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 ingeometric analysis that is motivated by the nonlinear supersymmetricsigma model of quantum ﬁeld theory. The corresponding action functionalinvolves two ﬁelds, a map from a Riemann surface into a Riemannianmanifold and a spinor coupled to the map. The ﬁrst ﬁelds has to satisfy asecond order elliptic system, which we turn into a parabolic system so asto apply heat ﬂow techniques. The spinor, however, satisﬁes a ﬁrst orderDirac type equation. We carry that equation as a nonlinear constraintalong the ﬂow.
With this novel scheme, in more technical terms, we can show theexistence of Dirac-harmonic maps from a compact spin Riemann surfacewith smooth boundary to a general compact Riemannian manifold via aheat ﬂow method when a Dirichlet boundary condition is imposed on themap and a chiral boundary condition on the spinor.