The existence of Dirac-harmonic maps with Dirichlet-chiral boundary conditions
Jürgen Jost, Lei Liu, and Miaomiao Zhu
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Submission date: 08. Jun. 2017
Keywords and phrases: Supersymmetric nonlinear sigma model, Dirac-harmonic maps, $\alpha$-Dirac-harmonic maps, $\alpha$-Dirac-harmonic map flow, Dirichlet-chiral boundary
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In this paper, we show the existence of Dirac-harmonic maps from a compact spin Riemann surface with smooth boundary to a general compact Riemannian manifold when a Dirichlet boundary condition is imposed on the map and a chiral boundary condition on the spinor. Our approach is based on the α-Dirac-harmonic map flow and the blow-up analysis of a sequence of Sacks-Uhlenbeck type approximations for Dirac-harmonic maps. In particular, when the target admits no nontrivial harmonic spheres, then the map part of the limit Dirac-harmonic map is in the same homotopy class of the initial map.