Asymptotically linear Dirac-harmonic maps into flat tori
Let \((M, g, \sigma)\) be a spin manifold with a spin structure \(\sigma\), and let \((N, h)\) be a Riemannian manifold. Consider a map \(\phi: M \to N\) and a twisted spinor \(\psi\) which is a section of the twisted spinor bundle \({\mathbb S}(M)\otimes TN \to M\), where \({\mathbb S}(M)\) is the spin bundle associated with the spin structure \(\sigma\). For such a pair \((\phi, \psi)\), we consider the following perturbed action functional \({\mathcal L}_H\) defined by \[ {\mathcal L}_H(\phi, \psi) = \frac{1}{2}\int_M |d\phi|^2 dv_g + \frac{1}{2}\int_M \langle D_\phi \psi, \psi\rangle dv_g - \int_M H(x, \phi(x), \psi(x))dv_g, \] where \(D_\phi\) is a canonical Dirac operator on the bundle \({\mathbb S}(M) \otimes \phi^*TN\) and \(H\) is a smooth function defined on the external tensor product \({\mathbb S}(M) \dot\times TN \to M \times N\). Critical points of \({\mathcal L}_H\) are called \(H\)-Dirac-harmonic maps. For the special cases \(H \equiv 0\), \(H\)-Dirac-harmonic maps are simply called Dirac-harmonic maps. To get supersymmetry and describe various types of interactions, \(H\) is assumed to be a nontrivial perturbation in this paper. Let \(\mathscr C\) be a free homotopy class of pairs \((\phi, \psi)\), where \(\phi \in C^\infty(M, N)\) and \(\psi \in C^\infty(M, {\mathbb S}(M)\otimes \phi^*TN)\). Since such a class is uniquely determined by the free homotopy class of \(\phi\), for a given free homotopy class \(\alpha\in [M, N]\) we denote by \({\mathscr C}_\alpha\) the corresponding class of pairs \((\phi, \psi)\) with \(\phi \in \alpha\). In this paper, the author studies the existence and compactness problem of perturbed Dirac-harmonic maps into flat tori. When \(N = {\mathbb T}^n_\Lambda:={\mathbb R}^n/\Lambda \,(\Lambda \subset {\mathbb R}^n\) is a lattice of rank \(n\)) with the flat metric, the Euler-Lagrange equations for \({\mathcal L}_H\) are given by \[ \tau(\phi) = \frac{1}{2}R^N(\phi,\psi) - \nabla_\phi H(x, \phi, \psi), \] \[D_\phi \psi = \nabla_\psi H(x, \phi, \psi), \] where \(\tau(\phi)\) is the tension field of \(\phi\) and \[ \sum_{1\le \beta \le m, 1\le i, j \le n}\langle \psi^i, e_\beta\cdot \psi^j\rangle R^{\phi^*TN} \left(\phi^*\left(\frac{\partial}{\partial y^i}\right), \phi^*\left(\frac{\partial}{\partial y^j}\right)\right)\phi_*e_\beta. \] Here \(\{e_\beta\} (1\le \beta\le n = \dim M)\) is a local orthonormal frame of \(TM\), and \(n = \dim N, \phi^j \in {\mathbb S}(M)\) with local coordinate \(y = (y^j)\) on \(N\). The author considers two types of asymptotically linear perturbations, non-resonance and resonance cases. Let \(\alpha \in [M, {\mathbb T}^n_\Lambda] \cong \mathrm{H}^1(M; {\mathbb Z}^n)\). If \(H \in C^{1, \gamma}({\mathbb S}(M)\dot\times T{\mathbb T}^n_\Lambda)\) satisfies some boundedness or growth condition on its gradient, the author proves that there exist at least \(n+1\) distinct \(H\)-Dirac-harmonic maps in \({\mathscr C}_\alpha\) in both cases. Moreover, in the case of non-resonance case, the author shows that the set of \(H\)-Dirac-harmonic maps in \({\mathscr C}_\alpha\) is compact.
scientific article; zbMATH DE number 7341147
scientific article