Communications in Mathematical Physics

, Volume 353, Issue 1, pp 413–468 | Cite as

Kitaev Lattice Models as a Hopf Algebra Gauge Theory

  • Catherine MeusburgerEmail author


We prove that Kitaev’s lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern–Simons theory for the Drinfeld double D(H). This shows that Kitaev models are a special case of the older and more general combinatorial models. This equivalence is an analogue of the relation between Turaev–Viro and Reshetikhin–Turaev TQFTs and relates them to the quantisation of moduli spaces of flat connections. We show that the topological invariants of the two models, the algebra of operators acting on the protected space of the Kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism, are isomorphic. This is established in a gauge theoretical picture, in which both models appear as Hopf algebra valued lattice gauge theories. We first prove that the triangle operators of a Kitaev model form a module algebra over a Hopf algebra of gauge transformations and that this module algebra is isomorphic to the lattice algebra in the combinatorial formalism. Both algebras can be viewed as the algebra of functions on gauge fields in a Hopf algebra gauge theory. The isomorphism between them induces an algebra isomorphism between their subalgebras of invariants, which are interpreted as gauge invariant functions or observables. It also relates the curvatures in the two models, which are given as holonomies around the faces of the lattice. This yields an isomorphism between the subalgebras obtained by projecting out curvatures, which can be viewed as the algebras of functions on flat gauge fields and are the topological invariants of the two models.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alekseev A., Grosse H., Schomerus V.: Combinatorial quantization of the Hamiltonian Chern–Simons theory I. Commun. Math. Phys. 172(2), 317–358 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alekseev A., Grosse H., Schomerus V.: Combinatorial quantization of the Hamiltonian Chern–Simons theory II. Commun. Math. Phys. 174(3), 561–604 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alekseev A., Schomerus V.: Representation theory of Chern–Simons observables. Duke Math. J. 85(2), 447–510 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alekseev A., Malkin A.: Symplectic structures associated to Lie–Poisson groups. Commun. Math. Phys. 162(1), 147–173 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Balsam B.: Kirillov Jr A.: Kitaev’s Lattice Model and Turaev–Viro TQFTs. arXiv preprint arXiv:1206.2308
  6. 6.
    Balsam B.: Turaev–Viro invariants as an extended TQFT II. arXiv preprint arXiv:1010.1222
  7. 7.
    Balsam B.: Turaev–Viro invariants as an extended TQFT III. arXiv preprint arXiv:1012.0560
  8. 8.
    Barrett J.: Quantum gravity as topological quantum field theory. J. Math. Phys. 36(11), 6161–6179 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Barrett J., Martins J., García-Islas J.: Observables in the Turaev–Viro and Crane–Yetter models. J. Math. Phys. 48(9), 093508 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Barrett J., Westbury B.: Invariants of piecewise-linear 3-manifolds. Trans. Am. Math. Soc. 348(10), 3997–4022 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Baskerville W., Majid S.: The braided Heisenberg group. J. Math. Phys. 34(8), 3588–3606 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Beverland M., Buerschaper O., Koenig R., Pastawski F., Preskill J., Sijher S.: Protected gates for topological quantum field theories. J. Math. Phys. 57(2), 022201 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bombin H., Martin-Delgado M.: A family of non-Abelian Kitaev models on a lattice: topological condensation and confinement. Phys. Rev. B 78(11), 115421 (2008)ADSCrossRefGoogle Scholar
  14. 14.
    Buerschaper O., Aguado M.: Mapping Kitaev’s quantum double lattice models to Levin and Wen’s string-net models. Phys. Rev. B 80(15), 155136 (2009)ADSCrossRefGoogle Scholar
  15. 15.
    Buerschaper O., Mombelli J. M., Christandl M., Aguado M.: A hierarchy of topological tensor network states. J. Math. Phys. 54(1), 012201 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Buffenoir E., Noui K., Roche Ph.: Hamiltonian quantization of Chern–Simons theory with \({SL(2,\mathbb C)}\) group. Class. Quantum Gravity 19, 4953–5016 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Buffenoir E., Roche Ph.: Two dimensional lattice gauge theory based on a quantum group. Commun. Math. Phys. 170(3), 669–698 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Buffenoir, E., Roche, Ph.: Link invariants and combinatorial quantization of hamiltonian Chern Simons theory. Commun. Math. Phys. 181(2), 331–365 (1996)Google Scholar
  19. 19.
    Bullock D., Frohman C., Kania-Bartoszýnska J.: Topological interpretations of lattice gauge field theory. Commun. Math. Phys. 198, 47–81 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Drinfeld V.: On almost cocommutative Hopf algebras. Len. Math. J. 1, 321–342 (1990)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ellis-Monaghan J., Moffat I.: Graphs on Surfaces: Dualities, Polynomials, and Knots, vol. 84. Springer, Berlin (2013)CrossRefGoogle Scholar
  22. 22.
    Etinghof P., Gelaki S.: Some properties of finite-dimensional semisimple Hopf algebras. Mathods Res. Lett. 5, 191–197 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fock V., Rosly A.: Poisson structure on moduli of flat connections and r-matrix. Am. Math. Soc. Transl. 191, 67–86 (1999)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kádár, Z., Marzuoli, A.: Rasetti M. Microscopic description of 2d topological phases, duality, and 3D state sums. Adv. Math. Phys. 2010, Article ID 671039 (2010)Google Scholar
  25. 25.
    Kádár Z., Marzuoli A., Rasetti M.: Braiding and entanglement in spin networks: a combinatorial approach to topological phases. Int. J. Quantum Inf. 7(supp), 195–203 (2009)CrossRefGoogle Scholar
  26. 26.
    Kassel C.: Quantum Groups,Vol. 155. Springer Science & Business Media, New York (2012)Google Scholar
  27. 27.
    Kirillov Jr., A., Balsam, B.: Turaev–Viro invariants as an extended TQFT. arXiv preprint arXiv:1004.1533
  28. 28.
    Kitaev A: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003)Google Scholar
  29. 29.
    Kitaev A., Kong L.: Models for gapped boundaries and domain walls. Commun. Math. Phys. 313(2), 351–373 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Koenig R. Kuperberg G. Reichardt B.: Quantum computation with Turaev–Viro codes. Ann. Phys. 325(12), 2707–2749 (2010)Google Scholar
  31. 31.
    Lando S.Zvonkin, A.: Graphs on Surfaces and Their Applications, vol 141. Springer Science & Business Media, New York (2013)Google Scholar
  32. 32.
    Larson G., Radford D.: Semisimple cosemisimple Hopf algebras. Am. J. Math 109, 187–195 (1987)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Levin M., Wen X.-G.: String-net condensation: a physical mechanism for topological phases. Phys. Rev. B 71(4), 045110 (2005)ADSCrossRefGoogle Scholar
  34. 34.
    Levin M., Wen X.-G.: Detecting topological order in a ground state wave function. Phys. Rev. Lett. 96(11), 110405 (2006)ADSCrossRefGoogle Scholar
  35. 35.
    Majid, S.: Algebras and Hopf algebras in braided categories. In: Advances in Hopf algebras. Lecture Notes in Pure and Appl. Math Dekker, New York 158, 55–105 (1994)Google Scholar
  36. 36.
    Majid S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  37. 37.
    Meusburger C., Noui K.: The Hilbert space of 3d gravity: quantum group symmetries and observables. Adv. Theor. Math. Phys 14(6), 1651–1716 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Meusburger C. Wise D.: Hopf algebra gauge theory on a ribbon graph. arXiv preprint arXiv:1512.03966
  39. 39.
    Montgomery S.: Hopf algebras and their actions on rings Montgomery. Am. Math. Soc. 82 (1993)Google Scholar
  40. 40.
    Radford D.: Minimal quasitriangular Hopf algebras. J. Algebra 157(2), 281–315 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Radford D.: Hopf Algebras Series on Knots and Everything, vol. 49. World Scientific, Singapore (2011)Google Scholar
  42. 42.
    Reshetikhin N., Turaev V.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103(1), 547–597 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Turaev, V., Virelizier, A.: On two approaches to 3-dimensional TQFTs. arXiv preprint arXiv:1006.3501
  44. 44.
    Turaev V., Viro O.: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31(4), 865–902 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department MathematikFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

Personalised recommendations