A(dφ, dφ) originates from the variation of a two-form. If we would assume that m = dim M ≥ 2 then this term would be proportional to |dφ| m .
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 with the constants κ 3 := |R N | L ∞ , c 7 := |E| L ∞ and c 8 := |F | L ∞ . Moreover, δ 7 and δ 8 are positive constants to be determined later. We set and in addition t := δ 2 + δ 4 . Adding up (4.1) and (4.2) we obtain This allows us to derive a first (similar to [21] for harmonic maps and [11] for Dirac-harmonic maps) Theorem 4.2. Let (φ, ψ) be a smooth solution of (2.2) and (2.3). Suppose that M is a closed Riemannian manifold with positive Ricci curvature and that the sectional curvature of N is bounded. If e(φ, ψ) < ε (4.5) for ε small enough, then φ is constant and ψ vanishes identically.
Lemma 4.4. Let (φ, ψ) be a smooth solution of (2.2) and (2.3). Moreover, suppose that the Ricci-curvature of M satisfies Ric M ≥ −κ 1 and the sectional curvature K N of N satisfies K N ≤ κ 2 . Then the following inequality holds: 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 (4.6) we find This energy inequality has the same analytic structure as in the case of harmonic maps.
To obtain a gradient estimate from (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/(2 √ d 1 ), thus 0 < ξ(R) < √ d 1 on the ball B R (y 0 ).  Proof. This follows from the Hessian Comparison theorem, see [19], p. 19,Prop. 2.20 and [17], p.93.
In addition, let r be the distance function from the point x 0 in M . Define the function 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.
Lemma 4.7. Suppose that (M, h) and (N, g) are complete Riemannian manifolds. Let (φ, ψ) be a smooth solution of (2.2) and (2.3) satisfying φ : M → B R (y 0 ) ⊂ N with R < π/(2 √ d 1 ). Moreover, suppose that the Ricci-curvature of M satisfies Ric M ≥ −κ 1 and the sectional curvature K N of N satisfies K N ≤ κ 2 . Then the following inequality holds: Proof. Differentiating log F at its maximum x max we obtain and also (4.14) Inserting (4.7) into (4.14) we find By squaring (4.13) we also get Combining (4.16) and (4.15) then gives the result.
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.
Moreover, we have |dξ| = d 1 | sin( √ d 1 ρ)| ≤ d 1 and to obtain a gradient estimate we set By the properties of the Riemannian distance function ρ on N , equation (4.17), the definition of L 1 and the estimate on Hess ξ we find Remark 4.9. If we consider the limiting case of harmonic maps in (4.19) then we obtain the same inequality leading to a gradient estimate as in [17].
First of all, let us consider the case that A(dφ, dφ) = 0 in (2.2), which means that c 1 = c 2 = δ 1 = δ 2 = 0 and we obtain for some positive number δ 9 . We require the coefficient in front of |dφ| 2 to be positive, which in this case can be expressed as Hence, we have to choose d 1 such that (4.21) holds. However, note that we have some freedom to choose δ 3 and δ 9 in (4.21). Again, to shorten the notation, we set  , where d 1 is determined by (4.21). Suppose that A(dφ, dφ) = 0 and B, C, E, F are bounded. Moreover, assume that the Ricci curvature of M satisfies Ric ≥ −κ 1 and that the sectional curvature K N of N satisfies K N ≤ κ 2 . Then for any x 0 ∈ B a (x 0 ) the following estimate holds |dφ| ≤ 4rd 1 d(a 2 − r 2 )ξ • φ + L 1 + L 2 |ψ| 4 d , (4.24) where L 1 is given by (4.18) and L 2 is given by (4.22).
In the case that A(dφ, dφ) = 0 it is more difficult to obtain an estimate on |dφ|. Let us again consider (4.19) 0 for some positive number δ 10 . Again, we require the coefficient in front of |dφ| 2 to be positive, which in this case can be expressed as However, it seems quite difficult to check if one can arrange all the constants above such that the inequality (4.26) holds.
(1) Due to the additional terms on the right hand side of (2.3) it is hard to say in which cases the estimate (4.24) is sharp.
(2) It becomes clear along the proof that we have a lot of freedom rearranging the constants involved in all the estimates. However, this does not change the general structure of the estimate (4.24).
(3) Our calculation shows that the magnitude of A(dφ, dφ) clearly has the strongest influence on the estimate on |dφ|. (4) For Dirac-harmonic maps gradient estimates have been established in [11], the authors used a Kato-Yau inequality to obtain the optimal constants in their estimates. However, this does not seem to help much here since we are considering a more complicated system as in [11].