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Quon Language: Surface Algebras and Fourier Duality

Abstract

Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality induced by the S matrix of the modular tensor category. The pictorial Fourier duality also coincides with the graphic duality on the sphere. For each pair of dual graphs, we obtain an algebraic identity related to the S matrix. These identities include well-known ones, such as the Verlinde formula; partially known ones, such as the 6j-symbol self-duality; and completely new ones.

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References

  1. Atiyah M.F.: Topological quantum field theory. Publications Mathématiques de l’IHÉS 68, 175–186 (1988)

    Article  MATH  Google Scholar 

  2. Barrett J.: Geometrical measurements in three-dimensional quantum gravity. Int. J. Mod. Phys. A 18(2), 97–113 (2003)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. Biamonte J.: Charged string tensor networks. Proc. Natl. Acad. Sci. 114(10), 2447–2449 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  4. Bisch D.: A note on intermediate subfactors. Pac. J. Math. 163, 201–216 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  5. Bisch D., Jones V.F.R.: Algebras associated to intermediate subfactors. Invent. Math. 128, 89–157 (1997)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. Barrett J.W., Westbury B.W.: Spherical categories. Adv. Math. 143(2), 357–375 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  7. Coecke B., Kissinger A.: Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, Cambridge (2017)

    Book  MATH  Google Scholar 

  8. Drinfeld V.G.: Quantum groups. Zapiski Nauchnykh Seminarov POMI 155, 18–49 (1986)

    Google Scholar 

  9. Evans D.E., Kawahigashi Y.: Quantum Symmetries on Operator Algebras, vol. 147. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  10. Freidel L., Noui K., Roche Ph.: 6j symbols duality relations. J. Math. Phys. 48(11), 113512 (2007)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. Gannon T.: Modular data: the algebraic combinatorics of conformal field theory. J. Algebr. Comb. 22(2), 211–250 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  12. Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bell’s theorem. In: Kafatos, M. (ed.) Bell’s Theorem, Quantum Theory and Conceptions of the Universe. Fundamental Theories of Physics, vol. 37. Springer, Dordrecht (1989)

  13. Jaffe A., Liu Z.: Mathematical picture language program. Proc. Natl. Acad. Sci. 115(1), 81–86 (2018)

    MathSciNet  Article  Google Scholar 

  14. Jaffe A., Liu Z., Wozniakowski A.: Holographic software for quantum networks. Sci. China Math. 61(4), 593–626 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  15. Jiang C., Liu Z., Wu J.: Noncommutative uncertainty principles. J. Funct. Anal. 270(1), 264–311 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  16. Jones, V.F.R.: Planar algebras, I. arXiv:math/9909027

  17. Jones, V.F.R.: The planar algebra of a bipartite graph. In: Knots in Hellas ’98 (Delphi). Series in Knots Everything, vol. 24, pp. 94–117. World Sci. Publ., River Edge (2000)

  18. Jones V.F.R.: Quadratic tangles in planar algebras. Duke Math. J. 161(12), 2257–2295 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  19. Jones V.F.R., Sunder V.S.: Introduction to Subfactors, vol. 234. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  20. Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219(3), 631–669 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. Lafont Y.: Towards an algebraic theory of boolean circuits. J. Pure Appl. Algebra 184(2), 257–310 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  22. Liu Z.: Exchange relation planar algebras of small rank. Trans. AMS 368, 8303–8348 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. Liu, Z., Morrison, S., Penneys, D.: Lifting shadings on symmetrically self-dual subfactor planar algebras. To appear Contemporary Mathematics arXiv:1709.05023

  24. Longo R., Rehren K.-H.: Nets of subfactors. Rev. Math. Phys. 7(04), 567–597 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  25. Liu Z., Wozniakowski A., Jaffe A.: Quon 3D language for quantum information. Proc. Natl. Acad. Sci. 114(10), 2497–2502 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  26. Liu, Z., Xu, F.: Jones-Wassermann subfactors for modular tensor categories (2016). arXiv:1612.08573

  27. Moore G., Seiberg N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123(2), 177–254 (1989)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. Müger M.: From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories. Journal of Pure and Applied Algebra 180, 81–157 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  29. Müger M.: On the structure of modular categories. Proc. Lond. Math. Soc. 87(2), 291–308 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  30. Ocneanu, A.: Quantized groups, string algebras, and Galois theory for algebras. In: Evans, D., Takesaki, M. (eds.) Operator Algebras and Applications (London Mathematical Society Lecture Note Series), pp. 119–172. Cambridge University Press, Cambridge (1989). https://doi.org/10.1017/CBO9780511662287.008

  31. Robert J.: Skein theory and turaev-viro invariants. Topology 34(4), 771–788 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  32. Reshetikhin N., Turaev V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103(1), 547–597 (1991)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. Turaev V.G.: Quantum Invariants of Knots and 3-manifolds, vol.~18. Walter de Gruyter GmbH & Co KG, Cambridge (2016)

    Book  MATH  Google Scholar 

  34. Turaev V.G., Viro O.Y.: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31(4), 865–902 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  35. Tambara D., Yamagami S.: Tensor categories with fusion rules of self-duality for finite abelian groups. J. Algebra 209(2), 692–707 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  36. Verlinde E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360–376 (1988)

    ADS  Article  MATH  Google Scholar 

  37. Wassermann A.: Operator algebras and conformal field theory III. Fusion of positive energy representations of LSU(N) using bounded operators. Invent. Math. 133(3), 467–538 (1998)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  38. Witten E.: Topological quantum field theory. Commun. Math. Phys. 117(3), 353–386 (1988)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. Xu F.: Jones-Wassermann subfactors for disconnected intervals. Commun. Contemp. Math. 2(03), 307–347 (2000)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The author would like to thank Terry Gannon, Arthur Jaffe, Vaughan F. R. Jones, Shamil Shakirov, Cumrun Vafa, Erik Verlinde, Jinsong Wu, and Feng Xu for helpful discussions. The author was supported by Grants TRT0080 and TRT0159 from the Templeton Religion Trust and an AMS-Simons Travel Grant. The author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Operator algebras: subfactors and their applications”.

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Liu, Z. Quon Language: Surface Algebras and Fourier Duality. Commun. Math. Phys. 366, 865–894 (2019). https://doi.org/10.1007/s00220-019-03361-3

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