ADHM and the 4d quantum Hall effect

  • Alec Barns-Graham
  • Nick Dorey
  • Nakarin Lohitsiri
  • David Tong
  • Carl Turner
Open Access
Regular Article - Theoretical Physics


Yang-Mills instantons are solitonic particles in d = 4 + 1 dimensional gauge theories. We construct and analyse the quantum Hall states that arise when these particles are restricted to the lowest Landau level. We describe the ground state wavefunctions for both Abelian and non-Abelian quantum Hall states. Although our model is purely bosonic, we show that the excitations of this 4d quantum Hall state are governed by the Nekrasov partition function of a certain five dimensional supersymmetric gauge theory with Chern-Simons term. The partition function can also be interpreted as a variant of the Hilbert series of the instanton moduli space, counting holomorphic sections rather than holomorphic functions.

It is known that the Hilbert series of the instanton moduli space can be rewritten using mirror symmetry of 3d gauge theories in terms of Coulomb branch variables. We generalise this approach to include the effect of a five dimensional Chern-Simons term. We demonstrate that the resulting Coulomb branch formula coincides with the corresponding Higgs branch Molien integral which, in turn, reproduces the standard formula for the Nekrasov partition function.


Matrix Models Solitons Monopoles and Instantons Topological States of Matter 


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]
    S.-C. Zhang and J.-p. Hu, A four-dimensional generalization of the quantum Hall effect, Science 294 (2001) 823 [cond-mat/0110572] [INSPIRE].
  2. [2]
    J.-p. Hu and S.-C. Zhang, Collective excitations at the boundary of a 4-D quantum Hall droplet, Phys. Rev. B 66 (2002) 125301 [cond-mat/0112432] [INSPIRE].
  3. [3]
    H. Elvang and J. Polchinski, The quantum Hall effect on R 4, hep-th/0209104 [INSPIRE].
  4. [4]
    M. Fabinger, Higher dimensional quantum Hall effect in string theory, JHEP 05 (2002) 037 [hep-th/0201016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    B.A. Bernevig, C.-H. Chern, J.-P. Hu, N. Toumbas and S.-C. Zhang, Effective field theory description of the higher dimensional quantum Hall liquid, Annals Phys. 300 (2002) 185 [cond-mat/0206164] [INSPIRE].
  6. [6]
    Y.-X. Chen, B.-Y. Hou and B.-Y. Hou, Noncommutative geometry of four-dimensional quantum Hall droplet, Nucl. Phys. B 638 (2002) 220 [hep-th/0203095] [INSPIRE].
  7. [7]
    D. Karabali and V.P. Nair, Quantum Hall effect in higher dimensions, Nucl. Phys. B 641 (2002) 533 [hep-th/0203264] [INSPIRE].
  8. [8]
    D. Karabali and V.P. Nair, Edge states for quantum Hall droplets in higher dimensions and a generalized WZW model, Nucl. Phys. B 697 (2004) 513 [hep-th/0403111] [INSPIRE].
  9. [9]
    D. Karabali and V.P. Nair, Quantum Hall effect in higher dimensions, matrix models and fuzzy geometry, J. Phys. A 39 (2006) 12735 [hep-th/0606161] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  10. [10]
    D. Karabali and V.P. Nair, Geometry of the quantum Hall effect: An effective action for all dimensions, Phys. Rev. D 94 (2016) 024022 [arXiv:1604.00722] [INSPIRE].
  11. [11]
    B.A. Bernevig, J.-p. Hu, N. Toumbas and S.-C. Zhang, The eight-dimensional quantum Hall effect and the octonions, Phys. Rev. Lett. 91 (2003) 236803 [cond-mat/0306045] [INSPIRE].
  12. [12]
    J.J. Heckman and L. Tizzano, 6D Fractional Quantum Hall Effect, arXiv:1708.02250 [INSPIRE].
  13. [13]
    H.M. Price, O. Zilberberg, T. Ozawa, I. Carusotto and N. Goldman, Four-Dimensional Quantum Hall Effect with Ultracold Atoms, Phys. Rev. Lett. 115 (2015) 195303 [arXiv:1505.04387] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    M. Lohse, C. Schweizer, H.M. Price, O, Zilberberg and I. Bloch, Exploring 4D Quantum Hall Physics with a 2D Topological Charge Pump, Nature 553 (2018) 55 [arXiv:1705.08371].
  15. [15]
    C.N. Yang, Generalization of Dirac’s Monopole to SU(2) Gauge Fields, J. Math. Phys. 19 (1978) 320 [INSPIRE].
  16. [16]
    B. Blok and X.G. Wen, Many body systems with nonAbelian statistics, Nucl. Phys. B 374 (1992) 615 [INSPIRE].
  17. [17]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Y. Tachikawa, Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, JHEP 02 (2004) 050 [hep-th/0401184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d \( \mathcal{N}=4 \) gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].
  21. [21]
    S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and Hall-Littlewood polynomials, JHEP 09 (2014) 178 [arXiv:1403.0585] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    S. Cremonesi, G. Ferlito, A. Hanany and N. Mekareeya, Coulomb Branch and The Moduli Space of Instantons, JHEP 12 (2014) 103 [arXiv:1408.6835] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  23. [23]
    A.P. Polychronakos, Quantum Hall states as matrix Chern-Simons theory, JHEP 04 (2001) 011 [hep-th/0103013] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    N. Dorey, D. Tong and C. Turner, Matrix model for non-Abelian quantum Hall states, Phys. Rev. B 94 (2016) 085114 [arXiv:1603.09688] [INSPIRE].
  25. [25]
    L. Susskind, The quantum Hall fluid and noncommutative Chern-Simons theory, hep-th/0101029 [INSPIRE].
  26. [26]
    D. Tong, A quantum Hall fluid of vortices, JHEP 02 (2004) 046 [hep-th/0306266] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    D. Tong and C. Turner, Quantum Hall effect in supersymmetric Chern-Simons theories, Phys. Rev. B 92 (2015) 235125 [arXiv:1508.00580] [INSPIRE].
  28. [28]
    S. Hellerman and M. Van Raamsdonk, Quantum Hall physics equals noncommutative field theory, JHEP 10 (2001) 039 [hep-th/0103179] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    D. Karabali and B. Sakita, Chern-Simons matrix model: Coherent states and relation to Laughlin wavefunctions, Phys. Rev. B 64 (2001) 245316 [hep-th/0106016] [INSPIRE].
  30. [30]
    D. Karabali and B. Sakita, Orthogonal basis for the energy eigenfunctions of the Chern-Simons matrix model, Phys. Rev. B 65 (2002) 075304 [hep-th/0107168] [INSPIRE].
  31. [31]
    T.H. Hansson, J. Kailasvuori and A. Karlhede, Charge and current in the quantum Hall matrix model, Phys. Rev. B 68 (2003) 035327 [cond-mat/0304271].
  32. [32]
    A. Cappelli and I.D. Rodriguez, Jain States in a Matrix Theory of the Quantum Hall Effect, JHEP 12 (2006) 056 [hep-th/0610269] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    A. Cappelli and M. Riccardi, Matrix model description of Laughlin Hall states, J. Stat. Mech. 0505 (2005) P05001 [hep-th/0410151] [INSPIRE].
  34. [34]
    I.D. Rodriguez, Edge excitations of the Chern Simons matrix theory for the FQHE, JHEP 07 (2009) 100 [arXiv:0812.4531] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    N. Dorey, D. Tong and C. Turner, A Matrix Model for WZW, JHEP 08 (2016) 007 [arXiv:1604.05711] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
  37. [37]
    S. Kim and S. Lee, The geometry of dyonic instantons in 5-dimensional supergravity, arXiv:0712.0090 [INSPIRE].
  38. [38]
    B. Collie and D. Tong, Instantons, Fermions and Chern-Simons Terms, JHEP 07 (2008) 015 [arXiv:0804.1772] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    S. Kim, K.-M. Lee and S. Lee, Dyonic Instantons in 5-dim Yang-Mills Chern-Simons Theories, JHEP 08 (2008) 064 [arXiv:0804.1207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    V.P. Nair and J. Schiff, A Kähler-Chern-Simons Theory and Quantization of Instanton Moduli Spaces, Phys. Lett. B 246 (1990) 423 [INSPIRE].
  41. [41]
    V.P. Nair and J. Schiff, Kähler Chern-Simons theory and symmetries of antiselfdual gauge fields, Nucl. Phys. B 371 (1992) 329 [INSPIRE].
  42. [42]
    M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Yu. I. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].
  43. [43]
    M.R. Douglas, Gauge fields and D-branes, J. Geom. Phys. 28 (1998) 255 [hep-th/9604198] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    N. Nekrasov, Four Dimensional Holomorphic Theories, Ph.D. Thesis, Princeton University, U.S.A.,
  45. [45]
    N. Dorey and A. Singleton, Instantons, Integrability and Discrete Light-Cone Quantisation, arXiv:1412.5178 [INSPIRE].
  46. [46]
    N. Nekrasov and A.S. Schwarz, Instantons on noncommutative R 4 and (2,0) superconformal six-dimensional theory, Commun. Math. Phys. 198 (1998) 689 [hep-th/9802068] [INSPIRE].
  47. [47]
    Y.-X. Chen, Matrix models of four-dimensional quantum Hall fluids, hep-th/0209182 [INSPIRE].
  48. [48]
    Y.-X. Chen, Quasiparticle excitations and hierarchies of four-dimensional quantum Hall fluid states in the matrix models, hep-th/0210059 [INSPIRE].
  49. [49]
    Yi. Lu and C. Wu, High-Dimensional Topological Insulators with Quaternionic Analytic Landau Levels, Phys. Rev. Lett. 110 (2013) 216802 [arXiv:1103.5422].
  50. [50]
    H. Nakajima, Lectures on Hilbert Scheme of Points on Surfaces, Am. Math. Soc. (1999).Google Scholar
  51. [51]
    A. Losev, G.W. Moore, N. Nekrasov and S. Shatashvili, Chiral Lagrangians, anomalies, supersymmetry and holomorphy, Nucl. Phys. B 484 (1997) 196 [hep-th/9606082] [INSPIRE].
  52. [52]
    S. Benvenuti, A. Hanany and N. Mekareeya, The Hilbert Series of the One Instanton Moduli Space, JHEP 06 (2010) 100 [arXiv:1005.3026] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  53. [53]
    E. Carlsson, N. Nekrasov and A. Okounkov, Five dimensional gauge theories and vertex operators, Moscow Math. J. 14 (2014) 39 [arXiv:1308.2465] [INSPIRE].MathSciNetMATHGoogle Scholar
  54. [54]
    H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, JHEP 12 (2011) 031 [arXiv:1110.2175] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
  56. [56]
    J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [INSPIRE].
  57. [57]
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University, Press (1995), Lect. Notes Math. 1271 (1987) 189 [Publ. I.R.M.A. (1988) 131].Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Alec Barns-Graham
    • 1
  • Nick Dorey
    • 1
  • Nakarin Lohitsiri
    • 1
  • David Tong
    • 1
  • Carl Turner
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeU.K.

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