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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Aharonov Y., Anandan J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596 (1987)
Aranson I.S., Kramer L.: The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002)
Baptista J.M.: Vortex equations in abelian gauged sigma-models. Commun. Math. Phys. 261, 161–194 (2006)
Baptista J.M.: On the L2-metric of vortex moduli spaces. Nucl. Phys. B. 844, 308–333 (2011)
Berry M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 392(1802), 45–57 (1984)
Bradlow S.B.: Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys. 135(1), 1–17 (1990)
Bokstedt M., Romao N.M.: On the curvature of vortex moduli spaces. Math. Z. 277, 549–573 (2014)
Chen H.Y., Manton M.S.: The Kähler potential of abelian Higgs vortices. J. Math. Phys. 46, 052305 (2005)
Dorigoni D., Dunajski M., Manton N.S.: Vortex motion on surfaces of small curvature. Ann. Phys. 339, 570–587 (2013)
Etesi G., Nagy Á..: S-duality in abelian gauge theory revisited. J. Geom. Phys. 61, 693–707 (2011)
Freed, D.S., Uhlenbeck, K.K.: Instantons and FOUR-MANIFOlds, Mathematical Sciences Research Institute Publications. vol. 1, 2nd edn. Springer, New York (1991)
Ginzburg V.L., Landau L.D.: On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950)
Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
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)
Ivanov D.A.: Non-abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86(2), 268 (2001)
Jaffe, A., Taubes, C.H.: Vortices and Monopoles. Progress in Physics. Birkhäuser, Boston, Mass (1980)
Kato T.: On the Adiabatic Theorem of Quantum Mechanics. J. Phys. Soc. Jpn. 5(6), 435–439 (1950)
Kohmoto M.: Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160(2), 343–354 (1985)
Kobayashi S., Nomizu K.: Foundations of Differential Geometry. Vol I. Interscience Publishers, New York-London (1963)
Maldonado R., Manton N.S.: Analytic vortex solutions on compact hyperbolic surfaces. J. Phys. A. 48(24), 245403 (2015)
Manton N.S., Nasir S.M.: Volume of vortex moduli spaces. Commun. Math. Phys. 199(3), 591–604 (1999)
Manton N.S., Speight J.M.: Asymptotic interactions of critically coupled vortices. Commun. Math. Phys. 236, 535–555 (2003)
Mundet i Riera, I.: A Hitchin–Kobayashi correspondence for Kähler fibrations. J. Reine Angew. Math. 528, 41–80 (2000)
Taubes C.H.: On the Yang–Mills–Higgs equations. Bull. Am. Math. Soc. (N.S.). 10(2), 295–297 (1984)
Taubes C.H.: GR = SW: counting curves and connections. J. Differ. Geom. 52(3), 453–609 (1999)
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Communicated by N. A. Nekrasov
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Nagy, Á. The Berry Connection of the Ginzburg–Landau Vortices. Commun. Math. Phys. 350, 105–128 (2017). https://doi.org/10.1007/s00220-016-2701-0
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DOI: https://doi.org/10.1007/s00220-016-2701-0