Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

On the origin and development of subfactors and quantum topology


Author: Vaughan Jones
Journal: Bull. Amer. Math. Soc. 46 (2009), 309-326
MSC (2000): Primary 46L37; Secondary 46L54
DOI: https://doi.org/10.1090/S0273-0979-09-01244-0
Published electronically: January 28, 2009
MathSciNet review: 2476415
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give an account of the beginning of subfactor theory and TQFT and some more recent developments.


References [Enhancements On Off] (What's this?)

  • [HaAs99] M. Asaeda, U. Haagerup, Exotic subfactors of finite depth with Jones indices $ (5+\sqrt{13})/2$ and $ (5+\sqrt{17})/2$, Comm. Math. Phys. 202 (1999), 1-63. MR 1686551 (2000c:46120)
  • [BaPo86] B. M. Baker, R. T. Powers, Product states of certain group-invariant AF-algebras, J. Operator Theory 16 (1986), no. 1, 3-50. MR 847331 (87m:46117)
  • [BN95] Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423-472. MR 1318886 (97d:57004)
  • [Bi75] J. Birman, Braids, links and mapping class groups, Princeton University Press, 1975. MR 0375281 (51:11477)
  • [BL93] Joan S. Birman, Xiao-Song Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993), no. 2, 225-270. MR 1198809 (94d:57010)
  • [Ba82] R. Baxter, Exactly solved models in statistical mechanics, Academic Press (Harcourt Brace Jovanovich, Publishers), London, 1982. MR 690578 (86i:82002a)
  • [BPZ80] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Physics B 241 (1980), 333-380. MR 757857 (86m:81097)
  • [BBJ97] Eiichi Bannai, Etsuko Bannai, François Jaeger, On spin models, modular invariance, and duality, J. Algebraic Combin. 6 (1997), 203-228. MR 1456578 (98f:05150)
  • [Bi01] Stephen J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), 471-486. MR 1815219 (2002a:20043)
  • [BiJo97] D. Bisch and V. F. R. Jones, Algebras associated to intermediate subfactors, Inventiones Mathematicae 128 (1997), 89-157. MR 1437496 (99c:46072)
  • [BHMV92] C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992), no. 4, 685-699. MR 1191373 (94a:57010)
  • [CFW08] Danny Calegari, Michael Freedman, and Kevin Walker, Positivity of the universal pairing in $ 3$ dimensions, arXiv:0802.3208
  • [Co94] A. Connes, Noncommutative geometry, Academic Press, San Diego, 1994. MR 1303779 (95j:46063)
  • [Co75] A. Connes, Outer conjugacy classes of automorphisms of factors, Annales Scientifiques de l'École Normale Supérieure 8 (1975), 383-419. MR 0394228 (52:15031)
  • [DHR74] S. Doplicher, R. Haag, and J. E. Roberts, Local observables and particle statistics. II. Comm. Math. Phys. 35 (1974), 49-85. MR 0334742 (48:13060)
  • [DHR71] S. Doplicher, R. Haag, and J. E. Roberts, Local observables and particle statistics. I. Comm. Math. Phys. 23 (1971), 199-230. MR 0297259 (45:6316)
  • [EKT03] Shalom Eliahou, Louis H. Kauffman, and Morwen B. Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42 (2003), no. 1, 155-169. MR 1928648 (2003g:57015)
  • [FW87] J. Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987), 97-108. MR 896009 (88k:57006)
  • [DFZ90] P. Di-Francesco and J.-B. Zuber, ``$ SU(N)$ lattice integrable models and modular invariance'', Recent developments in conformal field theories, 179-216, World Scientific, 1990. MR 1160374
  • [diF98] P. di Francesco, New integrable lattice models from Fuss-Catalan algebras, Nuclear Phys. B 532 (1998), no. 3, 609-634. MR 1657030 (99k:82020)
  • [FRS89] K. Fredenhagen, K.-H. Rehren, and B. Schroer, Superselection sectors with braid group statistics and exchange algebras, Communications in Mathematical Physics 125 (1989), 201-226. MR 1016869 (91c:81047)
  • [FKLW03] M. Freedman, A. Kitaev, M. Larsen, and Wang Zhenghan, Topological quantum computation. Mathematical challenges of the 21st century (Los Angeles, CA, 2000), Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 1, 31-38. MR 1943131 (2003m:57065)
  • [Ga07] Stavros Garoufalidis, Chern-Simons theory, analytic continuation and arithmetic, arXiv:0711.1716
  • [GrJ07] Pinhas Grossman and Vaughan F. R. Jones, Intermediate subfactors with no extra structure, J. Amer. Math. Soc. 20 (2007), no. 1, 219-265. MR 2257402 (2007h:46077)
  • [GJS07] A. Guionnet, V. F. R. Jones, and D. Shlyakhtenko, Random matrices, free probability, planar algebras and subfactors, arXiv:0712.2904
  • [Ha96] R. Haag, Local quantum physics, Springer-Verlag, Berlin-Heidelberg-New York, 1996. MR 1405610 (98b:81001)
  • [HOMFLY85] P. Freyd, D. Yetter, J. Hoste, W. Lickorish, K. Millet, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985), 239-246. MR 776477 (86e:57007)
  • [GHJ89] F. Goodman, P. de la Harpe, and V. F. R. Jones, Coxeter graphs and towers of algebras, MSRI Publications (Springer), vol. 14, 1989. MR 999799 (91c:46082)
  • [GJS07] A. Guionnet, V. F. R. Jones, and D. Shlyakhtenko, Random matrices, free probability, planar algebras and subfactors, arXiv:0712.2904
  • [Jon83] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25. MR 696688 (84d:46097)
  • [Jon99] V. F. R. Jones, Planar algebras, Preprint, Berkeley, 1999, math.QA/9909027.
  • [Jo85] V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), no. 1, 103-111. MR 766964 (86e:57006)
  • [Jo87] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Annals of Mathematics 126 (1987), 335-388. MR 908150 (89c:46092)
  • [Jo07] Vaughan F. R. Jones, ``In and around the origin of quantum groups'', Prospects in mathematical physics, 101-126, Contemp. Math., 437, Amer. Math. Soc., Providence, RI, 2007. MR 2354658 (2008k:17018)
  • [Kash96] Rinat M. Kashaev, ``Quantum hyperbolic invariants of knots'', Discrete integrable geometry and physics (Vienna, 1996), 343-359, Oxford Lecture Ser. Math. Appl., 16, Oxford Univ. Press, New York, 1999. MR 1676604 (2001a:57021)
  • [Kau87] L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), 395-407. MR 899057 (88f:57006)
  • [Kau90] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417-471. MR 958895 (90g:57007)
  • [Khov00] Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359-426. MR 1740682 (2002j:57025)
  • [KZ84] V. Knizhnik and A. Zamolodchikov, Current algebra and Weiss-Zumino models in two dimensions, Nuclear Physics B 247 (1984), 83-103. MR 853258 (87h:81129)
  • [Kon93] Maxim Kontsevich, ``Vassiliev's knot invariants'', I. M. Gelfand Seminar, 133-150, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993. MR 1237836 (94k:57014)
  • [Kr02] Daan Krammer, Braid groups are linear, Ann. of Math. 155 (2002), 131-156. MR 1888796 (2003c:20040)
  • [LM86] W. B. R. Lickorish and K. C. Millett, The reversing result for the Jones polynomial, Pacific J. Math. 124 (1986), no, 1, 173-176. MR 850674 (87k:57007)
  • [Li97] W. B. R. Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, 175, Springer-Verlag, New York, 1997. MR 1472978 (98f:57015)
  • [Lo89] R. Longo, Index of subfactors and statistics of quantum fields, I., Communications in Mathematical Physics 126 (1989), 217-247. MR 1027496 (91c:46097)
  • [MeTh93] W. Menasco and M. Thistlethwaite, The classification of alternating links, Ann. Math. 138 (1993), 113-171. MR 1230928 (95g:57015)
  • [Mor86] H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), 107-109. MR 809504 (87c:57006)
  • [Mur87] K. Murasugi, The Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), 187-194. MR 895570 (88m:57010)
  • [Oc87] A. Ocneanu, ``Quantized group, string algebras and Galois theory for algebras'', Operator algebras and applications, vol. 2, 119-172, L.M.S Lecture Note Series, 136, 1987. MR 996454 (91k:46068)
  • [Oc00] A. Ocneanu, ``The classification of subgroups of quantum $ {\rm SU}(N)$'', Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), 133-159, Contemp. Math., 294, Amer. Math. Soc., Providence, RI, 2002. MR 1907188 (2003h:81101)
  • [Po94] S. Popa, Classification of amenable subfactors of type II, Acta Mathematica 172 (1994), 163-255. MR 1278111 (95f:46105)
  • [Po95] S. Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Inventiones Mathematicae 120 (1995), 427-446. MR 1334479 (96g:46051)
  • [Ras05] Jacob Rasmussen, ``Knot polynomials and knot homologies'', Geometry and topology of manifolds, 261-280, Fields Inst. Commun., 47, Amer. Math. Soc., Providence, RI, 2005. MR 2189938 (2006i:57029)
  • [RT91] N. Yu. Reshetikhin and V. G. Turaev, Invariants of $ 3$-manifolds via link polynomials and quantum groups, Inventiones Mathematicae 103 (1991), 547-598. MR 1091619 (92b:57024)
  • [Ro88] M. Rosso, Groupes quantiques et modèles à vertex de V. Jones en théorie des nœuds. (French) [Quantum groups and V. Jones's vertex models for knots], C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), no. 6, 207-210. MR 956807 (90d:57009)
  • [Saw95] S. Sawin, Subfactors constructed from quantum groups, Amer. J. Math. 117 (1995), 1349-1369. MR 1363071 (96h:46097)
  • [PS86] A. Pressley and G. Segal, Loop groups, Oxford University Press, 1986. MR 900587 (88i:22049)
  • [TL70] H. N. V. Temperley and E. H. Lieb, Relations between the ``percolation'' and ``colouring'' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ``percolation'' problem, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1549, 251-280. MR 0498284 (58:16425)
  • [Th87] M. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), 297-309. MR 899051 (88h:57007)
  • [Th01] Morwen Thistlethwaite, Links with trivial Jones polynomial, J. Knot Theory Ramifications 10 (2001), no. 4, 641-643. MR 1831681 (2002a:57012)
  • [TK88] A. Tsuchiya and Y. Kanie, Vertex operators on conformal field theory on $ {\bf P}^1$ and monodromy representations of braid groups, Advanced Studies in Pure Mathematics 16 (1988), 297-372. MR 972998 (89m:81166)
  • [TV92] V. G. Turaev and and O. Y. Viro, State sum invariants of $ 3$-manifolds and quantum $ 6j$-symbols, Topology 31 (1992), 865-902. MR 1191386 (94d:57044)
  • [Va95] Victor A. Vassiliev, ``Topology of discriminants and their complements'', Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 209-226, Birkhäuser, Basel, 1995. MR 1403923 (97j:57041)
  • [Vo01] Pierre Vogel, ``Invariants de type fini'', Nouveaux invariants en gèomètrie et en topologie, 99-128, Panor. Synthéses, 11, Soc. Math. France, Paris, 2001. MR 1882446 (2002j:57028)
  • [Wa98] A. Wassermann, Operator algebras and conformal field theory. III. Fusion of positive energy representations of $ {\rm LSU}(N)$ using bounded operators, Invent. Math. 133 (1998), no. 3, 467-538. MR 1645078 (99j:81101)
  • [We98] H. Wenzl, $ C^*$ tensor categories from quantum groups, J. Amer. Math. Soc. 11 (1998), 261-282. MR 1470857 (98k:46123)
  • [We88] H. Wenzl, Hecke algebras of type $ A_n$ and subfactors, Invent. Math. 92 (1988), 349-383. MR 936086 (90b:46118)
  • [We87] H. Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada, 9 (1987), no. 1, 5-9. MR 873400 (88k:46070)
  • [Wt89] E. Witten, Quantum field theory and Jones polynomial. Comm. Math. Phys. 121 (1989), 351-399. MR 0990772 (90h:57009)
  • [Wo87] S. L. Woronowicz, Twisted $ {\rm SU}(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), no. 1, 117-181. MR 890482 (88h:46130)
  • [Xu98] Xu F., Standard $ \lambda$-lattices from quantum groups, Inventiones Mathematicae 134 (1998), 455-487. MR 1660937 (2000c:46123)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 46L37, 46L54

Retrieve articles in all journals with MSC (2000): 46L37, 46L54


Additional Information

Vaughan Jones
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Email: vfr@math.berkeley.edu

DOI: https://doi.org/10.1090/S0273-0979-09-01244-0
Keywords: Subfactors, planar algebras, TQFT, knots, CFT, statistical mechanics
Received by editor(s): November 12, 2008
Published electronically: January 28, 2009
Additional Notes: Supported by NSF under Grant No. DMS-0401734, Auckland University and the NZIMA
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society