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SUSY Quantum Hall Effect on Non-Anti-Commutative Geometry
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

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We review the recent developments of the SUSY quantum Hall effect [hep-th/0411137, hep-th/0503162, hep-th/0606007, arXiv:0705.4527]. We introduce a SUSY formulation of the quantum Hall effect on supermanifolds. On each of supersphere and superplane, we investigate SUSY Landau problem and explicitly construct SUSY extensions of Laughlin wavefunction and topological excitations. The non-anti-commutative geometry naturally emerges in the lowest Landau level and brings particular physics to the SUSY quantum Hall effect. It is shown that SUSY provides a unified picture of the original Laughlin and Moore–Read states. Based on the charge-flux duality, we also develop a Chern–Simons effective field theory for the SUSY quantum Hall effect.
Keywords: quantum hall effect; non-anti-commutative geometry; supersymmetry; Hopf map; Landau problem; Chern–Simons theory; charge-flux duality. 
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     author = {Kazuki Hasebe},
     title = {SUSY {Quantum} {Hall} {Effect} on {Non-Anti-Commutative} {Geometry}},
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Kazuki Hasebe. SUSY Quantum Hall Effect on Non-Anti-Commutative Geometry. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a22/

[1] Ezawa Z. F., Tsitsishvili G., Hasebe K., “Noncommutative geometry, extended $W_\infty$ algebra and Grassmannian solitons in multicomponent quantum Hall systems”, Phys. Rev. B, 67 (2003), 125314, 16 pp., ages ; hep-th/0209198 | DOI

[2] Zhang S. C., Hu J. P., “A four dimensional generalization of the quantum Hall effect”, Science, 294 (2001), 823–828 ; cond-mat/0110572 | DOI

[3] Karabali D., Nair V. P., “Quantum Hall effect in higher dimensions”, Nuclear Phys. B, 641 (2002), 533–546 ; hep-th/0203264 | DOI | MR | Zbl

[4] Bernevig B. A., Hu J. P., Toumbas N., Zhang S. C., “The eight dimensional quantum Hall effect and the octonions”, Phys. Rev. Lett., 91 (2003), 236803, 4 pp., ages ; cond-mat/0306045 | DOI

[5] Hasebe K., Kimura Y., “Dimensional hierarchy in quantum Hall effects on fuzzy spheres”, Phys. Lett. B, 602 (2004), 255–260 ; hep-th/0310274 | DOI | MR

[6] Nair V. P., Randjbar-Daemi S., “Quantum Hall effect on $S^3$, edge states and fuzzy $S^3\mathbf Z_2$”, Nuclear Phys. B, 679 (2004), 447–463 ; hep-th/0309212 | DOI | MR | Zbl

[7] Jellal A., “Quantum Hall effect on higher dimensional spaces”, Nuclear Phys. B, 725 (2005), 554–576 ; hep-th/0505095 | DOI | MR | Zbl

[8] Daoud M., Jellal A., Quantum Hall effect on the flag manifold F$_2$, hep-th/0610157 | MR

[9] Landi G., “Spin-Hall effect with quantum group symmetry”, Lett. Math. Phys., 75 (2006), 187–200 ; hep-th/0504092 | DOI | MR | Zbl

[10] Karabali D., Nair V. P., “Quantum Hall effect in higher dimensions, matrix models and fuzzy geometry”, J. Phys. A: Math. Gen., 39 (2006), 12735–12763 ; hep-th/0606161 | DOI | MR | Zbl

[11] Murakami Sh., Nagaosa N., Zhang S. C., “Dissipationless quantum spin current at room temperature”, Science, 301 (2003), 1348–1351 ; cond-mat/0308167 | DOI

[12] de Boer J., Grassi P. A., van Nieuwenhuizen P., “Non-commutative superspace from string theory”, Phys. Lett. B, 574 (2003), 98–104 ; hep-th/0302078 | DOI | MR | Zbl

[13] Ooguri H., Vafa C., “The $C$-deformation of gluino and non-planar diagrams”, Adv. Theor. Math. Phys., 7 (2003), 53–85 ; hep-th/0302109 | MR

[14] Seiberg N., “Noncommutative superspace, $N=1/2$ supersymmetry, field theory and string theory”, J. High Energy Phys., 2003:6 (2003), 010, 17 pp., ages ; hep-th/0305248 | DOI | MR

[15] Azuma T., Iso S., Kawai H., Ohwashi Y., “Supermatrix models”, Nuclear Phys. B, 610 (2001), 251–279 ; hep-th/0102168 | DOI | MR | Zbl

[16] Iso S., Umetsu H., “Gauge theory on noncommutative supersphere from supermatrix model”, Phys. Rev. D, 69 (2004), 105003, 7 pp., ages ; ; Iso S., Umetsu H., “Note on gauge theory on fuzzy supersphere”, Phys. Rev. D, 69 (2004), 105014, 7 pp., ages ; hep-th/0311005hep-th/0312307 | DOI | MR | DOI | MR

[17] Balachandran A. P., Kurkcuoglu S., Rojas E., “The star product on the fuzzy supersphere”, J. High Energy Phys., 2002:7 (2002), 056, 22 pp., ages ; hep-th/0204170 | DOI | MR

[18] Balachandran A. P., Pinzul A., Qureshi B., “SUSY anomalies break $N=2$ to $N=1$: the supersphere and the fuzzy supersphere”, J. High Energy Phys., 2005:12 (2005), 002, 14 pp., ages ; hep-th/0506037 | DOI | MR

[19] Kurkcuoglu S., “Non-linear sigma model on the fuzzy supersphere”, J. High Energy Phys., 2004:3 (2004), 062, 12 pp., ages ; hep-th/0311031 | DOI | MR

[20] Schunck A. F., Wainwright Ch., “A geometric approach to scalar field theories on the supersphere”, J. Math. Phys., 46 (2005), 033511, 34 pp., ages ; hep-th/0409257 | DOI | MR | Zbl

[21] Panero M., “Quantum field theory in a non-commutative space: theoretical predictions and numerical results on the fuzzy sphere”, SIGMA, 2 (2006), 081, 14 pp., ages ; hep-th/0609205 | MR | Zbl

[22] Ivanov E., Mezincescu L., Townsend P. K., Fuzzy $CP(n|m)$ as a quantum superspace, hep-th/0311159

[23] Ivanov E., Mezincescu L., Townsend P. K., A super-flag Landau model, hep-th/0404108 | MR

[24] Ivanov E., Mezincescu L., Townsend P. K., “Planar super-Landau models”, J. High Energy Phys., 2006:1 (2006), 143, 23 pp., ages ; hep-th/0510019 | DOI | MR

[25] Curtright T., Ivanov E., Mezincescu L., Townsend P. K., “Planar super-Landau models revisited”, J. High Energy Phys., 2007:4 (2007), 020, 25 pp., ages ; hep-th/0612300 | DOI | MR

[26] Bellucci S., Beylin A., Krivonos S., Nersessian A., Orazi E., “$N=4$ supersymmetric mechanics with nonlinear chiral supermultiplet”, Phys. Lett. B, 616 (2005), 228–232 ; hep-th/0503244 | DOI | MR

[27] Gates S. J. Jr., Jellal A., Saidi E. H., Schreiber M., “Supersymmetric embedding of the quantum Hall matrix model”, J. High Energy Phys., 2004:11 (2004), 075, 29 pp., ages ; hep-th/0410070 | DOI | MR

[28] Yu M., Zhang X., Supersymmetric Hamiltonian approach to edge excitations in $\nu=5/2$ fractional quantum Hall effect, arXiv:0706.1338

[29] Haldane F. D. M., “Fractional quantization of the Hall effect: a hierarchy of incompressible quantum fluid states”, Phys. Rev. Lett., 51 (1983), 605–608 | DOI | MR

[30] Frappat L., Sciarrino A., Sorba P., Dictionary on Lie algebras and superalgebras, Academic Press, San Diego, 2000, and references therein | MR

[31] Nakahara M., Geometry, topology and physics, IOP Publishing, Bristol, 2003 | MR | Zbl

[32] Bartocci C., Bruzzo U., Landi G., “Chern–Simons forms on principal superfiber bundles”, J. Math. Phys., 31 (1987), 45–54 | DOI | MR

[33] Landi G., “Projective modules of finite type over the supersphere $S^{2,2}$”, Differential Geom. Appl., 14 (2001), 95–111 ; math-ph/9907020 | DOI | MR | Zbl

[34] Hasebe K., Kimura Y., “Fuzzy supersphere and supermonopole”, Nuclear Phys. B, 709 (2005), 94–114 ; hep-th/0409230 | DOI | MR | Zbl

[35] Hasebe K., “Supersymmetric quantum Hall effect on a fuzzy supersphere”, Phys. Rev. Lett., 94 (2005), 206802, 4 pp., ages ; hep-th/0411137 | DOI

[36] Grosse H., Klimcik C., Presnajder P., “Field theory on a supersymmetric lattice”, Comm. Math. Phys., 185 (1997), 155–175 ; hep-th/9507074 | DOI | MR | Zbl

[37] Grosse H., Reiter G., “The fuzzy supersphere”, J. Geom. Phys., 28 (1998), 349–383 ; math-ph/9804013 | DOI | MR | Zbl

[38] Balachandran A. P., Kurkcuoglu S., Vaidya S., Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114

[39] Hatsuda M., Iso S., Umetsu H., “Noncommutative superspace, supermatrix and lowest Landau level”, Nuclear Phys. B, 671 (2003), 217–242 ; hep-th/0306251 | DOI | MR | Zbl

[40] Hasebe K., “Quantum Hall liquid on a noncommutative superplane”, Phys. Rev. D, 72 (2005), 105017, 9 pp., ages ; hep-th/0503162 | DOI

[41] Hasebe K., “Unification of Laughlin and Moore–Read states in SUSY quantum Hall effect”, Phys. Lett. A, 372 (2008), 1516–1520 ; arXiv:0705.4527 | DOI

[42] Moore G., Read N., “Nonabelions in the fractional quantum hall effect”, Nuclear Phys. B, 360 (1991), 362–396 | DOI | MR

[43] Greiter M., Wen X-G., Wilczek F., “Paired Hall state at half filling”, Phys. Rev. Lett., 66 (1991), 3205–3208 | DOI | MR

[44] 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 | DOI

[45] Lee D. H., Fisher M. P. A., “Anyon superconductivity and charge-vortex duality”, Internat. J. Modern Phys. B, 5 (1991), 2675–2699 | DOI | MR

[46] Hasebe K., “Supersymmetric Chern–Simons theory and supersymmetric quantum Hall liquid”, Phys. Rev. D, 74 (2006), 045026, 12 pp., ages ; hep-th/0606007 | DOI | MR

[47] Nissimov E., Pacheva S., “Phase transition and $1/N$ expansion in (2+1)-dimensional supersymmetric sigma models”, Lett. Math. Phys., 5 (1981), 67–73 ; Nissimov E., Pacheva S., “Parity-violating anomalies in supersymmetric gauge theories”, Phys. Lett. B, 155 (1985), 76–82 ; Nissimov E., Pacheva S., “Anomalous generation of Chern–Simons terms in $D=3$, $N=2$ supersymmetric gauge theories”, Lett. Math. Phys., 11 (1986), 43–49 | DOI | MR | DOI | MR | DOI | MR | Zbl

[48] Lee B. H., Lee C. K., Min H., “Supersymmetric Chern–Simons vortex systems and fermion zero modes”, Phys. Rev. D, 45 (1992), 4588–4599 | DOI | MR

[49] Ezawa K., Ishikawa A., “$Osp(1|2)$ Chern–Simons gauge theory as 2D $N=1$ induced supergravity”, Phys. Rev. D, 56 (1997), 2362–2368 ; hep-th/9612031 | DOI | MR

[50] Sparling G., Twistor theory and the four-dimensional quantum Hall effect of Zhang and Hu, cond-mat/0211679

[51] Mihai D., Sparling G., Tillman Ph., Non-commutative time, the quantum Hall effect and twistor theory, cond-mat/0401224