Analysis and Mathematical Physics

, Volume 5, Issue 1, pp 23–37 | Cite as

Magnetic Dirac-harmonic maps

  • Volker BrandingEmail author


We study a functional, whose critical points couple Dirac-harmonic maps from surfaces with a two form. The critical points can be interpreted as coupling the prescribed mean curvature equation to spinor fields. On the other hand, this functional also arises as part of the supersymmetric sigma model in theoretical physics. In two dimensions it is conformally invariant. We call critical points of this functional magnetic Dirac-harmonic maps. We study geometric and analytic properties of magnetic Dirac-harmonic maps including their regularity and the removal of isolated singularities.


Magnetic Dirac-harmonic map Regularity Removable singularity 

Mathematics Subject Classification (2000)

53C27 58E20 


  1. 1.
    Alvarez, O., Singer, I.M.: Beyond the elliptic genus. Nucl. Phys. B 633(3), 309–344 (2002)Google Scholar
  2. 2.
    Ammann, B., Ginoux, N.: Dirac-harmonic maps from index theory. Calc. Var. Partial Differ. Equ. 47(3–4), 739–762 (2013)Google Scholar
  3. 3.
    Bethuel, F.: Un résultat de régularité pour les solutions de l’équation de surfaces à courbure moyenne prescrite. C. R. Acad. Sci. Paris Sér. I Math. 314(13), 1003–1007 (1992)Google Scholar
  4. 4.
    Branding, V.: The evolution equations for Dirac-harmonic maps, Ph.D. thesis, Universitaet Potsdam (2013)Google Scholar
  5. 5.
    Chen, Q., Jost, J., Wang, G.: Liouville theorems for Dirac-harmonic maps. J. Math. Phys. 48(11), 113517, 13 (2007)Google Scholar
  6. 6.
    Chen, Q., Jost, J., Wang, G., Zhu, M.: The boundary value problem for Dirac-harmonic maps. J. EMS 15(3), 997–1031 (2013)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chen, Q., Jost, J., Li, J., Wang, G.: Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z. 251(1), 61–84 (2005)Google Scholar
  8. 8.
    Chen, Q., Jost, J., Li, J., Wang, G.: Dirac-harmonic maps. Math. Z. 254(2), 409–432 (2006)Google Scholar
  9. 9.
    Chen, Q., Jost, J., Wang, G.: Nonlinear Dirac equations on Riemann surfaces. Ann. Glob. Anal. Geom. 33(3), 253–270 (2008)Google Scholar
  10. 10.
    Chen, Q., Jost, J., Wang, G.: The maximum principle and the Dirichlet problem for Dirac-harmonic maps. Calc. Var. Partial Differ. Equ. 47(1–2), 87–116 (2013)Google Scholar
  11. 11.
    Choné, P.: A regularity result for critical points of conformally invariant functionals. Potential Anal. 4(3), 269–296 (1995)Google Scholar
  12. 12.
    Fuchs, J., Nikolaus, T., Schweigert, C., Waldorf, K.: Bundle gerbes and surface holonomy. In: European Congress of Mathematics. Eur. Math. Soc., Zürich, pp. 167–195 (2010)Google Scholar
  13. 13.
    Grüter, M.: Regularity of weak \(H\)-surfaces. J. Reine Angew. Math. 329, 1–15 (1981)Google Scholar
  14. 14.
    Grüter, M.: Conformally invariant variational integrals and the removability of isolated singularities. Manuscr. Math. 47(1–3), 85–104 (1984)Google Scholar
  15. 15.
    Hélein, F.: Harmonic maps, conservation laws and moving frames. In: Cambridge Tracts in Mathematics, vol. 150, 2nd edn. Cambridge University Press, Cambridge (2002). Translated from the 1996 French original, with a foreword by James EellsGoogle Scholar
  16. 16.
    Hull, C.M., Papadopoulos, G., Townsend, P.K.: Potentials for \((p,0)\) and \((1,1)\) supersymmetric sigma models with torsion. Phys. Lett. B 316(2–3), 291–297 (1993)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Koh, D.: The evolution equation for closed magnetic geodesics. Dissertation, Universitätsverlag Potsdam (2008)Google Scholar
  18. 18.
    Lawson, H.B., Jr., Michelsohn, M.-L.: Spin geometry. In: Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1989)Google Scholar
  19. 19.
    Rivière, T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168(1), 1–22 (2007)Google Scholar
  20. 20.
    Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of \(2\)-spheres. Ann. Math. (2) 113(1), 1–24 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Schneider, M.: Closed magnetic geodesics on \(S^2\). J. Differ. Geom. 87(2), 343–388 (2011)Google Scholar
  22. 22.
    Sharp, B., Zhu, M.: Regularity at the free boundary for Dirac-harmonic maps from surfaces. arXiv:1306.4260 (2013)
  23. 23.
    Waldorf, K.: Surface holonomy, handbook of pseudo-Riemannian geometry and supersymmetry. IRMA Lect. Math. Theor. Phys., vol. 16. Eur. Math. Soc., Zürich, pp. 653–682 (2010)Google Scholar
  24. 24.
    Wang, C., Xu, D.: Regularity of Dirac-harmonic maps. Int. Math. Res. Not. IMRN 2009(20), 3759–3792 (2009)Google Scholar
  25. 25.
    Xu, D., Chen, Z.: Regularity for Dirac-harmonic map with Ricci type spinor potential. Calc. Var. Partial Differ. Equ. 46(3–4), 571–590 (2013)Google Scholar
  26. 26.
    Zhao, L.: Energy identities for Dirac-harmonic maps. Calc. Var. Partial Differ. Equ. 28(1), 121–138 (2007)Google Scholar
  27. 27.
    Zhu, M.: Regularity for weakly Dirac-harmonic maps to hypersurfaces. Ann. Glob. Anal. Geom. 35(4), 405–412 (2009)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und GeometrieWienAustria

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