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Gravitating Vortices, Cosmic Strings, and the Kähler–Yang–Mills Equations

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Abstract

In this paper we construct new solutions of the Kähler–Yang–Mills equations, by applying dimensional reduction methods to the product of the complex projective line with a compact Riemann surface. The resulting equations, which we call gravitating vortex equations, describe abelian vortices on the Riemann surface with back reaction of the metric. As a particular case of these gravitating vortices on the Riemann sphere we find solutions of the Einstein–Bogomol’nyi equations, which physically correspond to Nielsen–Olesen cosmic strings in the Bogomol’nyi phase. We use this to provide a Geometric Invariant Theory interpretation of an existence result by Y. Yang for the Einstein–Bogomol’nyi equations, applying a criterion due to G. Székelyhidi.

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References

  1. Álvarez-Cónsul L., García-Fernández M., García-Prada O.: Coupled equations for Kähler metrics and Yang–Mills connections. Geom. Top. 17, 2731–2812 (2013)

    Article  MATH  Google Scholar 

  2. Álvarez-Cónsul, L., García-Fernández, M., García-Prada, O.: Gravitating vortices and the Einstein–Bogomol’nyi equations. Preprint arXiv:1606.07699 [math.DG] (2016)

  3. Atiyah M.F., Bott R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. A 308, 523–615 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Bradlow S.: Vortices in Holomorphic Line Bundles Over Closed Kähler Manifolds. Commun. Math. Phys. 135, 1–17 (1990)

    Article  ADS  MATH  Google Scholar 

  5. Chen X., Hastings S., McLeod J., Yang Y.: A nonlinear elliptic equation arising from gauge field theory and cosmology. Proc. R. Soc. Lond. Series. A 446, 453–458 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Comtet A., Gibbons G.: Bogomol’nyi bounds for cosmic strings. Nuc. Phys. B299, 719–733 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  7. Donaldson S.K.: A new proof of a theorem of Narasimhan and Seshadri. J. Differ. Geom. 18, 269–277 (1983)

    MathSciNet  MATH  Google Scholar 

  8. Donaldson S.K.: Anti-self-dual Yang–Mills connections on a complex algebraic surface and stable vector bundles. Proc. Lond. Math. Soc. 50, 1–26 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Donaldson S.K. : Remarks on gauge theory, complex geometry and 4-manifold topology. In: Atiyah, I. (eds) Fields Medallists’ Lectures, pp. 384–403. World Scientific, Singapore (1997)

  10. Fujiki A.: Moduli space of polarized algebraic manifolds and Kähler metrics. Sugaku Expos. 5, 173–191 (1992)

    MATH  Google Scholar 

  11. Garcia-Fernandez, M.: Coupled equations for Kähler metrics and Yang–Mills connections. PhD Thesis. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Madrid. arXiv:1102.0985 [math.DG] (2009)

  12. Garcia-Fernandez M., Tipler C.: Deformation of complex structures and the Coupled Kähler–Yang–Mills equations. J. Lond. Math. Soc. (2) 89, 779–796 (2013)

    Article  MATH  Google Scholar 

  13. García-Prada O.: Invariant connections and vortices. Commun. Math. Phys. 156, 527–546 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. García-Prada O.: A direct existence proof for the vortex equations over a compact Riemann surface. Bull. Lond. Math. Soc. 26, 88–96 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gauduchon, P.: Calabi’s extremal Kähler metrics: an elementary introduction. (2015)

  16. Ginzburg V.L., Landau L.D.: On the theory of superconductivity. Zh. Eksp. Theor. Fiz. 20, 1064 (1950)

    Google Scholar 

  17. Hartshorne R.: Ample vector bundles on curves. Nagoya Math. J. 43, 73–89 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jaffe A.M., Taubes C.H.: Vortices and Monopoles: Structure of Static Gauge Theories. Birkhäuser, Boston (1980)

    MATH  Google Scholar 

  19. Kazdan J.L., Warner F.W.: Curvature functions for compact 2-manifolds. Ann. Math. 99, 14–47 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Keller J., Tonnesen-Friedman C.: Non trivial examples of coupled equations for Kähler metrics and Yang–Mills connections. Cent. Eur. J. Math. (5) 10, 1673–1687 (2012)

    Article  MATH  Google Scholar 

  21. Kibble T.: Topology of cosmic domains and strings. J. Phys. A Math. Gen. 9, 1387 (1976)

    Article  ADS  MATH  Google Scholar 

  22. LeBrun C., Simanca R.: Extremal Kähler metrics and complex deformation theory. Geometr. Funct. Anal. 4, 298–336 (1994)

    Article  MATH  Google Scholar 

  23. Lichnerowicz, A.: Géometrie des groupes de transformation. Travaux et Recherches Mathématiques 3, Dunod (1958)

  24. Linet B.: A vortex-line model for infinite straight cosmic strings in equilibrium. Phys. Lett. A 124, 240–242 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  25. Linet B.: On the supermassive U(1) gauge cosmic strings. Class. Quantum Grav. 7, L75–L79 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Mumford D., Fogarty J., Kirwan F.: Geometric Invariant Theory, Third Enlarged Edition. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  27. Narasimhan M.S., Seshadri C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. 82, 540–567 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nielsen H.B., Olesen P.: Vortex-line models for dual strings. Nuclear Phys. B 61, 45–61 (1973)

    Article  ADS  Google Scholar 

  29. Noguchi M.: Yang–Mills–Higgs theory on a compact Riemann surface. J. Math. Phys. 28, 2343–2346 (1987)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Spruck, J., Yang, Y.: Cosmic String solutions of the Einstein-matter-gauge equations. Research Report No. 92-NA-032, (1992) (unpublished).

  31. Spruck J., Yang Y.: Regular stationary solutions of the cylindrically symmetric Einstein-matter-gauge equations. J. Math. Anal. Appl. 195, 160–190 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Székelyhidi G.: The Kähler-Ricci flow and K-polystability. Amer. J. Math. 132, 1077–1090 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Taubes C.H.: Arbitrary N-vortex solutions to the first order Ginzburg–Landau equations. Commun. Math. Phys. 72(3), 277–292 (1980)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Taubes C.H.: On the Equivalence of the First and Second Order Equations for Gauge Theories. Commun. Math. Phys. 75, 207–227 (1980)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Thomas, R.P.: Notes on GIT and symplectic reduction for bundles and varieties. In: Surveys in Differential Geometry 10: A Tribute to Professor S.-S. Chern. Preprint arXiv:math/0512411 (2006)

  36. Tipler C., van Coevering C.: Deformations of constant scalar curvature Sasakian metrics and K-stability. Int. Math. Res. Not. 22, 11566–11604 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Uhlenbeck K.K., Yau S.-T.: On the existence of hermitian–Yang–Mills connections on stable bundles over compact Kähler manifolds. Commun. Pure and Appl. Math. 39-S, 257–293 (1986)

    Article  Google Scholar 

  38. Uhlenbeck K.K., Yau S.-T.: On the existence of hermitian–Yang–Mills connections on stable bundles over compact Kähler manifolds. Commun. Pure Appl. Math. 42, 703–707 (1989)

    Article  MATH  Google Scholar 

  39. Witten E.: Some exact multipseudoparticle solutions of classical Yang–Mills theory. Phys. Rev. Lett. 38, 121 (1977)

    Article  ADS  Google Scholar 

  40. Yang Y.: An equivalence theorem for string solutions of the Einstein-matter-gauge equations. Lett. Math. Phys. 26, 79–90 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Yang Y.: Obstructions to the existence of static cosmic strings in an Abelian Higgs model. Phys. Rev. Lett. 73, 10–13 (1994)

    Article  ADS  Google Scholar 

  42. Yang Y.: Self duality of the gauge field equations and the cosmological constant. Commun. Math. Phys. 162, 481–498 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Yang Y.: Prescribing topological defects for the coupled Einstein and Abelian Higgs equations. Commun. Math. Phys. 170, 541–582 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Yang Y.: Static cosmic strings on S 2 and criticality. Proc. R. Soc. Lond. A 453, 581–591 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Yang Y.: Solitons in Field Theory and Nonlinear Analysis. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera 13–15, Cantoblanco, 28049, Madrid, Spain

    Luis Álvarez-Cónsul, Mario Garcia-Fernandez & Oscar García-Prada

Authors
  1. Luis Álvarez-Cónsul
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  2. Mario Garcia-Fernandez
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  3. Oscar García-Prada
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Corresponding author

Correspondence to Mario Garcia-Fernandez.

Additional information

Communicated by X. Yin

Partially supported by the Spanish MINECO under the ICMAT Severo Ochoa Grant No. SEV-2011-0087, and under Grant No. MTM2013-43963-P. The work of the second author has been partially supported by the Nigel Hitchin Laboratory under the ICMAT Severo Ochoa Grant. The research leading to these results has received funding from the European Union’s Horizon 2020 Programme (H2020-MSCA-IF-2014) under Grant agreement No. 655162, and by the European Commission Marie Curie IRSES MODULI Programme PIRSES-GA-2013-612534.

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Álvarez-Cónsul, L., Garcia-Fernandez, M. & García-Prada, O. Gravitating Vortices, Cosmic Strings, and the Kähler–Yang–Mills Equations. Commun. Math. Phys. 351, 361–385 (2017). https://doi.org/10.1007/s00220-016-2728-2

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