Existence of solutions of a mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor
Jürgen Jost, Lei Liu, and Miaomiao Zhu
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Submission date: 08. Jun. 2017 (revised version: October 2018)
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 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 ﬁelds 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.
With this novel scheme, in more technical terms, we can 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.