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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 via a heat flow method when a Dirichlet boundary condition is imposed on the map and a chiral boundary condition on the spinor. Technically, we solve a new elliptic-parabolic system arising in geometric analysis that is motivated by the nonlinear supersymmetric sigma model of quantum field theory. The corresponding action functional involves two fields, a map from a Riemann surface into a Riemannian manifold and a spinor coupled to the map. The first field has to satisfy a second order elliptic system, which we turn into a parabolic system so as to apply heat flow techniques. The spinor, however, satisfies a first order Dirac type equation. We carry that equation as a nonlinear constraint along the flow. In order to solve this system, we adapt the idea of Sacks-Uhlenbeck to raise the integrand of the harmonic map action to a power
Then we develop a general spectrum of methods (Pohozaev identity, three-circle method, blow-up analysis, energy identities, energy decay estimates etc.) for the compactness problem of the space of
We prove generalized energy identities for both the map part and the spinor part. We also show that the map parts of the