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Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3)

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Abstract

We complete the computation of spectral measures for SU(3) nimrep graphs arising in subfactor theory, namely the \({SU(3) \mathcal{ADE}}\) graphs associated with SU(3) modular invariants and the McKay graphs of finite subgroups of SU(3). For the SU(2) graphs the spectral measures distill onto very special subsets of the semicircle/circle, whilst for the SU(3) graphs the spectral measures distill onto very special subsets of the discoid/torus. The theory of nimreps allows us to compute these measures precisely. We have previously determined spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with all SU(2) modular invariants, all subgroups of SU(2), the torus \({\mathbb{T}^2,\,SU(3)}\), and some SU(3) graphs.

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References

  1. Banica T., Bisch D.: Spectral measures of small index principal graphs. Commun. Math. Phys. 269, 259–281 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Behrend R.E., Pearce P.A., Petkova V.B., Zuber J.-B.: Boundary conditions in rational conformal field theories. Nucl. Phys. B 579, 707–773 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Birkhoff G., Mac Lane S.: A survey of modern algebra. Third edition. The Macmillan Co., New York (1965)

    Google Scholar 

  4. Böckenhauer J., Evans D.E.: Modular invariants, graphs and α-induction for nets of subfactors. II. Commun. Math. Phys. 200, 57–103 (1999)

    Article  MATH  ADS  Google Scholar 

  5. Böckenhauer J., Evans D.E.: Modular invariants, graphs and α-induction for nets of subfactors. III. Commun. Math. Phys. 205, 183–228 (1999)

    Article  MATH  ADS  Google Scholar 

  6. Böckenhauer J., Evans D.E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors. Commun. Math. Phys. 213, 267–289 (2000)

    Article  MATH  ADS  Google Scholar 

  7. Böckenhauer, J., Evans, D.E.: Modular invariants and subfactors. In: Mathematical physics in mathematics and physics (Siena, 2000), Fields Inst. Commun. 30, Providence, RI: Amer. Math. Soc., 2001, pp. 11–37

  8. Böckenhauer, J., Evans, D.E.: Modular invariants from subfactors. In: Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math. 294, Providence, RI: Amer. Math. Soc., 2002, pp. 95–131

  9. Böckenhauer J., Evans D.E., Kawahigashi Y.: On α-induction, chiral generators and modular invariants for subfactors. Commun. Math. Phys. 208, 429–487 (1999)

    Article  MATH  ADS  Google Scholar 

  10. Böckenhauer J., Evans D.E., Kawahigashi Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733–784 (2000)

    Article  MATH  ADS  Google Scholar 

  11. Bovier A., Lüling M., Wyler D.: Finite subgroups of SU(3). J. Math. Phys. 22, 1543–1547 (1981)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Cappelli A., Itzykson C., Zuber J.-B.: The A-D-E classification of minimal and \({A\sp {(1)}\sb 1}\) conformal invariant theories. Commun. Math. Phys. 113, 1–26 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Desmier P.E., Sharp R.T., Patera J.: Analytic SU(3) states in a finite subgroup basis. J. Math. Phys. 23, 1393–1398 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Di Francesco P., Zuber J.-B.: SU(N) lattice integrable models associated with graphs. Nuclear Phys. B 338, 602–646 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  15. Escobar J.A., Luhn C.: The flavor group Δ(6n 2). J. Math. Phys. 50, 013524 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  16. Evans D.E.: Fusion rules of modular invariants. Rev. Math. Phys. 14, 709–731 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Evans, D.E.: Critical phenomena, modular invariants and operator algebras. In: Operator algebras and mathematical physics (Constanţa, 2001), Bucharest: Theta, 2003, pp. 89–113

  18. Evans, D.E., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford Mathematical Monographs. New York: The Clarendon Press Oxford University Press, 1998

  19. Evans D.E., Pugh M.: Ocneanu Cells and Boltzmann Weights for the \({SU(3) \mathcal{ADE}}\) Graphs. Münster J. Math. 2, 95–142 (2009)

    MATH  MathSciNet  Google Scholar 

  20. Evans D.E., Pugh M.: SU(3)-Goodman-de la Harpe-Jones subfactors and the realisation of SU(3) modular invariants. Rev. Math. Phys. 21, 877–928 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Evans D.E., Pugh M.: A 2-Planar Algebras I. Quantum Topol. 1, 321–377 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Evans D.E., Pugh M.: Spectral Measures for Nimrep Graphs in Subfactor Theory. Commun. Math. Phys. 295, 363–413 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Evans, D.E., Pugh, M.: The Nakayama permutation of the nimreps associated to SU(3) modular invariants. Preprint: arXiv:1008.1003 (math.OA), 2010

  24. Fairbairn W.M., Fulton T., Klink W.H.: Finite and disconnected subgroups of SU 3 and their application to the elementary-particle spectrum. J. Math. Phys. 5, 1038–1051 (1964)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Gannon T.: The classification of affine SU(3) modular invariant partition functions. Commun. Math. Phys. 161, 233–263 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Gepner D.: Fusion rings and geometry. Commun. Math. Phys. 141, 381–411 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Hanany, A., He, Y.-H.: Non-abelian finite gauge theories. J. High Energy Phys. 9902, 013 (1999), Paper 13, 31 pp. (electronic)

  28. Kawai T.: On the structure of fusion algebras. Phys. Lett. B 217, 47–251 (1989)

    Google Scholar 

  29. Kuperberg G.: The quantum G 2 link invariant. Internat. J. Math. 5, 61–85 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kuperberg G.: Spiders for rank 2 Lie algebras. Commun. Math. Phys. 180, 109–151 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  31. Luhn, C., Nasri, S., Ramond, P.: Flavor group Δ(3n 2). J. Math. Phys. 48, 073501, (2007), 21pp.

    Google Scholar 

  32. Miller G.A., Blichfeldt H.F., Dickson L.E.: Theory and applications of finite groups. Dover Publications Inc., New York (1961)

    MATH  Google Scholar 

  33. Ocneanu, A.: Higher Coxeter Systems. Talk given at MSRI, 2000. http://www.msri.org/publications/ln/msri/2000/subfactors/ocneanu

  34. Ocneanu, A.:The classification of subgroups of quantum SU(N). In: Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math. 294, Providence, RI: Amer. Math. Soc., 2002, pp. 133–159

  35. Wassermann A.: Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU(N) using bounded operators. Invent. Math. 133, 467–538 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  36. Xu F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys. 192, 349–403 (1998)

    Article  MATH  ADS  Google Scholar 

  37. Yau, S.-T., Yu, Y.: Gorenstein quotient singularities in dimension three. Mem. Amer. Math. Soc. 105 viii+88 (1993)

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  1. School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, Wales, United Kingdom

    David E. Evans & Mathew Pugh

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  1. David E. Evans
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  2. Mathew Pugh
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Correspondence to Mathew Pugh.

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Communicated by Y. Kawahigashi

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Evans, D.E., Pugh, M. Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3). Commun. Math. Phys. 301, 771–809 (2011). https://doi.org/10.1007/s00220-010-1157-x

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