Classification of Two-Dimensional Local Conformal Nets with c < 1 and 2-Cohomology Vanishing for Tensor Categories

Communications in Mathematical Physics

Volumn 244, Issue 1, 2004, Pages 63-97

  Kawahigashi, Yasuyuki ^a   Longo, Roberto ^b  

^a UNIVERSITY OF TOKYO   (Japan)

^b UNIVERSITY OF ROME TOR VERGATA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0742320193     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-003-0979-1     Document Type: Article

References (60)

1. Baumgärtel, H.: Operatoralgebraic Methods in Quantum Field Theory. Berlin: Akademie Verlag, 1995
2. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. 241, 333-380 (1984)
3. Böckenhauer, J., Evans, D.E.: Modular invariants, graphs and α-induction for nets of subfactors I. Commun. Math. Phys. 197, 361-386 (1998), II. 200, 57-103 (1999), III. 205, 183-228 (1999)
4. Böckenhauer, J., Evans, D.E.: Modular invariants, graphs and α-induction for nets of subfactors I. Commun. Math. Phys. 197, 361-386 (1998), II. 200, 57-103 (1999), III. 205, 183-228 (1999)
5. Böckenhauer, J., Evans, D.E.: Modular invariants, graphs and α-induction for nets of subfactors I. Commun. Math. Phys. 197, 361-386 (1998), II. 200, 57-103 (1999), III. 205, 183-228 (1999)
6. Böckenhauer, J., Evans, D.E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors. Commun. Math. Phys. 213, 267-289 (2000)
7. Böckenhauer, J., Evans, D.E., Kawahigashi, Y: On α-induction, chiral projectors and modular invariants for subfactors. Commun. Math. Phys. 208, 429-487 (1999)
8. Böckenhauer, J., Evans, D.E., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733-784 (2000)
9. Böckenhauer, J., Evans, D.E., Kawahigashi, Y.: Longo-Rehren subfactors arising from α-induction. Publ. RIMS, Kyoto Univ. 37, 1-35 (2001)
10. Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201-219 (1993)
11. (1) conformal invariant theories. Commun. Math. Phys. 113, 1-26 (1987)
12. Carpi, S.: On the representation theory of Virasoro Nets. To appear in Commun. Math. Phys., math.OA/0306425
13. Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Berlin-Heidelberg-New York: Springer-Verlag, 1996
14. Doplicher, S., Haag, R., Roberts, J. E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199-230 (1971), II. 35, 49-85 (1974)
15. Doplicher, S., Haag, R., Roberts, J. E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199-230 (1971), II. 35, 49-85 (1974)
16. Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Invent. Math. 98, 157-218 (1989)
17. Evans, D. E., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford: Oxford University Press, 1998
18. Fredenhagen, K., Jörß, M.: Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansion. Commun. Math. Phys. 176, 541-554 (1996)
19. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. I. Commun. Math. Phys. 125, 201-226 (1989), II. Rev. Math. Phys. Special issue, 113-157 (1992); Fröhlich, J.: Statistics of fields, the Yang-Baxter equation, and the theory of knots and links. In: Nonperturbative quantum field theory (Cargése, 1987), NATO Adv. Sci. Inst. Ser. B Phys., 185, NewYork: Plenum, 1988, pp. 71-100
20. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. I. Commun. Math. Phys. 125, 201-226 (1989), II. Rev. Math. Phys. Special issue, 113-157 (1992); Fröhlich, J.: Statistics of fields, the Yang-Baxter equation, and the theory of knots and links. In: Nonperturbative quantum field theory (Cargése, 1987), NATO Adv. Sci. Inst. Ser. B Phys., 185, NewYork: Plenum, 1988, pp. 71-100
21. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. I. Commun. Math. Phys. 125, 201-226 (1989), II. Rev. Math. Phys. Special issue, 113-157 (1992); Fröhlich, J.: Statistics of fields, the Yang-Baxter equation, and the theory of knots and links. In: Nonperturbative quantum field theory (Cargése, 1987), NATO Adv. Sci. Inst. Ser. B Phys., 185, NewYork: Plenum, 1988, pp. 71-100
22. Friedan, D., Qiu, Z., Shenker, S.: Details of the non-unitarity proof for highest weight representations of the Virasoro algebra. Commun. Math. Phys. 107, 535-542 (1986)
23. Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103, 105-119 (1986)
24. Goddard, P., Olive, D. (eds.): Kac-Moody and Virasoro algebras. A Reprint Volume for Physicists. Singapore: World Scientific, 1988
25. Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11-35 (1996)
26. 2/CFT correspondence. gr-qc/0212025, to appear in Ann. H. Poincaré
27. Haag, R.: Local Quantum Physics. 2nd ed., Berlin-Heidelberg-New York: Springer, 1996
28. Hislop, P.D., Longo, R.: Modular structure of the von Neumann algebras associated with the free massless scalar field theory. Commun. Math. Phys. 84, 71-85 (1982)
29. Izumi, M.: Application of fusion rules to classification of subfactors. Publ. RIMS, Kyoto Univ. 27, 953-994 (1991)
30. Izumi, M.: Subalgebras of infinite C*-algebras with finite Watatani indices II: Cuntz-Krieger algebras. Duke Math. J. 91, 409-461 (1998)
31. Izumi, M.: The structure of sectors associated with the Longo-Rehren inclusions. Commun. Math. Phys. 213, 127-179 (2000)
32. Izumi, M., Kosaki, H.: On a subfactor analogue of the second cohomology. Rev. Math. Phys. 14, 733-757 (2002)
33. Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1-25 (1983)
34. Kauffman, L., Lins, S.L.: Temperley-Lieb recoupling theory and invariants of 3-manifolds. Princeton, NJ: Princeton University Press, 1994
35. Kawahigashi, Y.: On flatness of Ocneanu's connections on the Dynkin diagrams and classification of subfactors. J. Funct. Anal. 127, 63-107 (1995)
36. Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case c<1. To appear in Ann. Math., math-ph/0201015
37. Kawahigashi, Y., Longo, R., Müger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631-669 (2001)
38. (2) conformal field theories. Adv. Math. 171, 183-227 (2002)
39. Köster, S.: Local nature of coset models. Preprint 2003, math-ph/0303054
40. Longo, R.: Index of subfactors and statistics of quantum fields I-II. Commun. Math. Phys. 126, 217-247 (1989) & 130, 285-309 (1990)
41. Longo, R.: Index of subfactors and statistics of quantum fields I-II. Commun. Math. Phys. 126, 217-247 (1989) & 130, 285-309 (1990)
42. Longo R.: A duality for Hopf algebras and for subfactors. Commun. Math. Phys. 159, 133-150 (1994)
43. Longo, R.: Conformal subnets and intermediate subfactors. Commun. Math. Phys. 237, 7-30 (2003), math.OA/0102196
44. Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567-597 (1995)
45. Longo, R., Roberts, J. E.: A theory of dimension. K-theory 11, 103-159 (1997)
46. Masuda, T.: An analogue of Longo's canonical endomorphism for bimodule theory and its application to asymptotic inclusions. Internat. J. Math. 8, 249-265 (1997)
47. Müger, M.: Extensions and modular invariants of rational conformal field theories. In preparation
48. Ocneanu, A.: Quantized group, string algebras and Galois theory for algebras. In: Operator algebras and applications, Vol. 2 (Warwick, 1987). D.E. Evans and M. Takesaki, (eds.), London Mathematical Society Lecture Note Series 36, Cambridge: Cambridge University Press, 1988, pp. 119-172.
49. Ocneanu, A.: Quantum symmetry, differential geometry of finite graphs and classification of subfactors. University of Tokyo Seminary Notes 45, Notes recorded by Y. Kawahigashi, 1991
50. Popa, S.: Symmetric enveloping algebras, amenability and AFD properties for subfactors. Math. Res. Lett. 1, 409-425 (1994)
51. Popa, S.: Classification of subfactors and of their endomorphisms. CBMS Regional Conference Series. Amer. Math. Soc. 86 (1995)
52. Rehren, K.-H.: Braid group statistics and their superselection rules. In: The Algebraic Theory of Superselection Sectors, D. Kastler, (ed.), Singapore: World Scientific, 1990
53. Rehren, K.-H.: Chiral observables and modular invariants. Commun. Math. Phys. 208, 689-712 (2000)
54. Rehren, K.-H.: Canonical tensor product subfactors. Commun. Math. Phys. 211, 395-406 (2000)
55. Rehren, K.-H.: Locality and modular invariance in 2D conformal QFT. In: Mathematical Physics in Mathematics and Physics, R. Longo, (ed.), Fields Inst. Commun. 30, AMS Publications, Providence, RI: AMS, 2001 pp. 341-354, math-ph/0009004
56. Turaev, V.G.: Quantum invariants of knots and 3-manifolds. Berlin-New York: Walter de Gruyter, 1994
57. Takesaki, M.: Theory of Operator Algebras. Vol. I, II, III, Springer Encyclopaedia of Math. Sci. 124 (2002), 125, 127 (2003)
58. Xu, F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys. 192, 347-403 (1998)
59. Xu, F.: Algebraic coset conformal field theories I. Commun. Math. Phys. 211, 1-44 (2000)
60. Xu, F.: Strong additivity and conformal nets. Preprint 2003, math.QA/0303266