Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE-𝒪 Classification
Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 45 (1993), Exposé no. 6, 79 p. 
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     author = {Fr\"ohlich, J\"urg and Thiran, Emmanuel},
     title = {Integral {Quadratic} {Forms,} {Kac-Moody} {Algebras,} and {Fractional} {Quantum} {Hall} {Effect.} {An} {ADE-}$\mathcal {O}$ {Classification}},
     journal = {Les rencontres physiciens-math\'ematiciens de Strasbourg -RCP25},
     note = {talk:6},
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     publisher = {Institut de Recherche Math\'ematique Avanc\'ee - Universit\'e Louis Pasteur},
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Fröhlich, Jürg; Thiran, Emmanuel. Integral Quadratic Forms, Kac-Moody Algebras, and Fractional Quantum Hall Effect. An ADE-$\mathcal {O}$ Classification. Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 45 (1993), Exposé no. 6, 79 p. http://archive.numdam.org/item/RCP25_1993__45__71_0/

[1] K. Von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980).

[2] D. C. Tsui, H. L. Stormer and A. C. Gossard, Phys. Rev. B 48, 1559 (1982).

[3] R. Tao and Y.-S. Wu, Phys. Rev. B 31, 6859 (1985). | MR

[4] R. L. Willet, J. P. Eisenstein, H. L. Stormer, D. C. Tsui, A. C. Gossard and J. H. English, Phys. Rev. Lett. 59, 1776 (1987)

J. P. Eisenstein, R. L. Willet, H. L. Stormer, D. C. Tsui, A. C. Gossard and J. H. English, Phys. Rev. Lett. 61, 997 (1988); and references therein

J. P. Eisenstein, R. L. Willet, H. L. Stormer, L. N. Pfeiffer and K. W. West, Surf. Science 229, 31 (1990).

[5] Y. W. Suen, L. W. Engel, M. B. Santos, M. Shayegan and D. C. Tsui, Phys. Rev. Lett. 68, 1379 (1992);

J. P. Eisenstein, G. S. Boeblinger, L. N. Pfeiffer, K. W. West and Song He, Phys. Rev. Lett. 68, 1383 (1992).

[6] R. G. Clark, J. R. Mallet, S. R. Haynes, J. J. Harris and C. T. Foxon, Phys. Rev. Lett. 60, 1747 (1988);

A. M. Chang and J. E. Cunningham, Solid State Commun. 72, 652 (1989);

R. G. Clark et al., cited in [8] for σ H =2/3 study

[7] J. A. Simmons, H. P. Wei, L. W. Engel, D. C. Tsui and M. Shayegan, Phys. Rev. Lett. 63, 1731 (1989);

S. W. Hwang, J. A. Simmons, D. C. Tsui and M. Shayegan, Surf. Science 263, 72 (1992).

[8] Experimental results for tilted magnetic field transition at σ H =8 5 and σ H =5 3 : J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer and K. W. West, Phys. Rev. Lett. 62, 1540 (1989);

Experimental results for tilted magnetic field transition at σ H =8 5 and σ H =5 3 : J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer and K. W. West Surf. Science 229, 21 (1990)

at σ H =4 3 : R. G. Clark, S. R. Haynes, A. M. Suckling, J. R. Mallet, P. A. Wright, J. J. Harris and C. T. Foxon, Phys. Rev. Lett. 62, 1536 (1989)

at σ H =2 3 :J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer and K. W. West, Phys. Rev. B 41, 7910 (1990);

R. G. Clark, S. R. Haynes, J. V. Branch, A. M. Suckling, P. A. Wright, P. M. W. Oswald, J. J. Harris and C. T. Foxon, Surf. Science 229, 25 (1990)

at σ H =5 2 : see [4]; See also D. A. Syphers and J. E. Furneaux, Surf. Science 196, 252 (1988); Solid State Commun. 65, 1513 (1988);

J. Haug, K. Von Klitzing, R. J. Nicholas, J. C. Maan and G. Weimann, Phys. Rev. B 36, 4528 (1987).

[9] For reviews of the quantum Hall effect and comprehensive compilations of references, see e.g. R. E. Prange and S. M. Girvin, eds., "The Quantum Hall Effect", Second Edition, Graduate Texts in Contemporary Physics (Springer, New York, 1990);

G. Morandi, "Quantum Hall Effect" (Bibliopolis, Napoli, 1988); | MR | Zbl

T. Chakraborty and P. Pietiläinen, "The Fractional Quantum Hall Effect: Properties of an Incompressible Quantum Fluid", Springer Series in Solid State Science 85 (Springer, Berlin, 1988);

M. Stone, ed., "Quantum Hall Effect" (World Scientific, Singapore, 1992). See also: | MR | Zbl

F. Wilczeck, "Fractional Statistics and Anyon Superconductivity" (World Seientific, Singapore, 1990).

[10] For IQH: R. B. Laughlin, Phys. Rev. B 23, 5632 (1981);

for FQH: R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983);

R. B. Laughlin Phys. Rev. B 27, 3383 (1983);

R. B. Laughlin Surf. Science 141, 11 (1984); see also [28].

X. G. Wen, Phys. Rev. B 40, 7387 (1989);

X. G. Wen Phys. Rev. Lett. 64, 2206 (1990);

X. G. Wen Phys. Rev. B 41, 12838 (1990);

X. G. Wen Phys. Rev. B 43, 11025 (1991);

X. G. Wen Phys. Rev. Lett. 66, 802 (1991); | Zbl

X. G. Wen Int. J. Mod. Phys. B 6, 1711 (1992);

X. G. Wen and Q. Niu, Phys. Rev., B 41, 9377 (1990);

B. Block and X. G. Wen, Phys. Rev. B 42, 8133, 8145 (1990);

X. G. Wen and A. Zee, Phys. Rev. B 46, 2290 (1992).

[12] J. Fröhlich and T. Kerler, Nucl. Phys. B 354, 369 (1991);

J. Fröhlich and A. Zee, Nucl. Phys. B 364, 517 (1991);

J. Fröhlich and U. M. Studer, Commun. Math. Phys. 148, 553 (1992); | Zbl

J. Fröhlich and U. M. Studer Int. J. Mod. Phys. B6, 2201 (1992); "Incompressible Quantum Fluids, Gauge Invariance and Current Algebra", in J. Fröhlich et al., eds., New Symmetry Principles in Quantum Field Theory, Cargèse Lectures 1991 (Plenum Press New York, 1992); | Zbl

J. Fröhlich and U. M. Studer "Gauge Invariance and Current Algebra in Non-Relativistic Many-Body Theory", Rev. Mod. Phys. 65, 733 (1993).

[13] M. Stone, Int. J. Mod. Phys. B5, 509 (1991);

M. Stone Ann. Phys. (New York) 207, 38 (1991).

A. V. Balatsky, Phys. Rev. B 43, 1257 (1991).

[14] See J. Fröhlich and T. Kerler, cited in [12].

[15] J. H. Conway, N. J. A. Sloane, "Sphere Packings, Lattices and Groups" (Springer-Verlag, New York, 1988); | Zbl

J. H. Conway, F. R. S. Sloane and N. J. A. Sloane, Proc. R. Soc. Lond. A418, 17 (1988); ibid, 419, 29 (1988); ibid 418, 43 (1988); ibid 419, 259 (1988).

[16] J. W. S. Cassels, "Rational Quadratic Forms", (Academic Press, 1978). | Zbl

[17] E. Fradkin, "Field theories of Condensed Matter Systems", Frontiers in Physics, Vol. 82 (Addison-Wesley, Redwood City, 1991). | Zbl

[18] J. Fröhlich, U. M. Studer, cited in [12].

[19] R. E. Prange and S. M. Girvin, eds. "The Quantum Hall Effect", cited in [9].

[20] M. Stone, ed., "Quantum Hall Effect", cited in [9]. | Zbl

[21] B. I. Halperin, Phys. Rev. B 25, 2185 (1982).

[22] P. Goddard and D. Olive, Int. J. Mod. Phys. A1, 303 (1986).

[23] P. Ginsparg, "Applied Conformai Field Theory" in E. Brézin and J. Zinn-Justin, eds., "Fields, Strings and Critical Phenomena", Les Houches, Session XLIX, 1988

P. Ginsparg (North Holland, Elsevier Science Publishers B. V., 1990).

[24] E. Witten, Commun. Math. Phys. 121, 351 (1989). | Zbl

[25] J. Fröhlich and C. King, Commun. Math. Phys. 126, 167 (1989); | Zbl

J. Fröhlich and C. King Int. J. Mod. Phys. A4, 5321 (1989).

[26] S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Nucl. Phys. B 326, 108 (1989).

[27] J. Fröhlich and T. Kerler (1991), cited in [12]; J. Fröhlich and Z. Zee (1991), ibid.

[28] R. B. Laughlin, Chap. 7 in R. E. Prange and S. M. Girvin, eds., cited in [9]; F. D. M. Haldane, Chap. 8, ibid.

[29] K. Gawedzki, in G. 't Hooft et al., eds., "Non perturbative Quantum Field Theory" (Plenum, New York, 1988).

[30] X. G. Wen, Int. J. Mod. Phys., B 4, 239 (1990).

[31] G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991);

G. Moore and N. Read Prog. Theor. Phys. Suppl. 107, 157 (1992).

[32] X. G. Wen and A. Zee, Phys. Rev. Lett. 69, 953 (1992).

[33] N. Read, Phys. Rev. Lett. 65, 1502 (1990).

[34] V. V. Nikulin, Math. USSR-Izv. 14 (1979), n° 1, 103-167 (1980). | Zbl

[35] J. Fröhlich and P. A. Marchetti, Lett. Math. Phys. 16, 347 (1988); | Zbl

J. Fröhlich and P. A. Marchetti Commun. Math. Phys. 121, 177 (1989). | Zbl

[36] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B 241, 333 (1984). | Zbl

[37] J. Fröhlich, U. M. Studer and E. Thiran, "Gauge Symmetry, Integral Lattices and the Classification of Quantum Hall fluids", preprint KUL-TF-93/33; "An ADE-O Classification of Minimal Incompressible Quantum Hall fluids", to appear in the proceedings of the conference "On Three Levels", Leuven (Belgium), 1993, A. Verbeure et al., eds, Plenum.

[38] V. Kac, M. Wakimoto, Adv. Math. 70, 156 (1988); | Zbl

V. Kac, N. Sanielevici, Phys. Rev. D37, 2231 (1988);

M. A. Walton, Nucl. Phys. B322, 775 (1989);

D. Altschuler, M. Bauer and C. Itzykson, Commun. Math. Phys. 132, 349 (1990); | Zbl

D. Versiegen, Commun. Math. Phys. 137, 567 (1991); for the classification of Conformal Embeddings see [43].

[39] A. H. Chamseddine and J. Fröhlich, Commun. Math. Phys. 147, 549 (1992). | Zbl

[40] J. Fröhlich, T. Kerler and E. Thiran, in preparation.

[41] B. Block and X. G. Wen, Phys. Rev. B42, 8145 (1990).

[42] R. Slansky, Phys. Rep. 79, 1 (1981).

[43] F. A. Bais and P. Bouwknegt, Nucl. Phys. B279, 561 (1987);

A. N. Schellekens and N. P. Warner, Phys. Rev. D34, 3092 (1986).

[44] P. Engel, L. Michel and M. Senechal, Lattice Geometry (1993), in preparation.

[45] F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983);

B. I. Halperin, Phys. Rev. Lett. 51, 1583 (1983);

for a review see F. D. M. Haldane, cited in [28]; see also [33];

J. K. Jain and V. J. Goldman, Phys. Rev. B45, 1255 (1992);

see also Y. J. Chen, Phys. Rev. B46, 7941 (1992).

[46] R. C. Aschoori et al., Phys. Rev. B 45, 3894 (1992).