AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
The Classification of Subfactors with Index at Most $5 \frac {1}{4}$
About this Title
Narjess Afzaly, Scott Morrison and David Penneys
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 284, Number 1405
ISBNs: 978-1-4704-4712-0 (print); 978-1-4704-7443-0 (online)
DOI: https://doi.org/10.1090/memo/1405
Published electronically: March 21, 2023
Table of Contents
Chapters
- 1. Introduction
- 2. Background
- 3. The main theorem
- 4. Better combinatorics for graph enumeration
- 5. Weeds with branch factor $r=1$
- 6. Ruling out weeds using branch factor inequalities
- 7. Ruling out 4-spokes
- 8. Cyclotomicity of vines
- A. Appendices
Abstract
Subfactor standard invariants encode quantum symmetries. The small index subfactor classification program has been a rich source of interesting quantum symmetries. We give the complete classification of subfactor standard invariants to index $5\frac {1}{4}$, which includes $3+\sqrt {5}$, the first interesting composite index.- M. Asaeda and U. Haagerup, Exotic subfactors of finite depth with Jones indices $(5+\sqrt {13})/2$ and $(5+\sqrt {17})/2$, Comm. Math. Phys. 202 (1999), no. 1, 1–63. MR 1686551, DOI 10.1007/s002200050574
- Marta Asaeda and Seidai Yasuda, On Haagerup’s list of potential principal graphs of subfactors, Comm. Math. Phys. 286 (2009), no. 3, 1141–1157. MR 2472028, DOI 10.1007/s00220-008-0588-0
- N. Afzaly. Isomorph-free generation of graph classes. Australian National University, 2015.
- M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang. Symmetry, Defects, and Gauging of Topological Phases. arXiv:1410.4540. 2014.
- D. Bisch and U. Haagerup. Unpublished. 1994.
- Dietmar Bisch and Uffe Haagerup, Composition of subfactors: new examples of infinite depth subfactors, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 3, 329–383. MR 1386923
- D. Bisch and V. F. R. Jones. Algebras associated to intermediate subfactors. Invent. Math. 128(1):89–157, 1997. issn: 0020-9910. doi: 10.1007/s002220050137. url: http://dx.doi.org/10.1007/s002220050137.
- Stephen Bigelow, Emily Peters, Scott Morrison, and Noah Snyder, Constructing the extended Haagerup planar algebra, Acta Math. 209 (2012), no. 1, 29–82. MR 2979509, DOI 10.1007/s11511-012-0081-7
- Alain Bruguières and Sonia Natale, Exact sequences of tensor categories, Int. Math. Res. Not. IMRN 24 (2011), 5644–5705. MR 2863377, DOI 10.1093/imrn/rnq294
- Sebastian Burciu and Sonia Natale, Fusion rules of equivariantizations of fusion categories, J. Math. Phys. 54 (2013), no. 1, 013511, 21. MR 3059899, DOI 10.1063/1.4774293
- Alain Bruguières and Sonia Natale, Central exact sequences of tensor categories, equivariantization and applications, J. Math. Soc. Japan 66 (2014), no. 1, 257–287. MR 3161401, DOI 10.2969/jmsj/06610257
- Jocelyne Bion-Nadal, An example of a subfactor of the hyperfinite $\textrm {II}_1$ factor whose principal graph invariant is the Coxeter graph $E_6$, Current topics in operator algebras (Nara, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 104–113. MR 1193933
- Dietmar Bisch, Remus Nicoara, and Sorin Popa, Continuous families of hyperfinite subfactors with the same standard invariant, Internat. J. Math. 18 (2007), no. 3, 255–267. MR 2314611, DOI 10.1142/S0129167X07004011
- Paul Bruillard, Siu-Hung Ng, Eric C. Rowell, and Zhenghan Wang, Rank-finiteness for modular categories, J. Amer. Math. Soc. 29 (2016), no. 3, 857–881. MR 3486174, DOI 10.1090/jams/842
- P. Bruillard, S.-H. Ng, E. C. Rowell and Z. Wang. On classification of modular categories by rank. arXiv:1507.05139. 2015.
- Stephen Bigelow and David Penneys, Principal graph stability and the jellyfish algorithm, Math. Ann. 358 (2014), no. 1-2, 1–24. MR 3157990, DOI 10.1007/s00208-013-0941-2
- Arnaud Brothier and Stefaan Vaes, Families of hyperfinite subfactors with the same standard invariant and prescribed fundamental group, J. Noncommut. Geom. 9 (2015), no. 3, 775–796. MR 3420531, DOI 10.4171/JNCG/207
- Dietmar Bisch, Principal graphs of subfactors with small Jones index, Math. Ann. 311 (1998), no. 2, 223–231. MR 1625762, DOI 10.1007/s002080050185
- Dragoš Cvetković, Michael Doob, and Ivan Gutman, On graphs whose spectral radius does not exceed $(2+\sqrt {5})^{1/2}$, Ars Combin. 14 (1982), 225–239. MR 683990
- F. Calegari and Z. Guo. Abelian Spiders. arXiv:1502.00035. 2015.
- A. Coste and T. Gannon, Remarks on Galois symmetry in rational conformal field theories, Phys. Lett. B 323 (1994), no. 3-4, 316–321. MR 1266785, DOI 10.1016/0370-2693(94)91226-2
- Frank Calegari, Scott Morrison, and Noah Snyder, Cyclotomic integers, fusion categories, and subfactors, Comm. Math. Phys. 303 (2011), no. 3, 845–896. MR 2786219, DOI 10.1007/s00220-010-1136-2
- Paramita Das, Shamindra Kumar Ghosh, and Ved Prakash Gupta, Perturbations of planar algebras, Math. Scand. 114 (2014), no. 1, 38–85. MR 3178106, DOI 10.7146/math.scand.a-16639
- Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, On braided fusion categories. I, Selecta Math. (N.S.) 16 (2010), no. 1, 1–119. MR 2609644, DOI 10.1007/s00029-010-0017-z
- David E. Evans and Terry Gannon, The exoticness and realisability of twisted Haagerup-Izumi modular data, Comm. Math. Phys. 307 (2011), no. 2, 463–512. MR 2837122, DOI 10.1007/s00220-011-1329-3
- David E. Evans and Terry Gannon, Near-group fusion categories and their doubles, Adv. Math. 255 (2014), 586–640. MR 3167494, DOI 10.1016/j.aim.2013.12.014
- David E. Evans and Yasuyuki Kawahigashi, Quantum symmetries on operator algebras, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1642584
- Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), no. 2, 581–642. MR 2183279, DOI 10.4007/annals.2005.162.581
- Pavel Etingof, Dmitri Nikshych, and Victor Ostrik, Fusion categories and homotopy theory, Quantum Topol. 1 (2010), no. 3, 209–273. With an appendix by Ehud Meir. MR 2677836, DOI 10.4171/QT/6
- Sébastien Falguières and Sven Raum, Tensor $\textrm {C}^*$-categories arising as bimodule categories of $\mathrm {II}_1$ factors, Adv. Math. 237 (2013), 331–359. MR 3028581, DOI 10.1016/j.aim.2012.12.020
- Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR 999799, DOI 10.1007/978-1-4613-9641-3
- P. Grossman, M. Izumi and N. Snyder. The Asaeda-Haagerup fusion categories. arXiv:1501.07324. 2015.
- J. J. Graham and G. I. Lehrer, The representation theory of affine Temperley-Lieb algebras, Enseign. Math. (2) 44 (1998), no. 3-4, 173–218. MR 1659204
- Fumio Hiai and Masaki Izumi, Amenability and strong amenability for fusion algebras with applications to subfactor theory, Internat. J. Math. 9 (1998), no. 6, 669–722. MR 1644299, DOI 10.1142/S0129167X98000300
- Uffe Haagerup, Principal graphs of subfactors in the index range $4<[M:N]<3+\sqrt 2$, Subfactors (Kyuzeso, 1993) World Sci. Publ., River Edge, NJ, 1994, pp. 1–38. MR 1317352
- Richard Han, A Construction of the “2221” Planar Algebra, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–University of California, Riverside. MR 2822034
- Masaki Izumi, Vaughan F. R. Jones, Scott Morrison, and Noah Snyder, Subfactors of index less than 5, Part 3: Quadruple points, Comm. Math. Phys. 316 (2012), no. 2, 531–554. MR 2993924, DOI 10.1007/s00220-012-1472-5
- Masaki Izumi and Yasuyuki Kawahigashi, Classification of subfactors with the principal graph $D^{(1)}_n$, J. Funct. Anal. 112 (1993), no. 2, 257–286. MR 1213139, DOI 10.1006/jfan.1993.1033
- Masaki Izumi, Scott Morrison, and David Penneys, Quotients of $A_2\ast T_2$, Canad. J. Math. 68 (2016), no. 5, 999–1022. MR 3536926, DOI 10.4153/CJM-2015-017-4
- Masaki Izumi, Scott Morrison, David Penneys, Emily Peters, and Noah Snyder, Subfactors of index exactly 5, Bull. Lond. Math. Soc. 47 (2015), no. 2, 257–269. MR 3335120, DOI 10.1112/blms/bdu113
- Masaki Izumi, The structure of sectors associated with Longo-Rehren inclusions. II. Examples, Rev. Math. Phys. 13 (2001), no. 5, 603–674. MR 1832764, DOI 10.1142/S0129055X01000818
- Masaki Izumi. The classification of $3^n$ subfactors and related fusion categories. arXiv:1609.07604. 2016.
- Masaki Izumi, Application of fusion rules to classification of subfactors, Publ. Res. Inst. Math. Sci. 27 (1991), no. 6, 953–994. MR 1145672, DOI 10.2977/prims/1195169007
- Masaki Izumi, On type $\textrm {II}$ and type $\textrm {III}$ principal graphs of subfactors, Math. Scand. 73 (1993), no. 2, 307–319. MR 1269266, DOI 10.7146/math.scand.a-12473
- Masaki Izumi, On flatness of the Coxeter graph $E_8$, Pacific J. Math. 166 (1994), no. 2, 305–327. MR 1313457
- Masaki Izumi, Goldman’s type theorems in index theory, Operator algebras and quantum field theory (Rome, 1996) Int. Press, Cambridge, MA, 1997, pp. 249–269. MR 1491121, DOI 10.1016/0167-6687(96)90081-5
- Vaughan F. R. Jones, Scott Morrison, and Noah Snyder, The classification of subfactors of index at most 5, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 2, 277–327. MR 3166042, DOI 10.1090/S0273-0979-2013-01442-3
- Vaughan F. R. Jones and David Penneys, The embedding theorem for finite depth subfactor planar algebras, Quantum Topol. 2 (2011), no. 3, 301–337. MR 2812459, DOI 10.4171/QT/23
- Vaughan F. R. Jones, The annular structure of subfactors, Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38, Enseignement Math., Geneva, 2001, pp. 401–463. MR 1929335
- Vaughan F. R. Jones, Two subfactors and the algebraic decomposition of bimodules over $\rm II_1$ factors, Acta Math. Vietnam. 33 (2008), no. 3, 209–218. MR 2501843
- Vaughan F. R. Jones, Quadratic tangles in planar algebras, Duke Math. J. 161 (2012), no. 12, 2257–2295. MR 2972458, DOI 10.1215/00127094-1723608
- Vaughan F. R. Jones, Actions of finite groups on the hyperfinite type $\textrm {II}_{1}$ factor, Mem. Amer. Math. Soc. 28 (1980), no. 237, v+70. MR 587749, DOI 10.1090/memo/0237
- V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI 10.1007/BF01389127
- V. F. R. Jones, Subfactors of type $\textrm {II}_1$ factors and related topics, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 939–947. MR 934296
- V. F. R. Jones. Planar algebras, I. arXiv:math.QA/9909027. 1999.
- Yasuyuki Kawahigashi, Classification of paragroup actions in subfactors, Publ. Res. Inst. Math. Sci. 31 (1995), no. 3, 481–517. MR 1355948, DOI 10.2977/prims/1195164051
- Yasuyuki Kawahigashi, On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors, J. Funct. Anal. 127 (1995), no. 1, 63–107. MR 1308617, DOI 10.1006/jfan.1995.1003
- Yasuyuki Kawahigashi, Orbifold subfactors, central sequences, and the relative Jones invariant $\kappa$, Internat. Math. Res. Notices 3 (1995), 129–140. MR 1321700, DOI 10.1155/S1073792895000109
- Hideki Kosaki, Extension of Jones’ theory on index to arbitrary factors, J. Funct. Anal. 66 (1986), no. 1, 123–140. MR 829381, DOI 10.1016/0022-1236(86)90085-6
- Zhengwei Liu, Scott Morrison, and David Penneys, 1-supertransitive subfactors with index at most $6\frac {1}{5}$, Comm. Math. Phys. 334 (2015), no. 2, 889–922. MR 3306607, DOI 10.1007/s00220-014-2160-4
- Z. Liu and D. Penneys. The generator conjecture for $3^G$ subfactor planar algebras. arXiv:1507.04794. 2015.
- Hannah K. Larson, Pseudo-unitary non-self-dual fusion categories of rank 4, J. Algebra 415 (2014), 184–213. MR 3229513, DOI 10.1016/j.jalgebra.2014.05.032
- Zhengwei Liu, Composed inclusions of $A_3$ and $A_4$ subfactors, Adv. Math. 279 (2015), 307–371. MR 3345186, DOI 10.1016/j.aim.2015.03.017
- R. Longo. Index of subfactors and statistics of quantum fields. I. Comm. Math. Phys. 126(2):217–247, 1989. issn: 0010-3616. url: http://projecteuclid.org/getRecord?id=euclid.cmp/1104179850.
- G. Lusztig, Leading coefficients of character values of Hecke algebras, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 235–262. MR 933415, DOI 10.1007/bf01389157
- Brendan D. McKay and Adolfo Piperno, Practical graph isomorphism, II, J. Symbolic Comput. 60 (2014), 94–112. MR 3131381, DOI 10.1016/j.jsc.2013.09.003
- Scott Morrison and David Penneys, 2-supertransitive subfactors at index $3+\sqrt {5}$, J. Funct. Anal. 269 (2015), no. 9, 2845–2870. MR 3394622, DOI 10.1016/j.jfa.2015.06.023
- Scott Morrison and Emily Peters, The little desert? Some subfactors with index in the interval $(5,3+\sqrt {5})$, Internat. J. Math. 25 (2014), no. 8, 1450080, 51. MR 3254427, DOI 10.1142/S0129167X14500803
- Scott Morrison and David Penneys, Constructing spoke subfactors using the jellyfish algorithm, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3257–3298. MR 3314808, DOI 10.1090/S0002-9947-2014-06109-6
- Scott Morrison, David Penneys, Emily Peters, and Noah Snyder, Subfactors of index less than 5, Part 2: Triple points, Internat. J. Math. 23 (2012), no. 3, 1250016, 33. MR 2902285, DOI 10.1142/S0129167X11007586
- Scott Morrison and Noah Snyder, Subfactors of index less than 5, Part 1: The principal graph odometer, Comm. Math. Phys. 312 (2012), no. 1, 1–35. MR 2914056, DOI 10.1007/s00220-012-1426-y
- S. Morrison and K. Walker. The centre of the extended Haagerup subfactor has 22 simple objects. arXiv:1404.3955. 2014.
- Brendan D. McKay, Isomorph-free exhaustive generation, J. Algorithms 26 (1998), no. 2, 306–324. MR 1606516, DOI 10.1006/jagm.1997.0898
- Scott Morrison, An obstruction to subfactor principal graphs from the graph planar algebra embedding theorem, Bull. Lond. Math. Soc. 46 (2014), no. 3, 600–608. MR 3210716, DOI 10.1112/blms/bdu009
- Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-abelian anyons and topological quantum computation, Rev. Modern Phys. 80 (2008), no. 3, 1083–1159. MR 2443722, DOI 10.1103/RevModPhys.80.1083
- Masahiro Nakamura and Zirô Takeda, A Galois theory for finite factors, Proc. Japan Acad. 36 (1960), 258–260. MR 123925
- Adrian Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172. MR 996454
- Viktor Ostrik, Fusion categories of rank 2, Math. Res. Lett. 10 (2003), no. 2-3, 177–183. MR 1981895, DOI 10.4310/MRL.2003.v10.n2.a5
- Victor Ostrik, On formal codegrees of fusion categories, Math. Res. Lett. 16 (2009), no. 5, 895–901. MR 2576705, DOI 10.4310/MRL.2009.v16.n5.a11
- V. Ostrik. Pivotal fusion categories of rank 3. (with an Appendix written jointly with Dmitri Nikshych), arXiv:1309.4822. 2013.
- David Penneys and Emily Peters, Calculating two-strand jellyfish relations, Pacific J. Math. 277 (2015), no. 2, 463–510. MR 3402358, DOI 10.2140/pjm.2015.277.463
- Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106. MR 860811
- David Penneys and James E. Tener, Subfactors of index less than 5, Part 4: Vines, Internat. J. Math. 23 (2012), no. 3, 1250017, 18. MR 2902286, DOI 10.1142/S0129167X11007641
- David Penneys, Chirality and principal graph obstructions, Adv. Math. 273 (2015), 32–55. MR 3311757, DOI 10.1016/j.aim.2014.11.021
- Emily Peters, A planar algebra construction of the Haagerup subfactor, Internat. J. Math. 21 (2010), no. 8, 987–1045. MR 2679382, DOI 10.1142/S0129167X10006380
- Sorin Popa, Universal construction of subfactors, J. Reine Angew. Math. 543 (2002), 39–81. MR 1887878, DOI 10.1515/crll.2002.017
- S. Popa, Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), no. 1, 19–43. MR 1055708, DOI 10.1007/BF01231494
- Sorin Popa, Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math. 111 (1993), no. 2, 375–405. MR 1198815, DOI 10.1007/BF01231293
- Sorin Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163–255. MR 1278111, DOI 10.1007/BF02392646
- Sorin Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995), no. 3, 427–445. MR 1334479, DOI 10.1007/BF01241137
- Sorin Popa, Classification of subfactors and their endomorphisms, CBMS Regional Conference Series in Mathematics, vol. 86, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1995. MR 1339767, DOI 10.1090/cbms/086
- Sorin Popa, Some ergodic properties for infinite graphs associated with subfactors, Ergodic Theory Dynam. Systems 15 (1995), no. 5, 993–1003. MR 1356624, DOI 10.1017/S0143385700009731
- V. S. Sunder and A. K. Vijayarajan, On the nonoccurrence of the Coxeter graphs $\beta _{2n+1},\ D_{2n+1}$ and $E_7$ as the principal graph of an inclusion of $\textrm {II}_1$ factors, Pacific J. Math. 161 (1993), no. 1, 185–200. MR 1237144
- J. Schou. Commuting squares and index for subfactors. arXiv:1304.5907, Ph.D. thesis at Odense Universitet. 1990.
- Jacob Siehler, Near-group categories, Algebr. Geom. Topol. 3 (2003), 719–775. MR 1997336, DOI 10.2140/agt.2003.3.719
- Noah Snyder, A rotational approach to triple point obstructions, Anal. PDE 6 (2013), no. 8, 1923–1928. MR 3198588, DOI 10.2140/apde.2013.6.1923
- Hans Wenzl, Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. MR 936086, DOI 10.1007/BF01404457
- Hans Wenzl, $C^*$ tensor categories from quantum groups, J. Amer. Math. Soc. 11 (1998), no. 2, 261–282. MR 1470857, DOI 10.1090/S0894-0347-98-00253-7