Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 010, 31 pages      hep-th/0611282      http://dx.doi.org/10.3842/SIGMA.2007.010
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Modular Symmetry and Fractional Charges in N = 2 Supersymmetric Yang-Mills and the Quantum Hall Effect

Brian P. Dolan a, b
a) Department of Mathematical Physics, National University of Ireland, Maynooth, Ireland
b) School of Theoretical Physics, Dublin Institute for Advanced Studies, 10, Burlington Rd., Dublin, Ireland

Received September 29, 2006, in final form December 22, 2006; Published online January 10, 2007

Abstract
The parallel rôles of modular symmetry in N = 2 supersymmetric Yang-Mills and in the quantum Hall effect are reviewed. In supersymmetric Yang-Mills theories modular symmetry emerges as a version of Dirac's electric - magnetic duality. It has significant consequences for the vacuum structure of these theories, leading to a fractal vacuum which has an infinite hierarchy of related phases. In the case of N = 2 supersymmetric Yang-Mills in 3+1 dimensions, scaling functions can be defined which are modular forms of a subgroup of the full modular group and which interpolate between vacua. Infra-red fixed points at strong coupling correspond to θ-vacua with θ a rational number that, in the case of pure SUSY Yang-Mills, has odd denominator. There is a mass gap for electrically charged particles which can carry fractional electric charge. A similar structure applies to the 2+1 dimensional quantum Hall effect where the hierarchy of Hall plateaux can be understood in terms of an action of the modular group and the stability of Hall plateaux is due to the fact that odd denominator Hall conductivities are attractive infra-red fixed points. There is a mass gap for electrically charged excitations which, in the case of the fractional quantum Hall effect, carry fractional electric charge.

Key words: duality; modular symmetry; supersymmetry; quantum Hall effect.

pdf (780 kb)   ps (735 kb)   tex (819 kb)

References

  1. Dolan B.P., Duality in supersymmetric Yang-Mills and the quantum Hall effect, Modern Phys. Lett. A 21 (2006), 1567-1585.
  2. Shapere A., Wilczek F., Self-dual models with theta terms, Nuclear Phys. B 320 (1989), 669-695.
  3. Montonen C., Olive D., Magnetic monopoles as gauge particles, Phys. Lett. B 72 (1977), 117-120.
  4. Seiberg N., Witten E., Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nuclear Phys. B 426 (1994), 19-53, Erratum, Nuclear Phys. B 430 (1994), 169-169, hep-th/9407087.
  5. Seiberg N., Witten E., Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nuclear Phys. B 431 (1994), 484-550, hep-th/9408099.
  6. Cardy J.L., Rabinovici E., Phase structure of Zp models in the presence of a q parameter, Nuclear Phys. B 205 (1982), 1-16.
  7. Cardy J.L., Duality and the q parameter in Abelian lattice models, Nuclear Phys. B 205 (1982), 17-26.
  8. Asorey M., Esteve J.G., Salas J., Exact renormalization-group analysis of first-order phase transitions in clock models, Phys. Rev. B 48 (1993), 3626-3632.
  9. Callan C.G., Freed D., Phase diagram of the dissipative Hofstadter model, Nuclear Phys. B 374 (1992), 543-566, hep-th/9110046.
  10. Callan C.G., Felce A., Freed D., Critical theories of the dissipative Hofstadter model, Nuclear Phys. B 392 (1993), 551-592, hep-th/9202085.
  11. da Cruz W., The Hausdorff dimension of fractal sets and fractional quantum Hall effect, Chaos Solitons Fractals 17 (2003), 975-979, math-ph/0209028.
  12. da Cruz W., A quantum-geometrical description of fracton statistics, Internat. J. Modern Phys. A 18 (2003), 2213-2219, cond-mat/0212567.
  13. Lütken C.A., Ross G.G., Duality in the quantum Hall system, Phys. Rev. B 45 (1992), 11837-11845.
  14. Lütken C.A., Ross G.G., Delocalization, duality, and scaling in the quantum Hall system, Phys. Rev. B 48 (1993), 2500-2514.
  15. Lütken C.A., Global phase diagrams for charge transport in two dimensions, J. Phys. A: Math. Gen. 26 (1993), L811-L817.
  16. Lütken C.A., Geometry of renormalization group flows constrained by discrete global symmetries, Nuclear Phys. B 396 (1993), 670-692.
  17. Kivelson S., Lee D.-H., Zhang S.-C., Global phase diagram in the quantum Hall effect, Phys. Rev. B 46 (1992), 2223-2238.
  18. Figueroa-O'Farrill J.M., Electromagnetic duality for children, http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/EDC.pdf
  19. Dirac P.A.M., Quantized singularities in the electromagnetic field, Proc. Roy. Soc. A 133 (1931), 60-72.
  20. Wu T.T., Yang C.N., Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D 12 (1975), 3845-3857.
  21. Milton K.A., Theoretical and experimental status of magnetic monopoles, Rep. Prog. Phys. 69 (2006), 1637-1711, hep-ex/0602040.
  22. Schwinger J., Magnetic charge and quantum field theory, Phys. Rev. 144 (1966), 1087-1092.
  23. Zwanziger D., Local-Lagrangian quantum field theory of electric and magnetic charges, Phys. Rev. D 3 (1971), 880-891.
  24. Witten E., Dyons of charge [e/(2p)], Phys. Lett. B 86 (1979), 283-287.
  25. Witten E., On S-duality in Abelian gauge theory, Selecta. Math. (NS) 1 (1995), 383-410, hep-th/9505186.
  26. Verlinde E., Global aspects of electric-magnetic duality, Nuclear Phys. B 455 (1995), 211-225.
  27. Polyakov A.M., Compact gauge fields and the infrared catastrophe, Phys. Lett. B 59 (1975), 82-84.
  28. Belavin A.A., Polyakov A.M., Schwartz A., Tyupkin Y., Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975), 85-87.
  29. 't Hooft G., Symmetry breaking through Bell-Jackiw anomalies, Phys. Rev. Lett. 37 (1976), 8-11.
  30. Callan C., Dashen R., Gross D., The structure of the gauge theory vacuum, Phys. Lett. B 63 (1976), 334-340.
  31. Jackiw R., Rebbi C., Vacuum periodicity in a Yang-Mills quantum theory, Phys. Rev. Lett. 37 (1976), 172-175.
  32. 't Hooft G., Magnetic monopoles in unified gauge theories, Nuclear Phys. B 79 (1974), 276-284.
  33. Polyakov A.M., Particle spectrum in the quantum field theory, JETP Lett. 20 (1974), 194-195.
  34. Bogomol'nyi E.B., Stability of classical solutions, Soviet J. Nuclear Phys. 24 (1976), 449-454.
  35. Alvarez-Gaumé L., Hassan S.F., Introduction to S-duality in N = 2 supersymmetric gauge theory, Fortsch. Phys. 45 (1997), 159-236, hep-th/9701069.
  36. Peskin M.E., Duality in supersymmetric Yang-Mills theory, hep-th/9702094.
  37. Dorey N., Khoze V.V., Mattis M.P., Multi-instanton calculus in N = 2 supersymmetric gauge theory, Phys. Rev. D 54 (1996), 2921-2943, hep-th/9603136.
  38. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge University Press, 1940.
  39. Minahan J.A., Nemeschansky D., N = 2 super Yang-Mills and subgroups of SL(2,Z), Nuclear Phys. B 468 (1996), 72-84, hep-th/9601059.
  40. Ritz A., On the beta-function in N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 434 (1998), 54-60, hep-th/9710112.
  41. Dolan B.P., Renormalisation flow and geodesics on the moduli space of four dimensional N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 418 (1998), 107-110, hep-th/9710161.
  42. Carlino G., Konishi K., Maggiore N., Magnoli N., On the beta function in supersymmetric gauge theories, Phys. Lett. B 455 (1999), 171-178, hep-th/9902162.
  43. Konishi K., Renormalization group and dynamics of supersymmetric gauge theories, Internat. J. Modern Phys. A 16 (2001), 1861-1874, hep-th/0012122.
  44. D'Hoker E., Krichever I.M., Phong D.H., The effective prepotential of N = 2 supersymmetric SU(Nc) gauge theories, Nuclear Phys. B 489 (1997), 179-210, hep-th/9609041.
  45. D'Hoker E., Krichever I.M., Phong D.H., The renormalization group equation in N = 2 supersymmetric gauge theories, Nuclear Phys. B 494 (1997), 89-104, hep-th/9610156.
  46. Matone M., Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett. B 357 (1995), 342-348, hep-th/9506102.
  47. Rankin R.A., Modular forms and functions, Cambridge University Press, 1977.
  48. Dolan B.P., N=2 supersymmetric Yang-Mills and the quantum Hall effect, Internat. J. Modern Phys. A 21 (2006), 4807-4821, hep-th/0505138.
  49. Dolan B.P., Meromorphic scaling flow of N = 2 supersymmetric SU(2) Yang-Mills with matter, Nuclear Phys. B 737 (2006), 153-175, hep-th/0506088.
  50. Prange R.E., Girvin S.M. (Editors), The quantum Hall effect, Springer, New York, 1987.
  51. Heinonen O. (Editor), Composite fermions, World Scientific, 1998.
  52. Stone M., Quantum Hall effect, World Scientific, 1992.
  53. von Klitzing K., Dorda G., Pepper M., New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45 (1980), 494-497.
  54. Tsui D.C., Stormer H.L., Gossard A.C., Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48 (1982), 1559-1562.
  55. Tsui D.C., The strange physics of two-dimensional electrons in a strong magnetic field, Phys. B 164 (1990), 59-66.
  56. Landau L., Lifschitz E., A Course in theoretical physics, Vol. 8, 2nd ed., Pergamon Press, 1984.
  57. Jain J.K., Kivelson S.A., Trivedi N., Scaling theory of the fractional quantum Hall effect, Phys. Rev. Lett. 64 (1990), 1297-1300.
  58. Jain J.K., Goldman V.J., Hierarchy of states in the fractional quantum Hall effect, Phys. Rev. B 45 (1992), 1255-1258.
  59. Kogan I., Szabo R., Dual response models for the fractional quantum Hall effect, Phys. Rev. B 58 (1998), 7893-7897, cond-mat/9712233.
  60. Cappelli A., Zamba R., Modular invariant partition functions in the quantum Hall effect, Nuclear Phys. B 490 (1997), 595-632, hep-th/9605127.
  61. Jain J.K., Composite-fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63 (1989), 199-202.
  62. Jain J.K., Theory of the fractional quantum Hall effect, Phys. Rev. B 41 (1990), 7653-7665.
  63. Lopez A., Fradkin E., Fractional quantum Hall effect and Chern-Simons gauge theories, Phys. Rev. B 44 (1991), 5246-5262.
  64. Girvin S.M., Summary, omissions and unanswered questions, in The Quantum Hall Effect, Editors R.E. Prange and S.M. Girvin, Springer, New York, 1987, 381-399.
  65. Arovas D.P., Schrieffer J.R., Wilczek F., Zee A., Statistical mechanics of anyons, Nuclear Phys. B 251 (1985), 117-126.
  66. Zhang S.C., Hansson T.H., Kivelson S., Effective-field-theory model for the fractional quantum Hall effect, Phys. Rev. Lett. 62 (1989), 82-85.
  67. Fröhlich J., Zee A., Large scale physics of the quantum hall fluid, Nuclear Phys. B 364 (1991), 517-540.
  68. Balachandran A.P., Srivastava A.M., Chern-Simons dynamics and the quantum Hall effect, hep-th/9111006.
  69. Balatsky A., Fradkin E., Singlet quantum Hall effect and Chern-Simons theories, Phys. Rev. B 43 (1991), 10622-10634.
  70. Fradkin E., Kivelson S., Modular invariance, self-duality and the phase transition between quantum Hall plateaus, Nuclear Phys. B 474 (1996), 543-574.
  71. Witten E., SL(2,Z) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041.
  72. Leigh R.G., Petkou A.C., SL(2,Z) action on three-dimensional CFTs and holography, JHEP 0312 (2003), 020, 17 pages, hep-th/0309177.
  73. Zucchini R., Four dimensional Abelian duality and SL(2,Z) action in three dimensional conformal field theory, Adv. Theor. Math. Phys. 8 (2005), 895-937, hep-th/0311143.
  74. Yee H.-U., A note on AdS/CFT dual of SL(2,Z) action on 3D conformal field theories with U(1) symmetry, Phys. Lett. B 598 (2004), 139-148, hep-th/0402115.
  75. Wilczek F., Quantum mechanics of fractional-spin particles, Phys. Rev. Lett. 49 (1982), 957-959.
  76. Girvin S.M., MacDonald A.H., Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect, Phys. Rev. Lett. 58 (1987), 1252-1255.
  77. 't Hooft G., Topology of the gauge condition and new confinement phases in non-Abelian gauge theories, Nuclear Phys. B 190 (1981), 455-478.
  78. Laughlin R.B., Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50 (1983), 1395-1398.
  79. de-Picciotto R., Reznikov M., Heiblum M., Umansky V., Bunin G., Mahalu D., Direct observation of a fractional charge, Nature 389 (1997), 162-164.
  80. Saminadayar L., Glattli D.C., Jin Y., Etienne B., Observation of the e/3 fractionally charged Laughlin quasiparticle, Phys. Rev. Lett. 79 (1997), 2526-2529.
  81. Burgess C., Dolan B.P., Particle-vortex duality and the modular group: applications to the quantum Hall effect and other 2-D systems, Phys. Rev. B 63 (2001), 155309, 21 pages, hep-th/0010246.
  82. Burgess C., Dolan B.P., Duality and non-linear response for quantum Hall systems, Phys. Rev. B 65 (2002), 155323, 7 pages, cond-mat/0105621.
  83. Pan W., Stormer H.L., Tsui D.C., Pfeiffer L.N., Baldwin K.W., West K.W., Transition from an electron solid to the sequence of fractional quantum Hall states at very low Landau level filling factor, Phys. Rev. Lett. 88 (2002), 176802, 4 pages.
  84. Dolan B.P., Duality and the modular group in the quantum Hall effect, J. Phys. A: Math. Gen. 32 (1999), L243-L248, cond-mat/9805171.
  85. Taniguchi N., Nonperturbative renormalization group function for quantum Hall Plateau transitions imposed by global symmetries, cond-mat/9810334.
  86. Burgess C., Lütken A., One-dimensional flows in the quantum Hall system, Nuclear Phys. B 500 (1997), 367-378, cond-mat/9611118.
  87. Burgess C., Lütken A., On the implications of discrete symmetries for the beta function of quantum Hall systems, Phys. Lett. B 451 (1999), 365-371, cond-mat/9812396.
  88. Khmel'nitskii D.E., Quantisation of Hall conductivity, Pis'ma Zh. Eksp. Teor. Fiz. 38 (1983), 454-458 (English transl.: (JETP Lett. 38 (1983), 552-556).
  89. Pruisken A.M.M., Universal singularities in the integral quantum Hall effect, Phys. Rev. Lett. 61 (1988), 1297-1300.
  90. Wei H.P., Tsui D.C., Paalanen M.A., Pruisken A.M.M., Experiments on delocalization and universality in the integral quantum Hall effect, Phys. Rev. Lett. 61 (1988), 1294-1296.
  91. Engel L., Wei H.P., Tsui D.C., Shayegan M., Critical exponent in the fractional quantum Hall effect, Surf. Sci. 229 (1990), 13-15.
  92. Sondhi S.L., Girvin S.M., Carini J.P., Shankar D., Continuous quantum phase transitions, Rev. Modern Phys. 69 (1997), 315-333.
  93. Shahar D., Tsui D.C., Shayegan M., Shimshoni E., Sondhi S.L., A different view of the quantum Hall Plateau-to-Plateau transitions, Phys. Rev. Lett. 79 (1997), 479-482, cond-mat/9611011.
  94. Shahar D., Hilke M., Li C.C., Tsui D.C., Sondhi S.L., Razeghi M., A new transport regime in the quantum Hall effect, Solid State Commun. 107 (1998), 19-23, cond-mat/9706045.
  95. Zirnbauer M.R., Conformal field theory of the integer quantum Hall effect, hep-th/9905054.
  96. Dolan B.P., Modular invariance, universality and crossover in the quantum Hall effect, Nuclear Phys. B 554 (1999), 487-513, cond-mat/9809294.
  97. Pruisken A.M.M., Field theory, scaling and the localization problem, in The Quantum Hall Effect, Editors R.E. Prange and S.M. Girvin, Springer, New York, 1987, 117-173.
  98. Burgess C., Dolan B.P., Dib R., Derivation of the semi-circle law from the law of corresponding states, Phys. Rev. B 62 (2000), 15359-15362, cond-mat/9911476.
  99. Hilke M., Shahar D., Song S.H., Tsui D.C., Xie Y.H., Shayegan M., Semicircle: an exact relation in the integer and fractional quantum Hall effect, Eur. Phys. Lett. 46 (1999), 775-779, cond-mat/9810217.
  100. Dykhne A.M., Ruzin I.M., Theory of the fractional quantum Hall effect: the two-phase model, Phys. Rev. B 50 (1994), 2369-2379.
  101. Ruzin I., Feng S., Theory of the fractional quantum Hall effect: the two-phase model, Phys. Rev. Lett. 74 (1995), 154-157.
  102. Murzin S.S., Weiss M., Jansen A.G.M., Eberl K., Universal flow diagram for the magnetoconductance in disordered GaAs layers, Phys. Rev. B 66 (2002), 233314, 4 pages, cond-mat/0204206.
  103. Murzin S.S., Dorozhkin S.I., Maude D.K., Jansen A.G.M., Scaling flow diagram in the fractional quantum Hall regime of GaAs/AlGaAs heterostructures, Phys. Rev. B 72 (2005), 195317, 5 pages, cond-mat/0504235.
  104. Lütken C.A., Ross G.G., Anti-holomorphic scaling in the quantum Hall system, Phys. Lett. A 356 (2006), 382-384.
  105. Lütken C.A., Holomorphic anomaly in the quantum Hall system, Nuclear Phys. B 759 (2006), 343-369.
  106. Dolan B.P., Duality in the quantum Hall effect - the role of electron spin, Phys. Rev. B 62 (2000), 10278-10291, cond-mat/0002228.
  107. Huang C.F., Chang Y.H., Cheng H.H., Liang C.-T., Hwang G.J., A study on the universality of the magnetic-field-induced phase transitions in the two-dimensional electron system in an AlGaAs/GaAs heterostructure, Phys. E 22 (2004), 232-235, cond-mat/0404246.
  108. Hang D.R., Dunford R.B., Kim G.-H., Yeh H.D., Huang C.F., Ritchie D.A., Farrer I., Zhang Y.W., Liang C.-T., Chang Y.H., Effects of Zeeman spin splitting on the modular symmetry in the quantum Hall effect, Microelectron. J. 36 (2005), 469-471, cond-mat/0508577.
  109. Huang C.F., Chang Y.H., Cheng H.H., Yang Z.P., Yeh H.D., Hsu C.H., Liang C.-T., Hang D.R., Lin H.H., An experimental study on G(2) modular symmetry on the quantum Hall system with a small spin-splitting, J. Phys.: Condens. Matter 19 (2007), 026205, 8 pages, cond-mat/0609310.
  110. Flohr M.A.I., On a new universal class of phase transitions and quantum Hall effect, hep-th/9412053.
  111. Georgelin Y., Masson T., Wallet J.-C., Modular groups, visibility diagram and quantum Hall effect, J. Phys. A: Math. Gen. 30 (1997), 5065-5075.
  112. Georgelin Y., Masson T., Wallet J.-C., G(2) modular symmetry, renormalization group flow and the quantum hall effect, J. Phys. A: Math. Gen. 33 (2000), 39-55.
  113. Georgelin Y., Wallet J-C., Group G(2) and the fractional quantum Hall effect, Phys. Lett. A 224 (1997), 303-308.
  114. Pryadko L.P., Global symmetries of quantum Hall systems: lattice description, Phys. Rev. B 56 (1997), 6810-6822.
  115. Fisher M.P.A., Quantum phase transitions in disordered two-dimensional superconductors, Phys. Rev. Lett. 65 (1990), 923-926.
  116. Rey S.-J., Zee A., Self-duality of three-dimensional Chern-Simons theory, Nuclear Phys. B 352 (1991), 897-921.
  117. Berenstein D., Matrix model for a quantum Hall droplet with manifest particle-hole symmetry, Phys. Rev. D 71 (2005), 085001, 12 pages, hep-th/0409115.
  118. Lin H., Lunin O., Maldacena J., Bubbling AdS space and 1/2 BPS geometries, JHEP 0410 (2004), 025, 67 pages, hep-th/0409174.
  119. Ghodsi A., Mosaffa A.E., Saremi O., Sheikh-Jabbari M.M., LLL vs. LLM: half BPS sector of N = 4 SYM equals to quantum Hall system, Nuclear Phys. B 729 (2005), 467-491, hep-th/0505129.

Previous article   Next article   Contents of Volume 3 (2007)