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Journal ArticleDOI

Some aspects of Dirac-harmonic maps with curvature term

TL;DR: In this article, several geometric and analytic aspects of Dirac-harmonic maps with curvature term from closed Riemannian surfaces were studied, and the curvature terms were analyzed.
Abstract: We study several geometric and analytic aspects of Dirac-harmonic maps with curvature term from closed Riemannian surfaces.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the existence of a global weak solution of the heat flow for Dirac-harmonic maps from compact Riemann surfaces with boundary when the energy of the initial map and the L 2 -norm of the boundary values of the spinor are sufficiently small.

30 citations

10 Nov 2015
TL;DR: In this article, the authors derived a new exponential decay estimate of some weighted energy on neck domains for the spinor field based on some Hardy-type inequality and analyzed the blow-up of solutions.
Abstract: The mathematical version of the action functional of the nonlinear supersymmetric model of quantum field theory couples a map from a Riemann surface into a Riemannian manifold with a spinor field along the map. While a simplified version of the model, the so-called Dirac-harmonic map functional, has been extensively studied in the literature in recent years, the full model involves an additional curvature term. Handling the finer analytic aspects caused by this term requires new methods. These are developed in this paper. We analyze the blow-up of solutions. In particular, we show that the energy identities and no neck property hold during the blow-up process. In technical terms, we derive a new exponential decay estimate of some weighted energy on neck domains for the spinor field. This is based on some Hardy-type inequality.

20 citations


Cites background or result from "Some aspects of Dirac-harmonic maps..."

  • ...[2] V. Branding, Some aspects of Dirac-harmonic maps with curvature term, Differential Geometry and its Applications, 40 (2015), 1–13....

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  • ...9) are different from those in [2]) , we nevertheless need to go through all the details here....

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  • ...1) in [2] for a similar tensor where, however, the real part of the third term 〈ψ, eα · ∇̃eβψ〉 is not taken....

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  • ...The proof we give here is different from that in [2]....

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  • ...[3] V. Branding, Energy estimates for the supersymmetric nonlinear sigma model and applications, Potential Anal....

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Journal ArticleDOI
TL;DR: In this article, the gradient and energy estimates for critical points of the full supersymmetric sigma model were derived and several applications were discussed, including the use of the sigma models in particle physics.
Abstract: We derive gradient and energy estimates for critical points of the full supersymmetric sigma model and discuss several applications.

12 citations

Journal ArticleDOI
TL;DR: In this paper, a vanishing result for critical points of the supersymmetric nonlinear sigma model on complete non-compact Riemannian manifolds of positive Ricci curvature that admit an Euclidean type Sobolev inequality was proved.

10 citations


Cites background from "Some aspects of Dirac-harmonic maps..."

  • ...Its full action contains additional terms: Considering also a curvature term that is quartic in the spinors one is led to Dirac-harmonic maps with curvature term, see [3, 11] and also [7]....

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Journal ArticleDOI
TL;DR: In this article, the existence and multiplicity of Dirac-harmonic maps with superquadratic nonlinearities was shown for the case of arbitrary closed spin manifold and the target is a flat tori.
Abstract: Dirac-harmonic map is a mathematical model of supersymmetric $$\sigma $$ -model arising in quantum filed theory. In this paper, we prove the existence and multiplicity of Dirac-harmonic maps with superquadratic nonlinearities when the domain is an arbitrary closed spin manifold and the target is a flat tori. For general superquadratic nonlinearities, we show that there exist at least $$n+1$$ (irreducible) solutions in each homotopy class, where n is the dimension of the tori. In the case of odd nonlinearities, we show that there exist infinitely many (irreducible) solutions in each homotopy class.

9 citations

References
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Journal ArticleDOI
TL;DR: In this article, the theory of a self-coupled spinors field with L = λ ψ ψ Έ ω ψ (π) is solved in one space and one time dimension, and a representation of the eigenstates of the energy is given.

730 citations


"Some aspects of Dirac-harmonic maps..." refers background in this paper

  • ...Moreover, it should be noted that this equation also appears in the context of the spinorial representation of surfaces in R3 [16] and the Thirring model in quantum field theory [21]....

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Journal ArticleDOI

673 citations


"Some aspects of Dirac-harmonic maps..." refers methods in this paper

  • ...In addition, we want to apply Adam’s inequality on Morrey spaces (see [1], Theorem 3....

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  • ...In particular, if f = 0, then φ ∈ W 2,p loc for all p ∈ [1, 2) and φ ∈ W 1,q loc for all q ∈ [1,∞)....

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Book
01 Jan 2002
TL;DR: In this article, the authors present Harmonic maps with symmetries, Harmonic map with symmetric and non-symmetric functions, compensations and exotic function spaces, and Surfaces with mean curvature.
Abstract: Preface Introduction Acknowledgements Notations 1. Geometric and analytic setting 2. Harmonic maps with symmetries 3. Compensations and exotic function spaces 4. Harmonic maps without symmetries 5. Surfaces with mean curvature in L2 References.

379 citations


"Some aspects of Dirac-harmonic maps..." refers background in this paper

  • ...Using an estimate for harmonic functions (see [17], Lemma 3....

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Journal ArticleDOI
Henry C. Wente1

323 citations


"Some aspects of Dirac-harmonic maps..." refers methods in this paper

  • ...Using the symmetries of the target Sn it is possible to write the equation for φ in such a form that one may apply the classical Wente Lemma [24] yielding the continuity of the map φ similar to [11], Prop....

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  • ...Using the symmetries of the target S it is possible to write the equation for φ in such a form that one may apply the classical Wente Lemma [24] yielding the continuity of the map φ similar to [11], Prop....

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Book
01 May 1999
TL;DR: In this paper, Lagrangians and symmetries Supersymmetry in various dimensions Theories with two supersymmetries and theories with more supersymmetry References Index.
Abstract: What are fermions? Lagrangians and symmetries Supersymmetry in various dimensions Theories with two supersymmetries Theories with more supersymmetry References Index.

150 citations


"Some aspects of Dirac-harmonic maps..." refers background in this paper

  • ...However, the full nonlinear supersymmetric σ-model in physics contains additional terms, which are not captured by the analysis of Dirac-harmonic maps, see [15], [8] for the physical background....

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