Dirac-harmonic maps with the trivial index

Motivated by the supersymmetric nonlinear sigma model from quantum field theory, see [6], Dirac-harmonic maps from Riemann surfaces (with a fixed spin structure) into Riemannian manifolds were introduced in [2]. They generalize harmonic maps and harmonic spinors. From the variational point of view, they are critical points of a conformally invariant action functional. The Euler-Lagrange equation then is an elliptic system coupling a harmonic map type equation with a Dirac type equation.

The existence of Dirac-harmonic maps from closed spin manifolds is a very difficult problem. So far, there are only a few results in this direction. Most solutions found so far are uncoupled in the sense that the map part is harmonic. The existence result of [1] for uncoupled solutions depends on the index $I(M,u)$ being non-zero (see Definition 2.2). But when the domain and target are both closed Riemann surfaces, the index $I(M,u)$ always vanishes. In this case, an existence result about uncoupled Dirac-harmonic maps was proved in [3] by the Riemann-Roch formula. Later, this result was generalized to Kähler manifolds in [16]. A general existence result for Dirac-harmonic maps from closed Riemann surfaces to compact manifolds was first established in [9]. This implies the existence of Dirac-harmonic maps when the index $I(M,u)$ is nontrivial.

This then naturally raises the question of the existence of Dirac-harmonic maps in a given homotopy class $[u]$ between manifolds $M,N$ with trivial index $I(M,u)$. Of course, we should first identify conditions under which this index vanishes. When the domain $M$ is a closed Riemann surface with positive genus and the target $N$ is an odd-dimensional oriented manifold, there is always a spin structure on $M$ such that the index $I(M,u)$ is nontrivial. When, in contrast, $M$ is a closed Riemann surface and $N$ is an even-dimensional spin manifold, the index $I(M,u)$ is always zero. Therefore, here we consider the case where the target manifold $N$ is a Kähler spin manifold. In this case, it is necessary to use a new quantity that can replace the index $I(M,u)$. For this purpose, we introduce a candidate that uses the complex structure of the target manifold. More precisely, we first decompose the twisted Dirac operator as

according to the decomposition $(u^{*}TN)^{\mathbb{C}}=u^{*}T_{1,0}N\oplus u^{*}T_{1,0}N$. Then we just consider the kernel of one of the two operators, such as $\not{D}_{1,0}$. We define

for an even dimensional spin manifold $M$ whenever the complex dimension of the kernel of $\not{D}_{1,0}^{u}$ is even.

In order to be useful for our purposes, this should be homotopy invariant. Let us first look at an example. Suppose $M=\mathbb{C}{\rm P}^{1}$ and $N$ is a compact surface. Consider any map $u:M\to N$, the spinor bundle $\Sigma\mathbb{C}{\rm P}^{1}$ and the twisted bundle $\Sigma\mathbb{C}{\rm P}^{1}\otimes u^{*}T_{1,0}N$. Let $g_{N}$ be the genus of $N$ and $c_{1}\left(u^{*}T_{1,0}N\right)=a\gamma,a=2\deg(u)(1-g_{N})$, where $\gamma$ is the tautological bundle of $\mathbb{C}{\rm P}^{1}$. The unique spin structure of $\mathbb{C}{\rm P}^{1}$ is determined by $\gamma$ since $\Lambda^{1,0}\mathbb{C}{\rm P}^{1}=\gamma^{2}$. Then as a holomorphic bundle, we have

Since

we conclude that

Therefore, $\dim_{\mathbb{C}}\ker\not{D}^{u}_{1,0}$ is invariant in the homotopy class $[u]$. This implies the homotopy invariance of $\mathcal{I}(\mathbb{C}{\rm P}^{1},u^{*}T_{1,0}N)$, which is equal to $\left[|{\rm deg}(u)(g_{N}-1)|\right]_{\mathbb{Z}_{2}}$. Moreover, the dimension of the kernel of the Dirac operator is a constant in a given homotopy class. Then the following well-known fact follows from the first variational formula.

In general, we can give two different sufficient conditions to guarantee the homotopy invariance of $\mathcal{I}$.

Here $\alpha$-Dirac-harmonic maps are the critical points of the following functional

They are generalizations of Dirac-harmonic maps (i.e. the case of $\alpha=1$). As generalizations of harmonic maps, $\alpha$-harmonic maps are the critical points of the following functional

which was introduced by Sacks and Uhlenbeck in [14].

By the statement in [5, Theorem 2.2.2], such a commuting real structure in Theorem 1.2 always exists on $M$ if $m=0,6,7({\rm mod}\ 8)$. In particular, when $m=0,6({\rm mod}\ 8)$, we can get the existence of uncoupled Dirac-harmonic maps.

When $M$ is a closed Riemann surface, we can prove the short-time existence of $\alpha$-Dirac-harmonic map flow into a Kähler manifold, which generalizes the result in [10].

The rest of the paper is organized as follows: In Section 2, we recall some facts about Dirac-harmonic maps as well as the Dirac operator. In Section 3, we prove Theorem 1.2 and end this section by showing the density of minimal kernel. In Section 4, under the minimality assumption on the kernel of $\not{D}_{1,0}^{u_{0}}$, we prove the short time existence of $\alpha$-Dirac-harmonic map flow (Theorem 4.2) and the existence of Dirac-harmonic maps (Theorem 4.7). In the Appendix, we solve the constraint equation and prove Lipschitz continuity of the solution with respect to the map.

Let $(M,g)$ be a compact Riemann surface with a fixed spin structure $\chi$. On the complex spinor bundle $\Sigma M$, we denote the Hermitian inner product by $\langle\cdot,\cdot\rangle_{\Sigma M}$. For any $X\in\Gamma(TM)$ and $\xi\in\Gamma(\Sigma M)$, the Clifford multiplication satisfies the following skew-adjointness:

Let $\nabla$ be the Levi-Civita connection on $(M,g)$. There is a unique connection (also denoted by $\nabla$) on $\Sigma M$ compatible with $\langle\cdot,\cdot\rangle_{\Sigma M}$. Choosing a local orthonormal basis $\{e_{\beta}\}_{\beta=1,2}$ on $M$, the usual Dirac operator is defined as $\not{\partial}:=e_{\beta}\cdot\nabla_{\beta}$, where $\beta=1,2$. Here and in the sequel, we use the Einstein summation convention. One can find more about spin geometry in [11].

Let $u$ be a smooth map from $M$ to another compact Riemannian manifold $(N,h)$ of dimension $n\geq 2$. Let $u^{*}TN$ be the pull-back bundle of $TN$ by $u$ and consider the twisted bundle $\Sigma M\otimes_{\mathbb{R}}u^{*}TN$. On this bundle there is a metric $\langle\cdot,\cdot\rangle_{\Sigma M\otimes u^{*}TN}$ induced from the metric on $\Sigma M$ and $u^{*}TN$. Also, we have a connection $\tilde{\nabla}$ on this twisted bundle naturally induced from those on $\Sigma M$ and $u^{*}TN$. In local coordinates $\{y^{i}\}_{i=1,\dots,n}$, the section $\psi$ of $\Sigma M\otimes_{\mathbb{R}}u^{*}TN$ is written as

where each $\psi^{i}$ is a usual spinor on $M$. We also have the following local expression of $\tilde{\nabla}$

where $\Gamma^{i}_{jk}$ are the Christoffel symbols of the Levi-Civita connection of $N$. The Dirac operator along the map $u$ is defined as

which is self-adjoint [7]. Sometimes, we use $\not{D}^{u}$ to distinguish the Dirac operators defined on different maps. In [2], the authors introduced the functional

They computed the Euler-Lagrange equations of $L$:

where $\tau^{m}(u)$ is the $m$-th component of the tension field [7] of the map $u$ with respect to the coordinates on $N$, $\nabla u^{l}\cdot\psi^{j}$ denotes the Clifford multiplication of the vector field $\nabla u^{l}$ with the spinor $\psi^{j}$, and $R^{m}_{lij}$ stands for the components of the Riemann curvature tensor of the target manifold $N$. Denote

We can write (2.2) and (2.3) in the following global form:

and call the solutions $(u,\psi)$ Dirac-harmonic maps from $M$ to $N$.

With the aim to get a general existence scheme for Dirac-harmonic maps, the following heat flow for Dirac-harmonic maps was introduced in [4]:

When $M$ has boundary, the short time existence and uniqueness of (2.6)-(2.7) was shown in [4].

For a closed manifold $M$, the situation is more complicated because one cannot uniquely solve the second equation (2.7) and the kernel of the Dirac operator may jump along the flow. As we stated in the introduction, the short-time existence is only known in the minimal kernel case, i.e. ${\rm dim}_{\mathbb{H}}{\rm ker}\not{D}=1$. However, when the target manifold $N$ is an even-dimensional spin manifold, the index $\alpha(M,u)$ always vanishes for any map $u$ between $M$ and $N$. In order to deal with this case, we utilize the complex structure on $N$. We denote the complexification of $u^{*}TN$ by $(u^{*}TN)^{\mathbb{C}}$. Then we have

The pull-back metric $u^{*}g$ on $u^{*}TN$ could be naturally extended to a Hermitian product on $(u^{*}TN)^{\mathbb{C}}$. Moreover, there is a natural Hermitian product on $\Sigma M\otimes(u^{*}TN)^{\mathbb{C}}$ induced from those on $\Sigma M$ and $(u^{*}TN)^{\mathbb{C}}$, which is denoted by $\langle\cdot,\cdot\rangle_{\Sigma M\otimes(u^{*}TN)^{\mathbb{C}}}$.

For a general even-dimensional spin Riemannian manifold $M$, there is a parallel ${\mathbb{Z}}_{2}$-grading $G\in{\rm End}(\Sigma M)$ given by $G(\psi)=\left(\sqrt{-1}\right)^{m/2}e_{1}\cdot e_{2}\cdot\dotsc\cdot e_{m}\cdot\psi$ for a positively oriented orthonormal local frame $\{e_{1},e_{2},\dotsc,e_{m}\}$ where $m=\dim M$. Thus the spinor bundle can be decomposed as

where $\Sigma^{\pm}M$ are the eigenspaces of $G$ associated to the $\pm 1$, respectively. As $G$ is Hermitian and parallel, the decomposition is orthogonal in the complex sense and parallel. Consequently, we have

where we used $(u^{*}TN)^{\mathbb{C}}=u^{*}T_{1,0}N\oplus u^{*}T_{1,0}N$. Moreover, we also have the following decomposition for the Dirac operator:

where $\not{D}_{1,0}^{\pm}$ (resp. $\not{D}_{0,1}^{\pm}$) is obtained by restricting $\not{D}$ on $\Sigma^{\pm}M\otimes_{\mathbb{C}}u^{*}T_{1,0}N$ (resp. $\Sigma^{\pm}M\otimes_{\mathbb{C}}u^{*}T_{0,1}N$).

By [13], we can isometrically embed $N$ into $\mathbb{R}^{q}$. Then (2.4)-(2.5) is equivalent to the following system:

where $II$ is the second fundamental form of $N$ in $\mathbb{R}^{q}$, and

Here $P(\xi;\cdot)$ denotes the shape operator, defined by $\langle P(\xi;X),Y\rangle=\langle A(X,Y),\xi\rangle$ for $X,Y\in\Gamma(TN)$ and $Re(z)$ denotes the real part of $z\in\mathbb{C}$. Together with the nearest point projection:

where $N_{\delta}:=\{z\in\mathbb{R}^{q}|d(z,N)\leq\delta\}$, we can rewrite the evolution equation (2.6) as an equation in $\mathbb{R}^{q}$.

For future reference, we define

Note that for $u\in C^{1}(M,N)$ and $\psi\in\Gamma(\Sigma M\otimes u^{*}TN)$ we have

for all $p\in M$, where $\{e_{\alpha}\}$ is an orthonormal basis of $T_{p}M$.

Next, let us fix some notations, which will be used in the Section 4 and Appendix. For every $T>0$, we denote by $X_{T}$ the Banach space of bounded maps:

For any map $v\in X_{T}$, the closed ball with center $v$ and radius $R$ in $X_{T}$ is defined by

We denote by $P^{u_{t},v_{s}}=P^{u_{t},v_{s}}(x)$ the parallel transport of $N$ along the unique shortest geodesic from $\pi(u(x,t))$ to $\pi(v(x,s))$. We also denote by $P^{u_{t},v_{s}}$ the inducing mappings

and

We also define

and $\gamma_{t}(x):[0,2\pi]\to\mathbb{C}$ as

In general, we also denote by $\gamma$ the curve $\gamma(x):[0,2\pi]\to\mathbb{C}$ as

for some constant $\Lambda$ to be determined. Then the orthogonal projection onto ${\rm ker}(\not{D}^{\pi\circ u_{t}})$, which is the mapping

can be written via the resolvent by

where $R(\lambda,\not{D}^{\pi\circ u_{t}}):\Gamma_{L^{2}}\to\Gamma_{L^{2}}$ is the resolvent of $\not{D}^{\pi\circ u_{t}}:\Gamma_{W^{1,2}}\to\Gamma_{L^{2}}$.

In the end of this section, we recall the definition of the index.

In this section, we will prove Theorem 1.2 by constructing a commuting quaternionic structure on the twisted bundle $\Sigma M\otimes_{\mathbb{C}}u^{*}T_{1,0}N$ and show the density of the minimal kernel.

In the rest of this section, we will show the density of the minimal kernel. By the definition of $\mathcal{I}(M,u^{*}T_{1,0}N)$, we have

If equality holds above, then we say that $\not{D}_{1,0}^{u}$ has minimal kernel. Using the analyticity of $N$, one can prove the following density result for the minimal kernel.