- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Higher dimensional quantum Hall effect and the analog of W∞-algebra

Dimitra Karabali1,* and V. P. Nair2,†

  • *Contact author: dimitra.karabali@lehman.cuny.edu
  • †Contact author: vnair@ccny.cuny.edu
PDF

Phys. Rev. D 114, 065025 – Published 23 September, 2026

DOI: https://doi.org/10.1103/l8nj-kpgg

Abstract

We show that Abelian and non-Abelian gauge transformations are the analog of W∞ transformations for higher dimensional quantum Hall effect. The commutator anomaly and the extended algebra of such transformations on the edge modes of a droplet are obtained by purely topological arguments, basically utilizing the two-cocycle in the descent procedure for anomalies and using the fact that there is anomaly cancellation between the bulk and boundary actions. The method relies on the fact that bulk actions are easily constructed in general using the Dolbeault index theorem. The resulting algebras are shown to agree with explicit edge mode calculations for cases where they are available. We also comment on the similarities and differences in the nature of these transformations between two and higher dimensions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (27)

  1. R. E. Prange and S. M. Girvin, The Quantum Hall Effect, 2nd ed. (SpringerVerlag, Berlin, 2012); Z. F. Ezawa, Quantum Hall Effects (World Scientific, Singapore, 2008); T. H. Hansson, M. Hermanns, S. H. Simon, and S. F. Viefers, Rev. Mod. Phys. 89, 025005 (2017); D. Tong, arXiv:1606.06687.
  2. S. C. Zhang and J. P. Hu, Science 294, 823 (2001); J. P. Hu and S. C. Zhang, Phys. Rev. B 66, 125301 (2002).
  3. D. Karabali and V. P. Nair, Nucl. Phys. B641, 533 (2002).
  4. D. Karabali and V. P. Nair, Nucl. Phys. B679, 427 (2004).
  5. D. Karabali and V. P. Nair, Nucl. Phys. B697, 513 (2004).
  6. H. Elvang and J. Polchinski, C. R. Phys. 4, 405 (2003); B. A. Bernevig, C. H. Chern, J. P. Hu, N. Toumbas, and S. C. Zhang, Ann. Phys. (Amsterdam) 300, 185 (2002); B. A. Bernevig, J. P. Hu, N. Toumbas, and S. C. Zhang, Phys. Rev. Lett. 91, 236803 (2003); G. Meng, J. Phys. A 36, 9415 (2003); V. P. Nair and S. Randjbar-Daemi, Nucl. Phys. B679, 447 (2004); A. Jellal, B725, 554 (2005); K. Hasebe, B886, 952 (2014).
  7. H. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, Phys. Rev. Lett. 115, 195303 (2015); T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, Phys. Rev. A 93, 043827 (2016); H. M. Price, T. Ozawa, and N. Goldman, 95, 023607 (2017); J.-B. Bouhiron, A. Fabre, Q. Liu, Q. Redon, N. Mittal, T. Satoor, R. Lopes, and S. Nascim-bene, Science 384, 223 (2024).
  8. S. Iso, D. Karabali, and B. Sakita, Phys. Lett. B 296, 143 (1992); A. Cappelli, C. Trugenberger, and G. Zemba, Nucl. Phys. B396, 465 (1993).
  9. A. Dhar, G. Mandal, and S. R. Wadia, Mod. Phys. Lett. A 07, 3129 (1992); 08, 3557 (1993); Int. J. Mod. Phys. A 08, 325 (1993); S. R. Das, A. Dhar, G. Mandal, and S. R. Wadia, Mod. Phys. Lett. A 07, 71 (1992).
  10. A. Cappelli, G. Dunne, C. Trugenberger, and G. Zemba, Nucl. Phys. B398, 531 (1993); A. Cappelli, C. Trugenberger, and G. Zemba, Phys. Rev. Lett. 72, 1902 (1994).
  11. D. Karabali, Nucl. Phys. B419, 437 (1994); B428, 531 (1994); M. Flohr and R. Varnhagen, J. Phys. A 27, 3999 (1994).
  12. V. P. Nair, Phys. Rev. D 102, 025015 (2020).
  13. R. A. Bertlmann, Anomalies in Quantum Field Theory (Oxford University Press, New York, 1996).
  14. S. Treiman, R. Jackiw, B. Zumino, and E. Witten, Current Algebra and Anomalies (World Scientific, Singapore, 1985).
  15. L. D. Faddeev and S. L. Shatashvili, Teor. Mat. Fiz. 60, 206 (1984); Phys. Lett. 167B, 225 (1986); S. G. Jo, Phys. Rev. D 35, 3179 (1987); E. Langmann and J. Mickelsson, Phys. Lett. B 338, 241 (1994).
  16. D. Karabali, Nucl. Phys. B750, 265 (2006); 726, 407 (2005).
  17. D. Karabali and V. P. Nair, Phys. Rev. D 94, 024022 (2016).
  18. A. Agarwal, D. Karabali, and V. P. Nair, Phys. Rev. D 111, 205155 (2023).
  19. J. Wess and B. Zumino, Phys. Lett. 37B, 95 (1971); see also the lectures by Zumino in Ref. [14].
  20. C. G. Callan, Jr. and J. A. Harvey, Nucl. Phys. B250, 427 (1985).
  21. D. Karabali and V. P. Nair, J. Phys. A 39, 12735 (2006).
  22. T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. Rep. 66, 213 (1980).
  23. X. G. Wen, Phys. Rev. B 41, 12838 (1990); Mod. Phys. Lett. B 5, 39 (1991).
  24. V. P. Nair and J. Schiff, Phys. Lett. B 246, 423 (1990); Nucl. Phys. B371, 329 (1992).
  25. A. P. Polychronakos, Nucl. Phys. B705, 457 (2005); B711, 505 (2005).
  26. S. K. Wong, Nuovo Cimento A 65, 689 (1970); see also A. P. Balachandran, G. Marmo, and A. Stern, Nucl. Phys. B162, 385 (1980); A. P. Balachandran, G. Marmo, A. Stern, and B. S. Skagerstam, Phys. Lett. 89B, 1991 (1980); A. P. Balachandran, G. Marmo, B-S. Skagerstam, and A. Stern, Gauge Symmetries and Fibre Bundles, Lecture Notes in Physics Vol. 188 (Springer-Verlag, Berlin, 1982).
  27. W. A. Bardeen and B. Zumino, Nucl. Phys. B244, 421 (1984); L. Alvarez-Gaumé and P. Ginsparg, Ann. Phys. (N.Y.) 161, 423 (1985).

Outline

Information

Phys. Rev. D 114, 065025– Published 23 September, 2026
  • Received 17 July 2026
  • Accepted 3 September 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation