Classification of local conformal nets. Case c < 1

Annals of Mathematics

Volumn 160, Issue 2, 2004, Pages 493-522

  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: 15944387050     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.4007/annals.2004.160.493     Document Type: Article

References (76)

1. M. ASAEDA and U. HAAGERUP, Exotic subfactors of finite depth with Jones indices (5 + √13)/2 and (5 + √17)/2, Commun. Math. Phys. 202 (1999), 1-63.
2. A. A. BELAVIN, A. M. POLYAKOV, and A. B. ZAMOLODCHIKOV, Infinite conformal symmetry in two-dimensional quantum field theory, Nud. Phys. 241 (1984), 333-380.
3. J. BÖCKENHAUER and D. E. EVANS, Modular invariants, graphs and α-induction for nets of subfactors I, Commun. Math. Phys. 197 (1998), 361-386; II 200 (1999), 57-103; III 205 (1999), 183-228.
4. J. BÖCKENHAUER and D. E. EVANS, Modular invariants, graphs and α-induction for nets of subfactors I, Commun. Math. Phys. 197 (1998), 361-386; II 200 (1999), 57-103; III 205 (1999), 183-228.
5. J. BÖCKENHAUER and D. E. EVANS, Modular invariants, graphs and α-induction for nets of subfactors I, Commun. Math. Phys. 197 (1998), 361-386; II 200 (1999), 57-103; III 205 (1999), 183-228.
6. _, Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors, Commun. Math. Phys. 213 (2000), 267-289.
7. J. BÖCKENHAUER, D. E. EVANS, and Y. KAWAHIGASHI, On a-induction, chiral projectors and modular invariants for subfactors, Commun. Math. Phys. 208 (1999), 429-487.
8. _, Chiral structure of modular invariants for subfactors, Commun. Math. Phys. 210 (2000), 733-784.
9. _, Longo-Rehren subfactors arising from a-induction, Publ. RIMS, Kyoto Univ. 37 (2001), 1-35.
10. R. BRUNETTI, D. GUIDO, and R. LONGO, Modular structure and duality in conformal quantum field theory, Commun. Math. Phys. 156 (1993), 201-219.
11. D. BUCHHOLZ, G. MACK, and I. TODOROV, The current algebra on the circle as a germ of local field theories, Nud. Phys. B, Proc. Suppl. 5B (1988), 20-56.
12. D. BUCHHOLZ and H. SCHULZ-MIRBACH, Haag duality in conformal quantum field theory, Rev. Math. Phys. 2 (1990), 105-125.
13. (1) conformal invariant theories, Commun. Math. Phys. 113 (1987), 1-26.
14. S. CARPI, Absence of subsystems for the Haag-Kastler net generated by the energy-momentum tensor in two-dimensional conformal field theory, Lett. Math. Phys. 45 (1998), 259-267.
15. C. D'ANTONI, R. LONGO, and F. RADULESCU, Conformal nets, maximal temperature and models from free probability, J. Operator Theory 45 (2001), 195-208.
16. P. Di FRANCESCO, P. MATHIEU, and D. SÉNÉCHAL, Conformal Field Theory, Springer-Verlag, New York, 1996.
17. S. DOPLICHER, R. HAAG, and J. E. ROBERTS, Local observables and particle statistics, I. Commun. Math. Phys. 23 (1971), 199-230; II. 35 (1974), 49-85.
18. S. DOPLICHER, R. HAAG, and J. E. ROBERTS, Local observables and particle statistics, I. Commun. Math. Phys. 23 (1971), 199-230; II. 35 (1974), 49-85.
19. S. DOPLICHER and R. LONGO, Standard and split inclusions of von Neumann algebras, Invent. Math. 73 (1984), 493-536.
20. D. E. EVANS and Y. KAWAHIGASHI, Quantum Symmetries on Operator Algebras, Oxford University Press, Oxford, 1998.
21. K. FREDENHAGEN and M. JÖRSS, Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansion, Commun. Math. Phys. 176 (1996), 541-554.
22. K. FREDENHAGEN, K.-H. REHREN, and B. SCHROER, Superselection sectors with braid group statistics and exchange algebras, I. Commun. Math. Phys. 125 (1989), 201-226, II. Rev. Math. Phys. Special issue (1992), 113-157.
23. K. FREDENHAGEN, K.-H. REHREN, and B. SCHROER, Superselection sectors with braid group statistics and exchange algebras, I. Commun. Math. Phys. 125 (1989), 201-226, II. Rev. Math. Phys. Special issue (1992), 113-157.
24. D. FRIEDAN, Z. QIU, and S. SHENKER, Details of the nonunitarity proof for highest weight representations of the Virasoro algebra, Commun. Math. Phys. 107 (1986), 535-542.
25. J. FRÖHLICH and F. GABBIANI, Operator algebras and conformal field theory, Commun. Math. Phys. 155 (1993), 569-640.
26. T. CANNON, Modular data: the algebraic combinatorics of conformal field theory, preprint 2001, math.QA/0103044.
27. P. GODDARD, A. KENT, and D. OLIVE, Unitary representations of the Virasoro and super-Virasoro algebras, Commun. Math. Phys. 103 (1986), 105-119.
28. F. GOODMAN, P. DE LA HARPE, and V. F. R. JONES, Coxeter Graphs and Towers of Algebras, MSRI PUBLICATIONS 14 (1989), Springer, New York.
29. D. GUIDO and R. LONGO, Relativistic invariance and charge conjugation in quantum field theory, Commun. Math. Phys. 148 (1992), 521-551.
30. _, The conformal spin and statistics theorem, Commun. Math. Phys. 181 (1996), 11-35.
31. D. GUIDO, R. LONGO, and H.-W. WIESBROCK, Extensions of conformal nets and superselection structures, Commun. Math. Phys. 192 (1998), 217-244.
32. P. HISLOP and R. LONGO, Modular structure of the local algebras associated with the free massless scalar field theory, Commun. Math. Phys. 84 (1982), 71-85.
33. R. HAAG, Local Quantum Physics, Springer-Verlag, New York, 1996.
34. M. IZUMI, Subalgebras of infinite C*-algebras with finite Watatani indices II: CuntzKrieger algebras, Duke Math. J. 91 (1998), 409-461.
35. _, The structure of sectors associated with the Longo-Rehren inclusions, Commun. Math. Phys. 213 (2000), 127-179.
36. M. IZUMI and H. KOSAKI, On a subfactor analogue of the second cohomology, Rev. Math. Phys. 14 (2002), 733-757.
37. M. IZUMI, R. LONGO, and S. POPA, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Punct. Anal. 10 (1998), 25-63.
38. V. F. R. JONES, Index for subfactors, Invent. Math. 72 (1983), 1-25.
39. V. G. KAC and A. K. RAINA, Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientific 1987.
40. A. KATO, Classification of modular invariant partition functions in two dimensions, Modern Phys. Lett. A2 (1987), 585-600.
41. Y. KAWAHIGASHI, On flatness of Ocneanu's connections on the Dynkin diagrams and classification of subfactors, J. Fund. Anal. 127 (1995), 63-107.
42. _, Classification of paragroup actions on subfactors, Publ. RIMS, Kyoto Univ. 31 (1995), 481-517.
43. Y. KAWAHIGASHI and R. LONGO, Classification of two-dimensional local conformal nets with c < 1 and 2-cohomology vanishing for tensor categories, Commun. Math. Phys. 244 (2004), 63-97.
44. Y. KAWAHIGASHI, R. LONGO, and M. MÜGER, Multi-interval subfactors and modularity of representations in conformal field theory, Commun. Math. Phys. 219 (2001), 631-669.
45. (2) conformal field theories, Adv. Math. 171 (2002), 183-227.
46. 1), Thesis, University of Cambridge, 1994.
47. R. LONGO, Index of subfactors and statistics of quantum fields, I. Commun. Math. Phys. 126 (1989), 217-247, and II. 130 (1990), 285-309.
48. R. LONGO, Index of subfactors and statistics of quantum fields, I. Commun. Math. Phys. 126 (1989), 217-247, and II. 130 (1990), 285-309.
49. _, A duality for Hopf algebras and for subfactors, Commun. Math. Phys. 159 (1994), 133-150.
50. _, Conformal subnets and intermediate subfactors, Commun. Math. Phys. 237 (2003), 7-30; math.OA/0102196.
51. _, R. LONGO and K.-H. REHREN, Nets of subfactors, Rev. Math. Phys. 7 (1995), 567-597.
52. R. LONGO and J. E. ROBERTS, A theory of dimension, K-theory 11 (1997), 103-159.
53. A. OCNEANU, 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 University Press, Cambridge, 1988, 119-172.
54. A. OCNEANU, 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 University Press, Cambridge, 1988, 119-172.
55. _, Paths on Coxeter diagrams: from Platonic solids and singularities to minimal models and subfactors (Notes recorded by S. Goto), in Lectures on Operator Theory, (ed. B. V. Rajarama Bhat et al.), The Fields Institute Monographs, AMS Publications, 2000, 243-323.
56. M. PIMSNER and S. POPA, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. 19 (1986), 57-106.
57. S. POPA, Orthogonal pairs of *-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), 253-268.
58. _, Symmetric enveloping algebras, amenability and AFD properties for subfactors, Math. Res. Lett. I (1994), 409-425.
59. A. PRESSLEY and G. SEGAL, Loop Groups, Oxford University Press, Oxford, 1986.
60. K.-H. REHREN, Braid group statistics and their superselection rules, in The Algebraic Theory of Superselection Sectors (D. Kastler ed.), World Scientific, Singapore, 1990.
61. _, Space-time fields and exchange fields, Commun. Math. Phys. 132 (1990), 461-483.
62. _, A new view of the Virasoro algebra, Lett. Math. Phys. 30 (1994), 125-130.
63. _, Chiral observables and modular invariants, Commun. Math. Phys. 208 (2000), 689-712.
64. _, Canonical tensor product subfactors, Commun. Math. Phys. 211 (2000), 395-406.
65. _, Locality and modular invariance in 2D conformal QFT, in Mathematical Physics in Mathematics and Physics (R. Longo, ed.), Fields Inst. Commun. 30 (2001), AMS Publications, 341-354; math-ph/0009004.
66. M. TAKESAKI, Theory of Operator Algebras. I, Springer-Verlag, New York, 1979.
67. 2n, Thesis, University of Cambridge, 1997.
68. V. G. TURAEV, Quantum Invariants of Knots and 3-Manifolds, Walter de Gruyter, Berlin-New York, 1994.
69. A. WASSERMANN, Operator algebras and conformal field theory III: Fusion of positive energy representations of SU(N) using bounded operators, Invent. Math. 133 (1998), 467-538.
70. F. Xu, New braided endomorphisms from conformal inclusions, Commun. Math. Phys. 192 (1998), 347-403.
71. _, Applications of braided endomorphisms from conformal inclusions, Internat. Math. Res. Notices (1998), 5-23.
72. _, Jones-Wassermann subfactors for disconnected intervals, Commun. Contemp. Math. 2 (2000), 307-347.
73. _, Algebraic coset conformal field theories I, Commun. Math. Phys. 211 (2000), 1-44.
74. _, On a conjecture of Kac-Wakimoto, Publ. RIMS, Kyoto Univ. 37 (2001), 165-190.
75. _, 3-manifold invariants from cosets, preprint 1999, math.GT/9907077.
76. _, Algebraic orbifold conformal field theories, Proc. Nat. Acad. Sci. U.S.A. 97 (2000), 14069-14073.