Journal of High Energy Physics

, 2018:93 | Cite as

Bulk-boundary correspondence between charged, anyonic strings and vortices

  • Alexander Gußmann
  • Debajyoti Sarkar
  • Nico WintergerstEmail author
Open Access
Regular Article - Theoretical Physics


We discuss a unified framework of dealing with electrically charged, anyonic vortices in 2+1 dimensional spacetimes and extended, anyonic string-like vortices in one higher dimension. We elaborate on two ways of charging these topological objects and point out that in both cases the vortices and strings obey fractional statistics as a consequence of being electrically charged. The statistics of the charged vortices and strings can be obtained from the phase shift of their respective wave-functions under the classic Aharonov-Bohm type experiments. We show that for a manifold with boundary, where one can realize 2+1 dimensional vortices as endpoints of trivially extended 3+1 dimensional strings, there is a smooth limit where the phase shift of a bulk string-vortex goes over to the phase shift of the boundary vortex. This also enables one to read off the bulk statistics (arising essentially from either a QCD theta-type term or an external current along the string) just from the corresponding boundary statistics in a generic setting. Finally, we discuss various applications of these findings, and in particular their prospects for the AdS/CFT duality.


Anyons Holography and condensed matter physics (AdS/CMT) ChernSimons Theories Topological Field Theories 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    W. Pauli, The Connection Between Spin and Statistics, Phys. Rev. 58 (1940) 716 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    F. Wilczek, Quantum Mechanics of Fractional Spin Particles, Phys. Rev. Lett. 49 (1982) 957 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    F. Wilczek and A. Zee, Linking Numbers, Spin and Statistics of Solitons, Phys. Rev. Lett. 51 (1983) 2250 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    D.P. Arovas, J.R. Schrieffer, F. Wilczek and A. Zee, Statistical Mechanics of Anyons, Nucl. Phys. B 251 (1985) 117 [INSPIRE].
  5. [5]
    B. Julia and A. Zee, Poles with Both Magnetic and Electric Charges in Nonabelian Gauge Theory, Phys. Rev. D 11 (1975) 2227 [INSPIRE].
  6. [6]
    H.B. Nielsen and P. Olesen, Vortex Line Models for Dual Strings, Nucl. Phys. B 61 (1973) 45 [INSPIRE].
  7. [7]
    S.K. Paul and A. Khare, Charged Vortices in Abelian Higgs Model with Chern-Simons Term, Phys. Lett. B 174 (1986) 420 [Erratum ibid. B 177 (1986) 453] [INSPIRE].
  8. [8]
    H.J. de Vega and F.A. Schaposnik, Electrically Charged Vortices in Nonabelian Gauge Theories With Chern-Simons Term, Phys. Rev. Lett. 56 (1986) 2564 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    J. Fröhlich and P.A. Marchetti, Quantum Field Theories of Vortices and Anyons, Commun. Math. Phys. 121 (1989) 177 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Y.I. Manin and V.V. Schechtman, Arrangements of hyperplanes, higher braid groups and higher Bruhat orders, Adv. Stud. Pure Math. 17 (1989) 289.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    C. Aneziris, A.P. Balachandran, L. Kauffman and A.M. Srivastava, Novel Statistic for Strings and StringChern-SimonsTerms, Int. J. Mod. Phys. A 6 (1991) 2519 [INSPIRE].
  12. [12]
    C. Aneziris, Statistics of linked strings, Mod. Phys. Lett. A 7 (1992) 3789 [INSPIRE].
  13. [13]
    S.A. Hartnoll, Anyonic strings and membranes in AdS space and dual Aharonov-Bohm effects, Phys. Rev. Lett. 98 (2007) 111601 [hep-th/0612159] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    D. Roychowdhury, Chern-Simons vortices and holography, JHEP 10 (2014) 018 [arXiv:1407.3464] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M.H. Dehghani, A.M. Ghezelbash and R.B. Mann, Vortex holography, Nucl. Phys. B 625 (2002) 389 [hep-th/0105134] [INSPIRE].
  16. [16]
    M. Bergeron, G.W. Semenoff and R.J. Szabo, Canonical bf type topological field theory and fractional statistics of strings, Nucl. Phys. B 437 (1995) 695 [hep-th/9407020] [INSPIRE].
  17. [17]
    M.I. Polikarpov, U.J. Wiese and M.A. Zubkov, String representation of the Abelian Higgs theory and Aharonov-Bohm effect on the lattice, Phys. Lett. B 309 (1993) 133 [hep-lat/9303007] [INSPIRE].
  18. [18]
    F.S. Nogueira, Z. Nussinov and J. van den Brink, Josephson Currents Induced by the Witten Effect, Phys. Rev. Lett. 117 (2016) 167002 [arXiv:1607.04150] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    M. Dierigl and A. Pritzel, Topological Model for Domain Walls in (Super-)Yang-Mills Theories, Phys. Rev. D 90 (2014) 105008 [arXiv:1405.4291] [INSPIRE].
  20. [20]
    E. Witten, Superconducting Strings, Nucl. Phys. B 249 (1985) 557 [INSPIRE].
  21. [21]
    X.G. Wen and A. Zee, On the possibility of a statistics-changing phase transition, J. Phys. (France) 50 (1989) 1623.Google Scholar
  22. [22]
    Y. Kim and K.-M. Lee, Vortex dynamics in selfdual Chern-Simons Higgs systems, Phys. Rev. D 49 (1994) 2041 [hep-th/9211035] [INSPIRE].
  23. [23]
    X.G. Wen and A. Zee, Quantum Disorder, Duality and Fractional Statistics in (2+1)-dimensions, Phys. Rev. Lett. 62 (1989) 1937 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    A.S. Goldhaber, R. MacKenzie and F. Wilczek, Field corrections to induced statistics, Mod. Phys. Lett. A 4 (1989) 21 [INSPIRE].
  25. [25]
    M. Franz, Vortex-boson duality in four space-time dimensions, EPL 77 (2007) 47005 [cond-mat/0607310] [INSPIRE].
  26. [26]
    E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
  27. [27]
    E. Witten, Dyons of Charge e θ/2π, Phys. Lett. B 86 (1979) 283 [INSPIRE].
  28. [28]
    A.P. Balachandran, B.S. Skagerstam and A. Stern, Gauge Theory of Extended Objects, Phys. Rev. D 20 (1979) 439 [INSPIRE].
  29. [29]
    O.J.C. Dias, G.T. Horowitz, N. Iqbal and J.E. Santos, Vortices in holographic superfluids and superconductors as conformal defects, JHEP 04 (2014) 096 [arXiv:1311.3673] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
  31. [31]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].
  32. [32]
    E. Keski-Vakkuri and P. Kraus, Quantum Hall Effect in AdS/CFT, JHEP 09 (2008) 130 [arXiv:0805.4643] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    H. Nastase and F. Rojas, Vortices with source, FQHE and nontrivial statistics in 2+1 dimensions, arXiv:1610.08999 [INSPIRE].
  34. [34]
    M. Montull, A. Pomarol and P.J. Silva, The Holographic Superconductor Vortex, Phys. Rev. Lett. 103 (2009) 091601 [arXiv:0906.2396] [INSPIRE].
  35. [35]
    O. Domenech, M. Montull, A. Pomarol, A. Salvio and P.J. Silva, Emergent Gauge Fields in Holographic Superconductors, JHEP 08 (2010) 033 [arXiv:1005.1776] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  36. [36]
    L.M. Krauss and F. Wilczek, Discrete Gauge Symmetry in Continuum Theories, Phys. Rev. Lett. 62 (1989) 1221 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    J. Preskill and L.M. Krauss, Local Discrete Symmetry and Quantum Mechanical Hair, Nucl. Phys. B 341 (1990) 50 [INSPIRE].
  38. [38]
    S.R. Coleman, J. Preskill and F. Wilczek, Quantum hair on black holes, Nucl. Phys. B 378 (1992) 175 [hep-th/9201059] [INSPIRE].
  39. [39]
    M. Montull, O. Pujolàs, A. Salvio and P.J. Silva, Flux Periodicities and Quantum Hair on Holographic Superconductors, Phys. Rev. Lett. 107 (2011) 181601 [arXiv:1105.5392] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Arnold-Sommerfeld-Center for Theoretical PhysicsLudwig-Maximilians-UniversitätMünchenGermany
  2. 2.Albert Einstein Center for Fundamental Physics, Institute for Theoretical PhysicsUniversity of BernBernSwitzerland
  3. 3.The Niels Bohr InstituteUniversity of CopenhagenCopenhagen ØDenmark

Personalised recommendations