Asymptotically linear Dirac-harmonic maps into flat tori
type: Item
instance of: scholarly article
title: Asymptotically linear Dirac-harmonic maps into flat tori
zbMATH Open document ID: 1467.58009
author: Takeshi Isobe
published in: Differential Geometry and its Applications
publication date: 2021-04-29T00:00:00Z
review text: 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 = \…
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 81T20
Mathematics Subject Classification ID: 53C27
Mathematics Subject Classification ID: 53C28
zbMATH DE Number: 7341147
zbMATH Keywords: Dirac-harmonic map
zbMATH Keywords: asymptotically linear perturbation
zbMATH Keywords: twisted spinor bundle
zbMATH Keywords: action functional
zbMATH Keywords: Lusternik-Schnirelmann theory
MaRDI profile type: Publication
full work available at URL: j.difgeo.2020.101716
OpenAlex ID: W3119528763
Wikidata QID: Q115354636
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DOI: 10.1016/J.DIFGEO.2020.101716
description: scientific article; zbMATH DE number 7341147