Communications in Mathematical Physics

, Volume 350, Issue 1, pp 105–128 | Cite as

The Berry Connection of the Ginzburg–Landau Vortices

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Abstract

We analyze 2-dimensional Ginzburg–Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large area limit.

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References

  1. AA87.
    Aharonov Y., Anandan J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596 (1987)ADSMathSciNetCrossRefGoogle Scholar
  2. AK02.
    Aranson I.S., Kramer L.: The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. B06.
    Baptista J.M.: Vortex equations in abelian gauged sigma-models. Commun. Math. Phys. 261, 161–194 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. B11.
    Baptista J.M.: On the L2-metric of vortex moduli spaces. Nucl. Phys. B. 844, 308–333 (2011)ADSCrossRefMATHGoogle Scholar
  5. B84.
    Berry M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 392(1802), 45–57 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. B90.
    Bradlow S.B.: Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys. 135(1), 1–17 (1990)ADSCrossRefMATHGoogle Scholar
  7. BR14.
    Bokstedt M., Romao N.M.: On the curvature of vortex moduli spaces. Math. Z. 277, 549–573 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. CM05.
    Chen H.Y., Manton M.S.: The Kähler potential of abelian Higgs vortices. J. Math. Phys. 46, 052305 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. DDM13.
    Dorigoni D., Dunajski M., Manton N.S.: Vortex motion on surfaces of small curvature. Ann. Phys. 339, 570–587 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. EN11.
    Etesi G., Nagy Á..: S-duality in abelian gauge theory revisited. J. Geom. Phys. 61, 693–707 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. FU91.
    Freed, D.S., Uhlenbeck, K.K.: Instantons and FOUR-MANIFOlds, Mathematical Sciences Research Institute Publications. vol. 1, 2nd edn. Springer, New York (1991)Google Scholar
  12. GL50.
    Ginzburg V.L., Landau L.D.: On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950)Google Scholar
  13. H02.
    Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  14. HJS96.
    Hong, M., Jost, J. and Struwe, M.: Asymptotic limits of a Ginzburg–Landau type functional. In: Geometric analysis and the calculus of variations, pp. 99–123. Int. Press, Cambridge (1996)Google Scholar
  15. I01.
    Ivanov D.A.: Non-abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86(2), 268 (2001)ADSCrossRefGoogle Scholar
  16. JT80.
    Jaffe, A., Taubes, C.H.: Vortices and Monopoles. Progress in Physics. Birkhäuser, Boston, Mass (1980)Google Scholar
  17. K50.
    Kato T.: On the Adiabatic Theorem of Quantum Mechanics. J. Phys. Soc. Jpn. 5(6), 435–439 (1950)ADSCrossRefGoogle Scholar
  18. K85.
    Kohmoto M.: Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160(2), 343–354 (1985)ADSMathSciNetCrossRefGoogle Scholar
  19. KN63.
    Kobayashi S., Nomizu K.: Foundations of Differential Geometry. Vol I. Interscience Publishers, New York-London (1963)MATHGoogle Scholar
  20. MM15.
    Maldonado R., Manton N.S.: Analytic vortex solutions on compact hyperbolic surfaces. J. Phys. A. 48(24), 245403 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. MN99.
    Manton N.S., Nasir S.M.: Volume of vortex moduli spaces. Commun. Math. Phys. 199(3), 591–604 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. MS03.
    Manton N.S., Speight J.M.: Asymptotic interactions of critically coupled vortices. Commun. Math. Phys. 236, 535–555 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. R00.
    Mundet i Riera, I.: A Hitchin–Kobayashi correspondence for Kähler fibrations. J. Reine Angew. Math. 528, 41–80 (2000)Google Scholar
  24. T84.
    Taubes C.H.: On the Yang–Mills–Higgs equations. Bull. Am. Math. Soc. (N.S.). 10(2), 295–297 (1984)MathSciNetCrossRefMATHGoogle Scholar
  25. T99.
    Taubes C.H.: GR = SW: counting curves and connections. J. Differ. Geom. 52(3), 453–609 (1999)MathSciNetMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

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