Energy quantization for a nonlinear sigma model with critical gravitinos

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 614; the name “supersymmetric” comes from the fact that the action functional is invariant under certain transformations of the matter fields; see for instance 1318. 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 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 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 22, which are briefly recalled in the following. Let $(M,g)$ be an oriented closed Riemannian surface with a fixed spin structure, and let $S\to M$ be a spinor bundle, of real rank four, associated to the given spin structure. Note that the Levi-Civita connection $\nabla ^M$ on $M$ and the Riemannian metric $g$ induce a spin connection $\nabla ^s$ on $S$ in a canonical way and a spin metric $g_s$ which is a fiberwise real inner product;⁠¹ see 1926. The spinor bundle $S$ is a left module over the Clifford bundle $\operatorname {Cl}(M,-g)$ with the Clifford map being denoted by $\gamma \colon TM\to \operatorname {End}(S)$; 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 22.

✖

$$\begin{equation*} \gamma (X)\gamma (Y)+\gamma (Y)\gamma (X)=-2g(X,Y) \quad \forall X,Y\in \mathscr{X}(M). \end{equation*}$$

The Clifford action is compatible with the spinor metric and the spin connection, making $S$ into a Dirac bundle in the sense of 26. Therefore, the bundle $S\otimes TM$ is also a Dirac bundle over $M$, and a section $\chi \in \Gamma (S\otimes TM)$ is taken as a super partner of the Riemannian metric and called a gravitino. The Clifford multiplication gives rise to a map $\delta _\gamma \colon S\otimes TM\to S$, where $\delta _\gamma (s\otimes v)=\gamma (v)s=v\cdot s$ for $s\in \Gamma (S)$ and $v\in \Gamma (TM)$ and extending linearly. This map is surjective, and moreover the following short exact sequence splits:

$$\begin{equation*} 0\to \ker \to S\otimes TM \xrightarrow {\delta _\gamma } S \to 0. \end{equation*}$$

The projection map to the kernel is denoted by $Q\colon S\otimes TM\to S\otimes TM$. More explicitly, in a local oriented orthonormal frame $(e_\alpha )$ of $M$, a section $\chi \in \Gamma (S\otimes TM)$ can be written as $\chi ^\alpha \otimes e_\alpha$,⁠² and the $Q$-projection is given by

²

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

✖

$$\begin{equation*} \begin{split} Q\chi &\coloneq -\frac{1}{2}\gamma (e_\beta )\gamma (e_\alpha )\chi ^\beta \otimes e_\alpha \\ &=\frac{1}{2}\left( (\chi ^1+\omega \cdot \chi ^2)\otimes e_1 -\omega \cdot (\chi ^1+\omega \cdot \chi ^2)\otimes e_2\right), \end{split} \end{equation*}$$

where $\omega =e_1\cdot e_2$ is the real volume element in the Clifford bundle.

Let $(N,h)$ be a compact Riemannian manifold and let $\phi \colon M\to N$ be a map. One can consider the twisted spinor bundle $S\otimes \phi ^*TN$ with bundle metric $g_s\otimes \phi ^*h$ and connection $\widetilde{\nabla }\equiv \nabla ^{S\otimes \phi ^*TN}$, which is also a Dirac bundle, and the Clifford action on this bundle is also denoted by $\gamma$ or simply a dot. A section of this bundle is called a vector spinor, and it serves as a super partner of the map $\phi$ in this model. The twisted spin Dirac operator $\mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}$ is defined in the canonical way: let $(e_\alpha )$ be a local orthonormal frame of $M$. Then for any vector spinor $\psi \in \Gamma (S\otimes \phi ^*TN)$, define

$$\begin{equation*} \mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}\psi \coloneq \gamma (e_\alpha )\widetilde{\nabla }_{e_\alpha }\psi =e_\alpha \cdot \widetilde{\nabla }_{e_\alpha }\psi . \end{equation*}$$

It is elliptic and essentially self-adjoint with respect to the inner product in $L^2(S\otimes \phi ^*TN)$. In a local coordinate $(y^i)$ of $N$, write $\psi =\psi ^i\otimes \phi ^*(\frac{\partial }{\partial y^i})$. Then

$$\begin{equation*} \begin{split} \mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}\psi =\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}\psi ^i\otimes \phi ^*\left(\frac{\partial }{\partial y^i}\right) +\gamma (e_\alpha )\psi ^i\otimes \phi ^*\left(\nabla ^N_{T\phi (e_\alpha )}\frac{\partial }{\partial y^i}\right), \end{split} \end{equation*}$$

where $\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}$ is the spin Dirac operator on $S$. For later convention, we set

$$\begin{equation*} SR(\psi )\!\coloneq \langle \psi ^l,\psi ^j\rangle _{g_s} \psi ^k\otimes \phi ^*\left(\operatorname {R}^N(\frac{\partial }{\partial y^k},\frac{\partial }{\partial y^l})\frac{\partial }{\partial y^j}\right) \!=\! R^i_{\;jkl}(\phi )\langle \psi ^l,\psi ^j\rangle \psi ^k\otimes \phi ^*\left(\frac{\partial }{\partial y^i} \right), \end{equation*}$$

and $\operatorname {R}(\psi )\coloneq \langle SR(\psi ),\psi \rangle _{g_s\otimes \phi ^*h}$.

The action functional under consideration is given by

$$\begin{equation*} \begin{split} \mathbb{A}(\phi , \psi ;g, \chi )\coloneq \int _M & |\operatorname {d}\phi |_{g\otimes \phi ^*h}^2 + \langle \psi , \mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}\psi \rangle _{g_s\otimes \phi ^*h} -4\langle (\mathds{1}\otimes \phi _*)(Q\chi ), \psi \rangle _{g_s\otimes \phi ^*h}\\ & -|Q\chi |^2_{g_s\otimes g} |\psi |^2_{g_s\otimes \phi ^*h} -\frac{1}{6} \operatorname {R}(\psi ) \operatorname {d}vol_g. \end{split} \end{equation*}$$

From 22 we know that the Euler–Lagrange equations are

$$\begin{equation} \begin{split} \tau (\phi )=&\frac{1}{2}\operatorname {R}(\psi , e_\alpha \cdot \psi )\phi _* e_\alpha -\frac{1}{12}S\nabla R(\psi ) \\ & -(\langle \nabla ^s_{e_\beta }(e_\alpha \cdot e_\beta \cdot \chi ^\alpha ), \psi \rangle _{g_s} + \langle e_\alpha \cdot e_\beta \cdot \chi ^\alpha , \widetilde{\nabla }_{e_\beta } \psi \rangle _{g_s}), \\ \mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}\psi =& |Q\chi |^2\psi +\frac{1}{3}SR(\psi )+2(\mathds{1}\otimes \phi _*)Q\chi , \end{split} {\tag{1}} \end{equation}$$

where $S\nabla R(\psi )=\phi ^*(\nabla ^N R)_{ijkl}\langle \psi ^i,\psi ^k\rangle _{g_s} \langle \psi ^j,\psi ^l\rangle _{g_s}$ and

$$\begin{equation*} \begin{split} \operatorname {R}(\psi , e_\alpha \cdot \psi )\phi _* e_\alpha =&\langle \psi ^k,e_\alpha \cdot \psi ^l \rangle _{g_s} e_\alpha (\phi ^j) \phi ^*\left(\operatorname {R}\left(\frac{\partial }{\partial y^k},\frac{\partial }{\partial y^l}\right) \frac{\partial }{\partial y^j} \right) \\ =&R^i_{\;jkl}\langle \psi ^k,\nabla \phi ^j\cdot \psi ^l\rangle \otimes \phi ^*\left(\frac{\partial }{\partial y^i}\right). \end{split} \end{equation*}$$

One notices that this action functional can actually be defined for $(\phi ,\psi )$ that possess only little regularity. We only need integrability properties to make the action well defined; that is, $\phi \in W^{1,2}(M,N)$ and $\psi \in \Gamma ^{1,4/3}(S\otimes \phi ^*TN)$. The corresponding solutions of 1 in the sense of distributions are called weak solutions. When the Riemannian metric $g$ and the gravitino $\chi$ are assumed to be smooth parameters, it is shown in 22 that any weak solution $(\phi ,\psi )$ is actually smooth. We will show that these solutions have more interesting geometric and analytical properties. Embed $(N,h)$ isometrically into some Euclidean space $\mathbb{R}^K$. Then a solution can be represented by a tuple of functions $\phi =(\phi ^1,\ldots ,\phi ^K)$ taking values in $\mathbb{R}^K$ and a tuple of spinors $\psi =(\psi ^1,\ldots ,\psi ^K)$ where each $\psi ^i$ is a (pure) spinor, and together they satisfy the condition that at each point $\phi (x)$ in the image, for any normal vector $\nu =(\nu ^1,\ldots ,\nu ^K)\in T^\perp _{\phi (x)} N\subset T_{\phi (x)}\mathbb{R}^K$,

$$\begin{equation*} \sum _{i=1}^K \psi ^i(x)\nu ^i(\phi (x))=0. \end{equation*}$$

Moreover, writing the second fundamental form of the isometric embedding as $A=(A^i_{jk})$, the Euler–Lagrange equations can be written in the following form (see 22):

$$\begin{equation} \begin{split} \Delta \phi ^i =&A^i_{jk}\langle \nabla \phi ^j,\nabla \phi ^k\rangle +A^i_{jm}A^m_{kl}\langle \psi ^j,\nabla \phi ^k\cdot \psi ^l\rangle \\ & +Z^i(A,\nabla A)_{jklm} \langle \psi ^j,\psi ^l\rangle \langle \psi ^k,\psi ^m\rangle -\operatorname {div}V^i-A^i_{jk}\langle V^j,\nabla \phi ^k\rangle , \end{split} {\tag{2}} \end{equation}$$

$$\begin{equation} \begin{split} \mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}\psi ^i =&-A^i_{jk}\nabla \phi ^j\cdot \psi ^k+|Q\chi |^2\psi ^i +\frac{1}{3}A^i_{jm}A^m_{kl}\left(\langle \psi ^k,\psi ^l\rangle \psi ^j-\langle \psi ^j,\psi ^k\rangle \psi ^l\right)\\ & -e_\alpha \cdot \nabla \phi ^i\cdot \chi ^\alpha . \end{split} {\tag{3}} \end{equation}$$ Here the $V^i$’s are vector fields on $M$ locally given by

$$\begin{equation} V^i= \langle e_\alpha \cdot e_\beta \cdot \chi ^\alpha ,\psi ^i\rangle e_\beta . {\tag{4}} \end{equation}$$

One should note that there is some ambiguity here, because the second fundamental form maps tangent vectors of the submanifold $N$ to normal vectors, so the lower indices of $A^i_{jk}$ should be tangential indices, and the upper ones normal. However, one can extend the second fundamental form to a tubular neighborhood of $N$ in $\mathbb{R}^K$ such that all the $A^i_{jk}$’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 $A$ 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 $\mathbb{A}$ then reads

$$\begin{equation*} L_c(\phi ,\psi )=\int _M |\operatorname {d}\phi |^2+\langle \psi ,\mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}\psi \rangle -\frac{1}{6}\operatorname {R}(\psi )\operatorname {d}vol_g, \end{equation*}$$

whose critical points are known as Dirac-harmonic maps with curvature term. These were first introduced in 10 and further investigated in 4524. Furthermore, if the curvature term is also omitted, then we get the Dirac-harmonic map functional which was introduced in 78 and further explored from the perspective of geometric analysis in e.g. 9273133353637. 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 $f\colon (\widetilde{M},\tilde{g})\to (M,g)$ be a conformal diffeomorphism, with $f^*g=e^{2u}\tilde{g}$, and suppose the spin structure of $(\widetilde{M}, \tilde{g})$ is isomorphic to the pullback of the given one of $(M,g)$. There is an identification $B\colon S\to \tilde{S}$ which is an isomorphism and fiberwise isometry such that under the transformation

$$\begin{equation*} \begin{split} \phi &\mapsto \tilde{\phi }\coloneq \phi \circ f,\\ \psi &\mapsto \tilde{\psi }\coloneq e^{\frac{u}{2}}(B\otimes \mathds{1}_{\phi ^*TN})\psi , \\ \chi &\mapsto \tilde{\chi }\coloneq e^{\frac{3u}{2}}(B\otimes (f^{-1})_*)\chi , \\ g &\mapsto \tilde{g}, \end{split} \end{equation*}$$

each summand of the action functional stays invariant, and also

$$\begin{equation*} \int _M |\psi |^4 \operatorname {d}vol_g= \int _{\widetilde{M}} |\tilde{\psi }|^4 \operatorname {d}vol_{\tilde{g}}. \end{equation*}$$

Remark.

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

$$\begin{equation*} \int _{M}|\chi |^4\operatorname {d}x, \quad \int _M |\widetilde{\nabla }\psi |^{\frac{4}{3}} \operatorname {d}x, \quad \int _M |\widehat{\nabla }\chi |^{\frac{4}{3}}\operatorname {d}x, \end{equation*}$$

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

Example 1.

When the map $f$ is a rescaling by a constant $\lambda$ on the Euclidean space with the standard Euclidean metric $g_0$, then $f^*g_0=\lambda ^2 g_0$ and $(f^{-1})_*$ is a rescaling by $\lambda ^{-1}$. In this case the gravitino $\chi$ transforms to $\sqrt {\lambda }B\chi ^\alpha \otimes e_\alpha$, where $e_\alpha$ is a standard basis for $(\mathbb{R}^2,g_0)$.

For a given pair $(\phi ,\psi )$ and a domain $U\subset M$, the energy of this pair $(\phi ,\psi )$ on $U$ is suggested in 8 to be

$$\begin{equation*} E(\phi ,\psi ;U)\coloneq \int _U |\operatorname {d}\phi |^2+|\psi |^4\operatorname {d}vol_g, \end{equation*}$$

and when $U$ is the entire manifold we write $E(\phi ,\psi )$, omitting $U$. Similarly, the energies of the map $\phi$, respectively, the vector spinor $\psi$, on $U$ is defined by

$$\begin{align*} E(\phi ;U)\coloneq \int _U |\operatorname {d}\phi |^2\operatorname {d}vol_g, & &\text{respectively, $E(\psi ;U)\coloneq \int _U |\psi |^4\operatorname {d}vol_g$}. \end{align*}$$

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 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 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 $(e_\alpha )$ by

$$\begin{equation*} \begin{split} T =&\big \{2\langle \phi _*e_\alpha ,\phi _* e_\beta \rangle -|\operatorname {d}\phi |^2 g_{\alpha \beta } +\frac{1}{2}\left\langle \psi ,e_\alpha \cdot \widetilde{\nabla }_{e_\beta }\psi +e_\beta \cdot \widetilde{\nabla }_{e_\alpha }\psi \right\rangle -\langle \psi ,\mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}_g\psi \rangle g_{\alpha \beta }\\ & +\langle e_\eta \cdot e_\alpha \cdot \chi ^\eta \otimes \phi _*e_\beta +e_\eta \cdot e_\beta \cdot \chi ^\eta \otimes \phi _*e_\alpha , \psi \rangle +4\langle (\mathds{1}\otimes \phi _*)Q\chi ,\psi \rangle g_{\alpha \beta }\\ &\qquad + |Q\chi |^2 |\psi |^2 g_{\alpha \beta } +\frac{1}{6}\operatorname {R}(\psi )g_{\alpha \beta } \big \}e^\alpha \otimes e^\beta , \end{split} \end{equation*}$$

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 $z=x+iy$, this differential reads

$$\begin{equation*} T(z)\operatorname {d}z^2\coloneq (T_{11}-iT_{12})(\operatorname {d}x+i\operatorname {d}y)^2, \end{equation*}$$

with

$$\begin{equation*} \begin{split} T_{11} &=\left|\frac{\partial \phi }{\partial x}\right|^2-\left|\frac{\partial \phi }{\partial y}\right|^2 +\frac{1}{2}\left(\langle \psi ,\gamma (\partial _x)\widetilde{\nabla }_{\partial _x}\psi \rangle -\langle \psi ,\gamma (\partial _y)\widetilde{\nabla }_{\partial _y}\psi \rangle \right) +F_{11},\\ T_{12} &=\left\langle \frac{\partial \phi }{\partial x},\frac{\partial \phi }{\partial y}\right\rangle _{\phi ^*h} +\langle \psi ,\gamma (\partial _x)\widetilde{\nabla }_{\partial _y}\psi \rangle +F_{12}, \end{split} \end{equation*}$$

where in a local chart $\chi =\chi ^x\otimes \partial _x+\chi ^y\otimes \partial _y$ and

$$\begin{equation*} \begin{split} F_{11}&=2\langle -\chi ^x\otimes \phi _*(\partial _x)-\gamma (\partial _x)\gamma (\partial _y)\chi ^y\otimes \phi _*(\partial _x),\psi \rangle +2\langle (\mathds{1}\otimes \phi _*)Q\chi , \psi \rangle g(\partial _x,\partial _x), \\ F_{12}&=2\langle -\chi ^x\otimes \phi _*(\partial _y)-\gamma (\partial _x)\gamma (\partial _y)\chi ^y\otimes \phi _*(\partial _y),\psi \rangle . \end{split} \end{equation*}$$

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 $(\phi ,\psi )$ be a smooth solution of 1 on $B_1^*:=B_1\backslash \{0\}$ with $\chi$ being a critical gravitino which is smooth on $B_1$. Assume that $(\phi ,\psi )$ has finite energy on $B_1$. Then for any $0<r<1$,

$$\begin{equation} \begin{split} &\int _0^{2\pi }\left|\frac{\partial \phi }{\partial r}\right|^2-\frac{1}{r^2}\left|\frac{\partial \phi }{\partial \theta }\right|^2 \operatorname {d}\theta \\ &\qquad =\int _0^{2\pi } -\langle \psi , \gamma (\partial _r)\widetilde{\nabla }_{\partial _r}\psi \rangle +\frac{1}{6}\operatorname {R}(\psi ) -(F_{11}\cos {2\theta }+F_{12}\sin {2\theta })\operatorname {d}\theta \\ &\qquad =\int _0^{2\pi } \left\langle \psi ,\frac{1}{r^2}\gamma (\partial _\theta )\widetilde{\nabla }_{\partial _\theta }\psi \right\rangle -\frac{1}{6}\operatorname {R}(\psi )- (F_{11}\cos {2\theta }+F_{12}\sin {2\theta })\operatorname {d}\theta . \end{split} {\tag{5}} \end{equation}$$

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

Finally, for a sequence of solutions $(\phi _k,\psi _k)$ with uniformly bounded energies defined on $(M,g)$ with respect to critical gravitinos $\chi _k$ which converge in $W^{1,\frac{4}{3}}$ to some smooth limit $\chi$, a subsequence can be extracted which converges weakly in $W^{1,2}\times L^4$ to a solution defined on $(M,g)$. 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 $\mathbb{S}^2$ with target manifold $(N,h)$, known as “bubbles”. Moreover, the energies pass to the limit; i.e., the energy identities hold.

Theorem 1.3 (Energy identities).

Let $(\phi _k,\psi _k)$ be a sequence of solutions of 1 with respect to smooth critical gravitinos $\chi _k$ which converge in $W^{1,\frac{4}{3}}$ to a smooth limit $\chi$, and assume their energies are uniformly bounded:

$$\begin{equation*} E(\phi _k,\psi _k)\le \Lambda <\infty . \end{equation*}$$

Then passing to a subsequence if necessary, the sequence $(\phi _k,\psi _k)$ converges weakly in the space $W^{1,2}(M,N)\times L^4(S\otimes \mathbb{R}^K)$ to a smooth solution $(\phi ,\psi )$ with respect to $\chi$. Moreover, the blow-up set

$$\begin{equation*} \mathcal{S}\coloneq \bigcap _{r>0}\left\{p\in M\Big | \liminf _{k\to \infty }\int _{B_{r}(p)}|\nabla \phi _k|^2+|\psi _k|^4 \operatorname {d}vol_g\ge \varepsilon _0\right\} \end{equation*}$$

is a finite (possibly empty) set of points $\{p_1,\dots ,p_I\}$ and correspondingly a finite set (possibly empty) of Dirac-harmonic maps with curvature term $(\sigma ^l_i,\xi ^l_i)$ defined on $\mathbb{S}^2$ with target manifold $(N,h)$, for $l=1,\dots ,L_i$ and $i=1,\dots , I$, such that the following energy identities hold:

$$\begin{equation*} \lim _{k\to \infty } E(\phi _k)=E(\phi )+\sum _{i=1}^{I} \sum _{l=1}^{L_i} E(\sigma _i^l), \end{equation*}$$

$$\begin{equation*} \lim _{k\to \infty } E(\psi _k)=E(\psi )+\sum _{i=1}^{I} \sum _{l=1}^{L_i} E(\xi _i^l). \end{equation*}$$

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. 720242935, 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 582430, 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 31116 or more adapted versions in 1.

Theorem 2.1 ($\varepsilon _1$-regularity theorem).

Consider the local model defined on the Euclidean unit disk $B_1\subset \mathbb{R}^2$, and the target manifold is a submanifold $(N,h)\hookrightarrow \mathbb{R}^K$ with second fundamental form $A$. For any $p_1\in (1,\frac{4}{3})$ and $p_2\in (1,2)$ there exists an $\varepsilon _1=\varepsilon _1(A,p_1,p_2)\in (0,1)$ such that if the gravitino $\chi$ and a solution $(\phi ,\psi )$ of 1 satisfy

$$\begin{align*} E(\phi ,\psi ;B_1)=\int _{B_1}|\nabla \phi |^2+|\psi |^4\operatorname {d}x\le \varepsilon _1, & & \int _{B_1}|\chi |^4+|\widehat{\nabla }\chi |^{\frac{4}{3}}\operatorname {d}x\le \varepsilon _1, \end{align*}$$

then for any $U\Subset B_1$, the following estimates hold:

$$\begin{equation*} \|\phi \|_{W^{2,p_1}(U)} \le C\big (|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|_{W^{1,\frac{4}{3}}(B_1)}\big ), \end{equation*}$$

$$\begin{equation*} \|\psi \|_{W^{1,p_2}(U)} \le C\big (|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|_{W^{1,\frac{4}{3}}(B_1)}\big ), \end{equation*}$$ where $C=C(p_1, p_2,U,N)>0$.

Remark.

Note that if the second fundamental form $A$ vanishes identically, then $N$ is a totally geodesic submanifold of the Euclidean space $\mathbb{R}^K$. Hence there are no curvatures on $N$, and the model is then reduced to the scalar case and is not of interest in this article. So we will assume that $A\neq 0$, and without loss of generality, we assume $|A|\equiv \|A\|\ge 1$. For some $C(p)$ depending on the value of $p$ to be chosen later, the small barrier constant $\varepsilon _1$ will be required to satisfy

$$\begin{equation} C(p)|A|^2\sqrt {\varepsilon _1}\le \frac{1}{8}, \qquad C(p)|A| |\nabla A| \varepsilon _1^{3/4}\le \frac{1}{8}, {\tag{6}} \end{equation}$$

where $|\nabla A|\equiv \|\nabla A\|$. These restrictions will be explained in the proof.

Remark.

Note also that since the domain is the Euclidean disk $B_1$, the connection $\widehat{\nabla }$ is actually equivalent to $\nabla ^s$.

Proof of Theorem 2.1.

Since $N$ is taken as a compact submanifold of $\mathbb{R}^K$, we may assume that it is contained in a ball of radius $C_N$ in $\mathbb{R}^K$, which implies that $\|\phi \|_{L^\infty }\le C_N$. Moreover, as we are dealing with a local solution $(\phi ,\psi )$, we may assume that $\int _{B_1}\phi \operatorname {d}x=0$, so that the Poincaré inequalities hold: for any $p\in [1,\infty ]$,

$$\begin{equation*} \|\phi \|_{L^p(B_1)}\le C_p\|\nabla \phi \|_{L^p(B_1)}. \end{equation*}$$

Let $(U_k)_{k\ge 1}$ be a sequence of nonempty disks such that

$$\begin{equation*} B_1\Supset U_1\Supset U_2\Supset U_3\Supset \cdots . \end{equation*}$$

Take a smooth cutoff function $\eta \colon B_1\to \mathbb{R}$ such that $0\le \eta \le 1$, $\eta |_{U_1}\equiv 1$, and $\operatorname {supp}\eta \subset B_1$. Then $\eta \psi$ satisfies

$$\begin{equation*} \begin{split} \mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}(\eta \psi ^i) =&\nabla \eta \cdot \psi ^i+\eta \mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}\psi ^i \\ =&\nabla \eta \cdot \psi ^i-A^i_{jk}\nabla \phi ^j\cdot (\eta \psi ^k)+|Q\chi |^2(\eta \psi ^i) \\ &+\frac{1}{3}A^i_{jm}A^m_{kl}\left(\langle \psi ^k,\psi ^l\rangle (\eta \psi ^j) -\langle \psi ^j,\psi ^k\rangle (\eta \psi ^l)\right) \\ & -e_\alpha \cdot \nabla (\eta \phi ^i)\cdot \chi ^\alpha +e_\alpha \cdot \phi ^i\nabla \eta \cdot \chi ^\alpha . \end{split} \end{equation*}$$

Then one has

$$\begin{equation*} \begin{split} |\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}(\eta \psi )|\le &|\nabla \eta ||\psi |+|A||\nabla \phi |\cdot |\eta \psi |+|Q\chi |^2|\eta \psi |+|A|^2|\psi |^2\cdot |\eta \psi | \\ &+|Q\chi ||\nabla (\eta \phi )|+|\phi ||\nabla \eta ||Q\chi |. \end{split} \end{equation*}$$

Consider the $L^{p}$-norm (where $p\in (1,2)$) of the left hand side:

$$\begin{equation*} \begin{split} \|\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}(\eta \psi )\|_{L^{p}(B_1)} \le & \|\nabla \eta \|_{L^{\frac{4p}{4-p}}(B_1)}\|\psi \|_{L^4(B_1)} +|A|\|\nabla \phi \|_{L^2(B_1)}\|\eta \psi \|_{L^\frac{2p}{2-p}(B_1)}\\ & +\|Q\chi \|^2_{L^4(B_1)}\|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)} +|A|^2\|\psi \|^2_{L^4(B_1)} \|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)}\\ & +\|Q\chi \|_{L^4(B_1)}\|\nabla (\eta \phi )\|_{L^{\frac{4p}{4-p}}(B_1)} +C_N\|\nabla \eta \|_{L^{\frac{4p}{4-p}}(B_1)}\|Q\chi \|_{L^4(B_1)}. \end{split} \end{equation*}$$

Assume that $\|\nabla \eta \|_{L^{\frac{4p}{4-p}}(B_1)}$ is bounded by some constant $C'=C'(U_1,p)$. Since $\eta \psi$ vanishes on the boundary and $\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}$ is an elliptic operator of order one, we have

$$\begin{equation*} \|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)}\le C(p)\|\nabla ^s(\eta \psi )\|_{L^p(B_1)} \le C(p)\|\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}(\eta \psi )\|_{L^p(B_1)}. \end{equation*}$$

Then from

$$\begin{equation*} \begin{split} \|\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}(\eta \psi )\|_{L^p(B_1)} & \le \left(|A|\|\nabla \phi \|_{L^2(B_1)}+\|Q\chi \|^2_{L^4(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}\right) \|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)} \\ & +\|Q\chi \|_{L^4(B_1)}\|\nabla (\eta \phi )\|_{L^{\frac{4p}{4-p}}(B_1)} +C_N C'\left(\|\psi \|_{L^4(B_1)} +\|Q\chi \|_{L^4(B_1)}\right) \end{split} \end{equation*}$$

together with the fact that

$$\begin{equation*} |\widetilde{\nabla }(\eta \psi )|\le |\nabla ^s(\eta \psi )|+|A||\eta \psi ||\nabla \phi |, \end{equation*}$$

it follows that

$$\begin{equation} \begin{split} & \|\widetilde{\nabla }(\eta \psi )\|_{L^p(B_1)} \\ & \qquad \le 2C(p)\left(\|Q\chi \|_{L^4(B_1)}\|\nabla (\eta \phi )\|_{L^{\frac{4p}{4-p}}(B_1)} +C_N C'\left(\|\psi \|_{L^4(B_1)} +\|Q\chi \|_{L^4(B_1)}\right)\right), \end{split} {\tag{7}} \end{equation}$$

provided that 6 is satisfied.

Now consider the map $\phi$. The equations for $\eta \phi$ are

$$\begin{equation*} \begin{split} \Delta (\eta \phi ^i) =&\eta \Delta \phi ^i+2\langle \nabla \eta ,\nabla \phi ^i\rangle +(\Delta \eta )\phi ^i \\ =&\eta \big (A^i_{jk}\langle \nabla \phi ^j,\nabla \phi ^k\rangle +A^i_{jm}A^m_{kl}\langle \psi ^j,\nabla \phi ^k\cdot \psi ^l\rangle \\ & +Z^i(A,\nabla A)_{jklm} \langle \psi ^j,\psi ^l\rangle \langle \psi ^k,\psi ^m\rangle \\ & -\operatorname {div}V^i-A^i_{jk}\langle V^j,\nabla \phi ^k\rangle \big )+2\langle \nabla \eta ,\nabla \phi ^i\rangle +(\Delta \eta )\phi ^i. \end{split} \end{equation*}$$

Using $\eta \nabla \phi ^i=\nabla (\eta \phi ^i)-\phi ^i(\nabla \eta )$, we can rewrite it as

$$\begin{equation} \begin{split} \Delta (\eta \phi ^i) =&A^i_{jk}\langle \nabla \phi ^j,\nabla (\eta \phi ^k)\rangle +A^i_{jm}A^m_{kl}\langle \psi ^j,\nabla (\eta \phi ^k)\cdot \psi ^l\rangle \\ & +Z^i(A,\nabla A)_{jklm} \langle \psi ^j,\psi ^l\rangle \langle \psi ^k,\eta \psi ^m\rangle \\ &-\operatorname {div}(\eta V^i)-A^i_{jk}\langle V^j,\nabla (\eta \phi ^k)\rangle +2\langle \nabla \eta ,\nabla \phi ^i\rangle +(\Delta \eta )\phi ^i \\ &-A^i_{jk}\langle \nabla \phi ^j,\phi ^k\nabla \eta \rangle \!-\! A^i_{jm}A^m_{kl}\langle \psi ^j,\phi ^k\nabla \eta \cdot \psi ^l\rangle \!+\! \langle \nabla \eta ,V^i\rangle \!+\! A^i_{jk}\langle V^j,\phi ^k\nabla \eta \rangle . \end{split} {\tag{8}} \end{equation}$$

Notice that $\eta \phi ^i\in C^\infty _0(B_1)$. Split it as $\eta \phi ^i=u^i+v^i$, where $u^i\in C^\infty _0(B_1)$ uniquely solves (see e.g. 12, Chap. 8)

$$\begin{equation*} \Delta u^i=-\operatorname {div}(\eta V^i). \end{equation*}$$

Since $\eta V^i\in L^{\frac{4p}{4-p}}(B_1)$, it follows from the $L^p$ theory of Laplacian operators that

$$\begin{equation} \|u\|_{W^{1,\frac{4p}{4-p}}(B_1)}\le C(p)\|Q\chi \|_{L^4(B_1)}\|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)}. {\tag{9}} \end{equation}$$

Then $v^i\in C^\infty _0(B_1)$ satisfies

$$\begin{equation*} \begin{split} \Delta v^i &=\Delta (\eta \phi ^i)-\Delta u^i \\ &=A^i_{jk}\langle \nabla \phi ^j,\nabla (\eta \phi ^k)\rangle +A^i_{jm}A^m_{kl}\langle \psi ^j,\nabla (\eta \phi ^k)\cdot \psi ^l\rangle \\ &{}+Z^i(A,\nabla A)_{jklm} \langle \psi ^j,\psi ^l\rangle \langle \psi ^k,\eta \psi ^m\rangle -A^i_{jk}\langle V^j,\nabla (\eta \phi ^k)\rangle +2\langle \nabla \eta ,\nabla \phi ^i\rangle +(\Delta \eta )\phi ^i \\ &{}-A^i_{jk}\langle \nabla \phi ^j,\phi ^k\nabla \eta \rangle -A^i_{jm}A^m_{kl}\langle \psi ^j,\phi ^k\nabla \eta \cdot \psi ^l\rangle +\langle \nabla \eta ,V^i\rangle +A^i_{jk}\langle V^j,\phi ^k\nabla \eta \rangle . \end{split} \end{equation*}$$

From 22, $\|Z(A,\nabla A)\|\le |A||\nabla A|$. Thus the $L^{\frac{4p}{4+p}}$ norm of $\Delta v$ can thus be estimated by

$$\begin{equation*} \begin{split} \lefteqn{\|\Delta v\|_{L^{\frac{4p}{4+p}}(B_1)} \le |A|\|\nabla \phi \|_{L^2(B_1)}\|\nabla (\eta \phi )\|_{L^{\frac{4p}{4-p}}(B_1)} +|A|^2\|\psi \|^2_{L^4(B_1)}\|\nabla (\eta \phi )\|_{L^{\frac{4p}{4-p}}(B_1)}}\quad & \\ &{}+|A||\nabla A|\|\psi \|^3_{L^4(B_1)}\|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)} +|A|\|Q\chi \|_{L^4(B_1)}\|\psi \|_{L^4(B_1)}\|\nabla (\eta \phi )\|_{L^{\frac{4p}{4-p}}(B_1)} \\ &{}+2\|\nabla \eta \|_{L^{\frac{4p}{4-p}}(B_1)}\|\nabla \phi \|_{L^2(B_1)} +\|\Delta \eta \|_{L^{\frac{4p}{4-p}}(B_1)}\|\phi \|_{L^2(B_1)}\\ &{}+|A|\|\nabla \phi \|_{L^2(B_1)}\|\phi \nabla \eta \|_{L^{\frac{4p}{4-p}}(B_1)} +|A|^2\|\psi \|^2_{L^4(B_1)}\|\phi \nabla \eta \|_{L^{\frac{4p}{4-p}}(B_1)} \\ &{}+\|\nabla \eta \|_{L^{\frac{4p}{4-p}}(B_1)}\|Q\chi \|_{L^4(B_1)}\|\psi \|_{L^4(B_1)} +|A|\|\phi \nabla \eta \|_{L^{\frac{4p}{4-p}}(B_1)}\|Q\chi \|_{L^4(B_1)}\|\psi \|_{L^4(B_1)}. \end{split} \end{equation*}$$

As before assume that $\|\nabla \eta \|_{L^{\frac{4p}{4-p}}(B_1)}$ and $\|\Delta \eta \|_{L^{\frac{4p}{4-p}}(B_1)}$ are bounded by $C'=C'(U_1,p)$. Collecting the terms, we get

$$\begin{equation*} \begin{split} & \|\Delta v\|_{L^{\frac{4p}{4+p}}(B_1)} \\ & \quad \le \begin{aligned}[t] &\big (|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^2(B_1)}+|A|\|Q\chi \|_{L^4(B_1)}\|\psi \|_{L^4(B_1)}\big ) \|\nabla (\eta \phi )\|_{L^{\frac{4p}{4-p}}(B_1)}\\ & {} +|A||\nabla A|\|\psi \|^3_{L^4(B_1)}\|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)} \\ & {} +C' C_N \big (2\|\nabla \phi \|_{L^2(B_1)}+\|\phi \|_{L^2(B_1)}+|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)} \\ &\qquad \qquad \qquad {} +\|Q\chi \|_{L^4(B_1)}\|\psi \|_{L^4(B_1)}+|A|\|Q\chi \|_{L^4(B_1)}\|\psi \|_{L^4(B_1)}\big ). \end{aligned} \end{split} \end{equation*}$$

By Sobolev embedding,

$$\begin{equation} \begin{split} \lefteqn{\|v\|_{W^{1,\frac{4p}{4-p}}(B_1)}}\qquad & \\ &\le \begin{aligned}[t] & C(p)\big (|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|^2_{L^4(B_1)}\big ) \|\nabla (\eta \phi )\|_{L^{\frac{4p}{4-p}}(B_1)} \\ & {} +C(p)|A||\nabla A|\|\psi \|^3_{L^4(B_1)}\|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)} \\ & {} + 4 C(p) C' C_N\big (|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|^2_{L^4(B_1)}\big ). \end{aligned} \end{split} {\tag{10}} \end{equation}$$

Since $\eta \phi =u+v$, combining 9 and 10, we get

$$\begin{equation} \begin{split} &\|\eta \phi \|_{W^{1,\frac{4p}{4-p}}(B_1)} \\ &\qquad \le \begin{aligned}[t] & C(p)\big (|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|^2_{L^4(B_1)}\big ) \|\nabla (\eta \phi )\|_{L^{\frac{4p}{4-p}}(B_1)} \\ &{} +C(p)|A||\nabla A|\|\psi \|^3_{L^4(B_1)}\|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)} +C(p)\|Q\chi \|_{L^4(B_1)}\|\eta \psi \|_{L^{\frac{2p}{2-p}}(B_1)}\\ &{} +4 C(p) C' C_N\big (|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|^2_{L^4(B_1)}\big ). \end{aligned} \end{split} {\tag{11}} \end{equation}$$

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

$$\begin{equation*} \begin{split} \|\eta \phi \|_{W^{1,\frac{4p}{4-p}}(B_1)} \le &2C(p)\big (|A||\nabla A|\|\psi \|^3_{L^4(B_1)}+\|Q\chi \|_{L^4(B_1)}\big )\|\eta \psi \|_{W^{1,p}(B_1)}\\ & + 8 C(p) C' C_N\big (|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|^2_{L^4(B_1)}\big ). \end{split} \end{equation*}$$

The estimates 7 and 11, together with the small energy assumption, imply that for any $p\in (1,2)$,

$$\begin{equation} \begin{split} &\|\eta \phi \|_{W^{1,\frac{4p}{4-p}}(B_1)}+\|\eta \psi \|_{W^{1,p}(B_1)}\\ &\qquad \le C(p,\eta ,N)\big (|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|^2_{L^4(B_1)}\big ). \end{split} {\tag{12}} \end{equation}$$

Note that as $p\nearrow 2$, $\frac{4p}{4-p}\nearrow 4$. Thus, $\eta \phi$ is almost a $W^{1,4}$ map, and $\eta \psi$ is almost a $W^{1,2}$ vector spinor.

Now $\chi \in W^{1,\frac{4}{3}}$; thus in the equations for the map $\phi$, the divergence terms can be reconsidered. Take another cutoff function, still denoted by $\eta$, such that $0\le \eta \le 1$, $\eta |_{U_2}\equiv 1$, and $\operatorname {supp}\eta \subset U_1$. Then $\eta \phi$ satisfies equations of the same form as 8, and $\operatorname {div}(\eta V^i)\in L^p(B_1)$ for any $p\in [1,\frac{4}{3})$. For example, we take $p=\frac{8}{7}$. Then

$$\begin{equation*} \|\operatorname {div}(\eta V^i)\|_{L^{\frac{8}{7}}(B_1)} \le C(\eta )\|\psi \|_{W^{1,\frac{8}{5}}(U_1)}\|Q\chi \|_{W^{1,\frac{4}{3}}(B_1)}, \end{equation*}$$

and note that $\|\psi \|_{W^{1,\frac{8}{5}}(U_1)}$ is under control by 12. Recalling 8 we have the estimate

$$\begin{equation*} \begin{split} &\|\Delta (\eta \phi )\|_{L^{\frac{8}{7}}(B_1)} \\ &\quad \le \begin{aligned}[t] & |A|\|\nabla \phi \|_{L^2(B_1)}\|\nabla (\eta \phi )\|_{L^{\frac{8}{3}}(B_1)} +|A|^2\|\psi \|^2_{L^4(B_1)}\|\nabla (\eta \phi )\|_{L^{\frac{8}{3}}(B_1)}\\ &{} +|A||\nabla A|\|\psi \|^3_{L^4(B_1)}\|\eta \psi \|_{L^8(B_1)} +|A|\|Q\chi \|_{L^4(B_1)}\|\psi \|_{L^4(B_1)}\|\nabla (\eta \phi )\|_{L^{\frac{8}{3}}(B_1)} \\ &{} +\|\operatorname {div}(\eta V)\|_{L^{\frac{8}{7}}(B_1)}+2\|\nabla \eta \|_{L^{\frac{8}{3}}(B_1)}\|\nabla \phi \|_{L^2(B_1)} +\|\Delta \eta \|_{L^{\frac{8}{3}}(B_1)}\|\phi \|_{L^2(B_1)} \\ &{} +|A|\|\nabla \phi \|_{L^2(B_1)}\|\phi \nabla \eta \|_{L^{\frac{8}{3}}(B_1)} +|A|^2\|\psi \|^2_{L^4(B_1)}\|\phi \nabla \eta \|_{L^{\frac{8}{3}}(B_1)} \\ &{} +\|\nabla \eta \|_{L^{\frac{8}{3}}(B_1)}\|Q\chi \|_{L^4(B_1)}\|\psi \|_{L^4(B_1)} +|A|\|\phi \nabla \eta \|_{L^{\frac{8}{3}}(B_1)}\|Q\chi \|_{L^4(B_1)}\|\psi \|_{L^4(B_1)}. \end{aligned} \end{split} \end{equation*}$$

As before we assume that $\|\nabla \eta \|_{L^{\frac{8}{3}}(B_1)}$ and $\|\Delta \eta \|_{L^{\frac{8}{3}}(B_1)}$ are bounded by $C''= {}C''(U_2,U_1)$. Then

$$\begin{equation*} \begin{split} \|\Delta (\eta \phi )\|_{L^{\frac{8}{7}}(B_1)} \le & \left(|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|^2_{L^4(B_1)}\right) \|\nabla (\eta \phi )\|_{L^{\frac{8}{3}}(B_1)} \\ &+|A||\nabla A|\|\psi \|^3_{L^4(B_1)}\|\eta \psi \|_{L^8(B_1)} \!+C(\eta )\|\psi \|_{W^{1,\frac{8}{5}}(U_1)}\|Q\chi \|_{W^{1,\frac{4}{3}}(B_1)} \\ &+4 C_N C''\left(|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|^2_{L^4(B_1)}\right). \end{split} \end{equation*}$$

By the smallness assumptions and the $L^p$ theory for Laplacian operator (here $p=\frac{8}{7}$) we get

$$\begin{equation*} \begin{split} \|\eta \phi \|_{W^{2,\frac{8}{7}}(B_1)} \le & C(p,U_2,N)\left(|A|\|\nabla \phi \|_{L^2(B_1)}+|A|^2\|\psi \|^2_{L^4(B_1)}+\|Q\chi \|^2_{W^{1,\frac{4}{3}}(B_1)}\right). \end{split} \end{equation*}$$

One can check that similar estimates hold for $\|\eta \phi \|_{W^{1,p}(B_1)}$ for any $p\in (1,\frac{4}{3})$. This accomplishes the proof.

■

Recall the Sobolev embeddings

$$\begin{equation*} W^{2,\frac{8}{7}}_0(B_1)\hookrightarrow W^{1,\frac{8}{3}}_0(B_1)\hookrightarrow C^{1/4}_0(B_1). \end{equation*}$$

Thus we see that the map $\phi$ is Hölder continuous with

$$\begin{equation*} \|\eta \phi \|_{C^{1/4}(B_1)}\le C\|\eta \phi \|_{W^{2,\frac{8}{7}}(B_1)}. \end{equation*}$$

In particular, when the energies of $(\phi ,\psi )$ and certain norms of the gravitino are small, say smaller than $\varepsilon$ (where $\varepsilon \le \varepsilon _1$), the $\frac{1}{4}$-Hölder norm of the map in the interior is also small, with the estimate

$$\begin{equation} \|\phi \|_{C^{1/4}(U)}\le C(N,U, |A|)\sqrt {\varepsilon }. {\tag{13}} \end{equation}$$

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 7, Theorem 3.1, 11, Lemma 4.1, 4, Lemmas 4.8 and 4.9 and 24, Proposition 5.2.

Proposition 2.2 (Energy gap property).

Suppose that $(\phi ,\psi )$ is a solution to 1 defined on an oriented closed surface $(M,g)$ with target manifold $(N,h)$. Suppose that the spinor bundle $S\to (M,g)$ doesn’t admit any nontrivial harmonic spinors. Then there exists an $\varepsilon _0=\varepsilon _0(M,g,A)\in (0,1)$ such that if

$$\begin{equation} E(\phi ,\psi )+\|Q\chi \|_{W^{1,\frac{4}{3}}(M)}\le \varepsilon _0, {\tag{14}} \end{equation}$$

then $(\phi ,\psi )$ 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 $\mathbb{S}^2$ with arbitrary Riemannian metric and the torus $\mathbb{T}^2$ with a nontrivial spin structure, and many others. For more information on harmonic spinors one can refer to 217.

Proof of Proposition 2.2.

When the spinor bundle $S$ doesn’t admit nontrivial harmonic spinors, the Dirac operator is “invertible”, in the sense that for any $1<p<\infty$, there holds

$$\begin{equation*} \|\sigma \|_{L^p(M)}\le C(p)\|\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}\sigma \|_{L^p(M)} \quad \forall \sigma \in \Gamma (S). \end{equation*}$$

See e.g. 11 for a proof.⁠³ As $\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}$ is an elliptic operator of first order, one has

³

There they show a proof for $p=\frac{4}{3}$, but it is easily generalized to a general $p\in (1,\infty )$.

✖

$$\begin{equation*} \|\nabla ^s\psi ^a\|_{L^{\frac{8}{5}}(M)}\le C\left(\|\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}\psi ^a\|_{L^{\frac{8}{5}}(M)}+\|\psi ^a\|_{L^{\frac{8}{5}}(M)}\right),\qquad 1\le a\le K. \end{equation*}$$

It follows that

$$\begin{equation} \|\psi \|_{W^{1,\frac{8}{5}}(M)}\le C\|\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}\psi \|_{L^{\frac{8}{5}}(M)}+|A|\|\psi \|_{L^4(M)}\|\nabla \phi \|_{L^{\frac{8}{3}}(M)}. {\tag{15}} \end{equation}$$

From 3 one gets

$$\begin{equation*} \begin{split} \|\mathchoice{\vcenter{\img[][5pt][9pt][{$\displaystyle\slashed{\partial}$}]{Images/img6da7275ff54ccc8724b95c1dd7733110.svg}}}{\vcenter{\img[][5pt][9pt][{$\textstyle\slashed{\partial}$}]{Images/img324cd598be8821aa08bb275862371607.svg}}}{\vcenter{\img[][4pt][7pt][{$\scriptstyle\slashed{\partial}$}]{Images/imgfa0aaac828e8730190b22e2706a50699.svg}}}{\vcenter{\img[][4pt][6pt][{$\scriptscriptstyle\slashed{\partial}$}]{Images/imge5e5947b6e26cdf299de1ebb62b2adcc.svg}}}\psi \|_{L^{\frac{8}{5}}(M)} \le & |A|\|\nabla \phi \|_{L^2(M)}\|\psi \|_{L^8(M)} +\|Q\chi \|^2_{L^4(M)}\|\psi \|_{L^8(M)} \\ & \quad +|A|^2\|\psi \|^2_{L^4(M)}\|\psi \|_{L^8(M)}+\|Q\chi \|_{L^4(M)}\|\nabla \phi \|_{L^{\frac{8}{3}}(M)}. \end{split} \end{equation*}$$

Since 14 holds, using 15 one obtains

$$\begin{equation} \|\psi \|_{W^{1,\frac{8}{5}}(M)}\le C\left(\|Q\chi \|_{L^4(M)}+\|\psi \|_{L^4(M)}\right)\|\nabla \phi \|_{L^{\frac{8}{3}}(M)}. {\tag{16}} \end{equation}$$

Next we deal with the map $\phi$. From 2 it follows that

$$\begin{equation*} \begin{split} \lefteqn{\|\Delta \phi \|_{L^{\frac{8}{7}}(M)} \le |A|\|\nabla \phi \|_{L^2(M)}\|\nabla \phi \|_{L^\frac{8}{3}(M)} +|A|^2\|\psi \|^2_{L^4(M)}\|\nabla \phi \|_{L^{\frac{8}{3}}(M)}}\qquad & \\ &{} +|A||\nabla A|\|\psi \|^3_{L^4(M)}\|\psi \|_{L^8(M)} +\left(\|\widehat{\nabla }Q\chi \|_{L^{\frac{4}{3}}(M)}+C\|Q\chi \|_{L^{\frac{4}{3}}(M)}\right)\|\psi \|_{L^8(M)} \\ &{} +\|Q\chi \|_{L^4(M)}\|\widetilde{\nabla }\psi \|_{L^{\frac{8}{5}}(M)} +|A|\|Q\chi \|_{L^4(M)}\|\psi \|_{L^4(M)}\|\nabla \phi \|_{L^{\frac{8}{3}}(M)}. \end{split} \end{equation*}$$

Combining with 14 this gives

$$\begin{equation*} \|\nabla \phi \|_{L^{\frac{8}{3}}(M)}\le C\varepsilon _0^{\frac{3}{4}} \|\psi \|_{W^{1,\frac{8}{5}}(M)} \le C\varepsilon _0^{\frac{3}{4}}\left(\|Q\chi \|_{L^4(M)}+\|\psi \|_{L^4(M)}\right)\|\nabla \phi \|_{L^{\frac{8}{3}}(M)}. \end{equation*}$$

Therefore, when $\varepsilon _0$ is sufficiently small, this implies that $\nabla \phi \equiv 0$; that is, $\phi =\mathrm{const}$. Then 16 says that $\psi$ is also trivial.

■

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 $S$ doesn’t admit nontrivial harmonic spinors to obtain the estimate 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 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 $\chi$ is also critical for the action functional with respect to variations; that is, for any smooth family $(\chi _t)_t$ of gravitinos with $\chi _0=\chi$, it holds that

$$\begin{equation*} \frac{\operatorname {d}}{\operatorname {d}t}\Big |_{t=0}\mathbb{A}(\phi ,\psi ;g,\chi _t)=0. \end{equation*}$$

One can conclude from this by direct calculation that the supercurrent $J=J^\alpha \otimes e_\alpha$ vanishes (or see 23), where

$$\begin{equation*} J^\alpha =2\langle \phi _*e_\beta , e_\beta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h}+|\psi |^2 e_\beta \cdot e_\alpha \cdot \chi ^\beta . \end{equation*}$$

Equivalently it can be formulated as

$$\begin{equation*} |\psi |^2 e_\beta \cdot e_\alpha \cdot \chi ^\beta =-2\langle \phi _*e_\beta ,e_\beta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h} \quad \forall \alpha . \end{equation*}$$

Recall that $Q\chi =-\frac{1}{2}e_\beta \cdot e_\alpha \cdot \chi ^\beta \otimes e_\alpha$. Thus

$$\begin{equation} |\psi |^2Q\chi =-\frac{1}{2}|\psi |^2 e_\beta \cdot e_\alpha \cdot \chi ^\beta \otimes e_\alpha =\langle \phi _* e_\beta , e_\beta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h}\otimes e_\alpha . {\tag{17}} \end{equation}$$

It follows that

$$\begin{equation*} \begin{split} |Q\chi |^2|\psi |^2=\langle \chi , |Q\chi |^2\chi \rangle _{\phi ^*h} &=\left\langle \chi ^\eta \otimes e_\eta , \langle \phi _*e_\beta ,e_\beta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h}\otimes e_\alpha \right\rangle _{g_s\otimes g} \\ &=\langle \chi ^\alpha \otimes \phi _*e_\beta , e_\beta \cdot e_\alpha \cdot \psi \rangle _{g_s\otimes \phi ^*h} \\ &=\left\langle e_\alpha \cdot e_\beta \cdot \chi ^\alpha \otimes \phi _* e_\beta ,\psi \right\rangle _{g_s\otimes \phi ^*h} \\ &=-2\left\langle (\mathds{1}\otimes \phi _*)Q\chi ,\psi \right\rangle . \end{split} \end{equation*}$$

Since the Euler–Lagrange equations for $\psi$ are

$$\begin{equation} \mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}\psi =\frac{1}{3}SR(\psi )+|Q\chi |^2\psi +2(\mathds{1}\otimes \phi _*)Q\chi , {\tag{18}} \end{equation}$$

if $\psi$ is critical, i.e., the above equation 18 holds, then

$$\begin{equation*} \langle \psi , \mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}\psi \rangle =\frac{1}{3}\langle SR(\psi ), \psi \rangle =\frac{1}{3}\operatorname {R}(\psi ). \end{equation*}$$

Therefore the following relation holds:

$$\begin{equation*} \langle \psi , e_2\cdot \widetilde{\nabla }_{e_2}\psi \rangle =-\langle \psi , e_1\cdot \widetilde{\nabla }_{e_1}\psi \rangle +\frac{1}{3}\operatorname {R}(\psi ). \end{equation*}$$

Lemma 3.1.

For any $\phi$ and $\psi$ and for any $\beta$,

$$\begin{equation*} e_\beta (|Q\chi |^2|\psi |^2) =2\langle \nabla ^s_{e_\beta }(e_\alpha \cdot e_\eta \cdot \chi ^\alpha )\otimes \phi _* e_\eta , \psi \rangle + 2|Q\chi |^2\langle \psi ,\widetilde{\nabla }_{e_\beta }\psi \rangle . \end{equation*}$$

Proof.

Since

$$\begin{equation*} e_\beta (|Q\chi |^2|\psi |^2)=e_\beta (|Q\chi |^2)|\psi |^2+2|Q\chi |^2 \langle \psi ,\widetilde{\nabla }_{e_\beta }\psi \rangle , \end{equation*}$$

it suffices to compute $e_{\beta }(|Q\chi |^2)|\psi |^2$. Note that

$$\begin{equation*} \begin{split} e_\beta (|Q\chi |^2) &=e_\beta \langle \chi , Q\chi \rangle =-\frac{1}{2} e_\beta \langle \chi ^\alpha , e_\eta \cdot e_\alpha \cdot \chi ^\eta \rangle \\ &=-\frac{1}{2}\big (\langle \nabla ^s_{e_\beta }\chi ^\alpha ,e_\eta \cdot e_\alpha \cdot \chi ^\eta \rangle +\langle \chi ^\alpha ,e_\eta \cdot e_\alpha \cdot \nabla ^s_{e_\beta }\chi ^\eta \rangle \big )\\ &=-\langle \nabla ^s_{e_\beta }\chi ^\alpha , e_\eta \cdot e_\alpha \cdot \chi ^\eta \rangle . \end{split} \end{equation*}$$

Therefore, by virtue of 17,

$$\begin{equation*} \begin{split} e_\beta (|Q\chi |^2)|\psi |^2 &=-\langle \nabla ^s_{e_\beta }\chi ^\alpha , |\psi |^2 e_\eta \cdot e_\alpha \cdot \chi ^\eta \rangle =2\big \langle \nabla ^s_{e_\beta }\chi ^\alpha ,\langle \phi _* e_\eta , e_\eta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h}\big \rangle _{g_s} \\ &=2\langle \nabla ^s_{e_\beta }\chi ^\alpha \otimes \phi _* e_\eta , e_\eta \cdot e_\alpha \cdot \psi \rangle =2\langle \nabla ^s_{e_\beta }(e_\alpha \cdot e_\eta \cdot \chi ^\alpha )\otimes \phi _* e_\eta , \psi \rangle . \end{split} \end{equation*}$$

The desired equality follows.

■

Lemma 3.2.

For any $\phi$ and $\psi$,

$$\begin{equation} \langle (\mathds{1}\otimes \phi _*)Q\chi , \omega \cdot \psi \rangle =0, {\tag{19}} \end{equation}$$

where $\omega =e_1\cdot e_2$ is the volume element.

Proof.

Since

$$\begin{equation*} |\psi |^2 (\mathds{1}\otimes \phi _*)Q\chi =-\frac{1}{2}|\psi |^2 e_\eta \cdot e_\alpha \cdot \chi ^\eta \otimes \phi _*e_\alpha =\langle \phi _* e_\eta , e_\eta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h} \otimes \phi _*e_\alpha , \end{equation*}$$

we have

$$\begin{equation*} \begin{split} |\psi |^2\langle (\mathds{1}\otimes \phi _*)Q\chi , e_1\cdot e_2\cdot \psi \rangle =&\left\langle \langle \phi _* e_\eta , e_\eta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h} \otimes \phi _*e_\alpha , e_1\cdot e_2\cdot \psi \right\rangle \\ =&\big \langle \langle \phi _* e_\eta ,e_\eta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h}, \langle \phi _* e_\alpha , e_1\cdot e_2\cdot \psi \rangle _{\phi ^*h} \big \rangle _{g_s}. \end{split} \end{equation*}$$

According to the Clifford relation it holds that

$$\begin{equation*} \begin{split} \big \langle \langle \phi _* e_\eta ,e_\eta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h}, &\langle \phi _* e_\alpha , e_1\cdot e_2\cdot \psi \rangle _{\phi ^*h} \big \rangle _{g_s} \\ =&\big \langle \langle \phi _* e_\eta ,e_2\cdot e_1\cdot e_\eta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h}, \langle \phi _* e_\alpha , \psi \rangle _{\phi ^*h} \big \rangle _{g_s} \\ =&\big \langle \langle \phi _* e_\eta ,e_\eta \cdot e_\alpha \cdot e_2\cdot e_1\cdot \psi \rangle _{\phi ^*h}, \langle \phi _* e_\alpha , \psi \rangle _{\phi ^*h} \big \rangle _{g_s} \\ =&\big \langle \langle \phi _* e_\eta , e_2\cdot e_1\cdot \psi \rangle _{\phi ^*h}, \langle \phi _* e_\alpha ,e_\alpha \cdot e_\eta \cdot \psi \rangle _{\phi ^*h} \big \rangle _{g_s} \\ =&-\big \langle \langle \phi _* e_\alpha , e_1\cdot e_2\cdot \psi \rangle _{\phi ^*h}, \langle \phi _* e_\eta ,e_\eta \cdot e_\alpha \cdot \psi \rangle _{\phi ^*h} \big \rangle _{g_s}. \end{split} \end{equation*}$$

It follows that

$$\begin{equation*} |\psi |^2\langle (\mathds{1}\otimes \phi _*)Q\chi , e_1\cdot e_2\cdot \psi \rangle =0. \end{equation*}$$

At any point $x\in M$, if $\psi (x)=0$, then 19 holds; and if $|\psi (x)|\neq 0$, then by the calculations above 19 also holds. This finishes the proof.

■

Remark.

More explicitly 19 is equivalent to

$$\begin{equation} \langle e_1\cdot e_2\cdot \chi ^1\otimes \phi _*e_1+\chi ^1\otimes \phi _*e_2-\chi ^2\otimes \phi _*e_1+e_1\cdot e_2\cdot \chi ^2\otimes \phi _*e_2,\psi \rangle =0. {\tag{20}} \end{equation}$$

From 23 we know the energy-momentum tensor is given by $T=T_{\alpha \beta }e^\alpha \otimes e^\beta$ where

$$\begin{equation} \begin{split} T_{\alpha \beta } =&2\langle \phi _*e_\alpha ,\phi _* e_\beta \rangle _{\phi ^*h}-|\operatorname {d}\phi |^2 g_{\alpha \beta } +\frac{1}{2}\left\langle \psi ,e_\alpha \cdot \widetilde{\nabla }_{e_\beta }\psi +e_\beta \cdot \widetilde{\nabla }_{e_\alpha }\psi \right\rangle _{g_s\otimes \phi ^*h} \\ & {}-\langle \psi ,\mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}_g\psi \rangle g_{\alpha \beta } +\langle e_\eta \cdot e_\alpha \cdot \chi ^\eta \otimes \phi _*e_\beta +e_\eta \cdot e_\beta \cdot \chi ^\eta \otimes \phi _*e_\alpha , \psi \rangle _{g_s\otimes \phi ^*h} \\ &\qquad {} +4\langle (\mathds{1}\otimes \phi _*)Q\chi ,\psi \rangle g_{\alpha \beta } + |Q\chi |^2 |\psi |^2 g_{\alpha \beta } +\frac{1}{6}\operatorname {R}(\psi )g_{\alpha \beta }. \end{split} {\tag{21}} \end{equation}$$

Suppose that $(\phi ,\psi )$ satisfies the Euler–Lagrange equations 1 and that the supercurrent $J$ vanishes. Then

$$\begin{equation*} \begin{split} T_{\alpha \beta } =&2\langle \phi _* e_\alpha ,\phi _* e_\beta \rangle -|\operatorname {d}\phi |^2 g_{\alpha \beta } +\frac{1}{2}\langle \psi , e_\alpha \cdot \widetilde{\nabla }_{e_\beta }\psi +e_\beta \cdot \widetilde{\nabla }_{e_\alpha }\psi \rangle -\frac{1}{2}\langle \psi ,\mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}\psi \rangle g_{\alpha \beta } \\ &+\langle e_\eta \cdot e_\alpha \cdot \chi ^\eta \otimes \phi _*e_\beta +e_\eta \cdot e_\beta \cdot \chi ^\eta \otimes \phi _*e_\alpha , \psi \rangle -\langle e_\theta \cdot e_\eta \cdot \chi ^\theta \otimes \phi _* e_\eta ,\psi \rangle g_{\alpha \beta }. \end{split} \end{equation*}$$

Clearly $T$ is symmetric and traceless. We will show it is also divergence free. Before this we rewrite it into a suitable form. Multiplying $\omega =e_1\cdot e_2$ on both sides of equations 18, we get

$$\begin{equation*} e_2\cdot \widetilde{\nabla }_{e_1}\psi -e_1\cdot \widetilde{\nabla }_{e_2}\psi =\frac{1}{3}\omega \cdot SR(\psi )+|Q\chi |^2\omega \cdot \psi +2\omega \cdot (\mathds{1}\otimes \phi _*)Q\chi . \end{equation*}$$

Note that the right hand side is perpendicular to $\psi$:

$$\begin{equation*} \begin{split} \langle \psi , \omega \cdot SR(\psi )\rangle &=R_{ijkl}\langle \psi ^j,\psi ^l\rangle \langle \psi ^i,\omega \cdot \psi ^k\rangle =0,\\ |Q\chi |^2\langle \psi ,\omega \cdot \psi \rangle &=0,\\ \langle 2\omega \cdot (\mathds{1}\otimes \phi _*)Q\chi ,\psi \rangle &=-2\langle (\mathds{1}\otimes \phi _*)Q\chi ,\omega \cdot \psi \rangle =0. \end{split} \end{equation*}$$

Hence $\langle \psi , e_2\cdot \widetilde{\nabla }_{e_1}\psi \rangle -\langle \psi , e_1\cdot \widetilde{\nabla }_{e_2}\psi \rangle =0$. Consequently,

$$\begin{equation*} \frac{1}{2}\langle \psi , e_\alpha \cdot \widetilde{\nabla }_{e_\beta }\psi +e_\beta \cdot \widetilde{\nabla }_{e_\alpha }\psi \rangle =\langle \psi , e_\alpha \cdot \widetilde{\nabla }_{e_\beta }\psi \rangle . \end{equation*}$$

Moreover, by 20,

$$\begin{equation*} \begin{split} &\langle e_\eta \cdot e_1\cdot \chi ^\eta \otimes \phi _*e_2,\psi \rangle -\langle e_\eta \cdot e_2\cdot \chi ^\eta \otimes \phi _* e_1,\psi \rangle \\ &\qquad {}=\langle -\chi ^1\otimes \phi _*e_2-e_1\cdot e_2\cdot \chi ^2\otimes \phi _*e_2-e_1\cdot e_2\cdot \chi ^1\otimes \phi _*e_1+\chi ^2\otimes \phi _*e_1,\psi \rangle =0. \end{split} \end{equation*}$$

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

$$\begin{equation} \begin{split} T_{\alpha \beta } =&2\langle \phi _* e_\alpha ,\phi _* e_\beta \rangle -|\operatorname {d}\phi |^2 g_{\alpha \beta } +\langle \psi , e_\alpha \cdot \widetilde{\nabla }_{e_\beta }\psi \rangle -\frac{1}{2}\langle \psi ,\mathchoice{\vcenter{\img[][8pt][9pt][{$\displaystyle\slashed{D}$}]{Images/imgf41df7f68ae0fd0cb901f8be531468f9.svg}}}{\vcenter{\img[][8pt][9pt][{$\textstyle\slashed{D}$}]{Images/img8d78fb8059ceb4636bd484669a210e2f.svg}}}{\vcenter{\img[][7pt][7pt][{$\scriptstyle\slashed{D}$}]{Images/img80fe900a31d6ed2bcc14afa63d6544ae.svg}}}{\vcenter{\img[][6pt][6pt][{$\scriptscriptstyle\slashed{D}$}]{Images/img81e3071e4ab7b01a817e69d59f54cd30.svg}}}\psi \rangle g_{\alpha \beta } \\ &+2\langle e_\eta \cdot e_\alpha \cdot \chi ^\eta \otimes \phi _*e_\beta , \psi \rangle -\langle e_\theta \cdot e_\eta \cdot \chi ^\theta \otimes \phi _* e_\eta ,\psi \rangle g_{\alpha \beta }. \end{split} {\tag{22}} \end{equation}$$

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

Proposition 3.3.

Let $(\phi ,\psi ,\chi )$ be critical. Then the tensor $T$ given by 21 or equivalently 22 is symmetric, traceless, and covariantly conserved.

Proof.

It remains to show that $T$ is covariantly conserved. Let $x\in M$ and take the normal coordinate at $x$ such that $\nabla e_\alpha (x)=0$. We will show that $\nabla _{e_\alpha }T_{\alpha \beta }(x)=0$. At the point $x$, making use of the Euler–Lagrange equations, one can calculate as follows.

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