Geodesic
Oriented Closed Polyhedral Maps and the Kitaev Model
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes $\Sigma$ on which the Hopf algebraic Kitaev model lives. We develop a theory of arrow presentations which underlines the role of the dual of the double $\mathcal{D}(\Sigma)^*$ of $\Sigma$ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on $\mathcal{D}(\Sigma)^*$ and facilitates computation of holonomy with values in the algebra of the Kitaev model. In this way, we can prove ribbon operator identities for arbitrary f.d. C$^*$-Hopf algebras and arbitrary oriented closed polyhedral maps. By means of a combinatorial notion of homotopy designed specially for ribbon curves, we can rigorously formulate “topological invariance” of states created by ribbon operators.
Keywords: Hopf algebra; polyhedral map; quantum double; ribbon operator; topological invariance 
@article{SIGMA_2024_20_a47,
     author = {Korn\'el Szlach\'anyi},
     title = {Oriented {Closed} {Polyhedral} {Maps} and the {Kitaev} {Model}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2024},
     volume = {20},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a47/}
}
TY  - JOUR
AU  - Kornél Szlachányi
TI  - Oriented Closed Polyhedral Maps and the Kitaev Model
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2024
VL  - 20
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a47/
LA  - en
ID  - SIGMA_2024_20_a47
ER  - 
%0 Journal Article
%A Kornél Szlachányi
%T Oriented Closed Polyhedral Maps and the Kitaev Model
%J Symmetry, integrability and geometry: methods and applications
%D 2024
%V 20
%U http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a47/
%G en
%F SIGMA_2024_20_a47
Kornél Szlachányi. Oriented Closed Polyhedral Maps and the Kitaev Model. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a47/

[1] Bombin H., Martin-Delgado M., “A family of non-Abelian Kitaev models on a lattice: topological condensation and confinement”, Phys. Rev. B, 78 (2008), 115421, 28 pp., arXiv: 0712.0190 | DOI

[2] Buchholz D., Fredenhagen K., “Locality and the structure of particle states”, Comm. Math. Phys., 84 (1982), 1–54 | DOI | MR | Zbl

[3] Buerschaper O., Mombelli J.M., Christandl M., Aguado M., “A hierarchy of topological tensor network states”, J. Math. Phys., 54 (2013), 012201, 46 pp., arXiv: 1007.5283 | DOI | MR | Zbl

[4] Cha M., Naaijkens P., Nachtergaele B., “On the stability of charges in infinite quantum spin systems”, Comm. Math. Phys., 373 (2020), 219–264, arXiv: 1804.03203 | DOI | MR | Zbl

[5] Cowtan A., Majid S., “Quantum double aspects of surface code models”, J. Math. Phys., 63 (2022), 042202, 49 pp., arXiv: 2107.04411 | DOI | MR | Zbl

[6] Doplicher S., Haag R., Roberts J.E., “Fields, observables and gauge transformations. I”, Comm. Math. Phys., 13 (1969), 1–23 | DOI | MR | Zbl

[7] Doplicher S., Haag R., Roberts J.E., “Fields, observables and gauge transformations. II”, Comm. Math. Phys., 15 (1969), 173–200 | DOI | MR | Zbl

[8] Doplicher S., Haag R., Roberts J.E., “Local observables and particle statistics. I”, Comm. Math. Phys., 23 (1971), 199–230 | DOI | MR

[9] Doplicher S., Haag R., Roberts J.E., “Local observables and particle statistics. II”, Comm. Math. Phys., 35 (1974), 49–85 | DOI | MR

[10] Hirmer A.K., Meusburger C., Categorical generalizations of quantum double models, arXiv: 2306.05950 | Zbl

[11] Jones G., Singerman D., “Maps, hypermaps and triangle groups”, The Grothendieck Theory of Dessins D'enfants (Luminy, 1993), London Math. Soc. Lecture Note Ser., 200, Cambridge University Press, Cambridge, 1994, 115–145 | DOI | MR | Zbl

[12] Kitaev A.Yu., “Fault-tolerant quantum computation by anyons”, Ann. Physics, 303 (2003), 2–30, arXiv: quant-ph/9707021 | DOI | MR | Zbl

[13] Lando S.K., Zvonkin A.K., Graphs on surfaces and their applications, Encyclopaedia Math. Sci., 141, Springer, Berlin, 2004 | DOI | MR | Zbl

[14] Larson R.G., Radford D.E., “Semisimple cosemisimple Hopf algebras”, Amer. J. Math., 110 (1988), 187–195 | DOI | MR | Zbl

[15] Meusburger C., “Kitaev lattice models as a Hopf algebra gauge theory”, Comm. Math. Phys., 353 (2017), 413–468, arXiv: 1607.01144 | DOI | MR | Zbl

[16] Meusburger C., Wise D.K., “Hopf algebra gauge theory on a ribbon graph”, Rev. Math. Phys., 33 (2021), 2150016, 93 pp., arXiv: 1512.03966 | DOI | MR | Zbl

[17] Montgomery S., Hopf algebras and their actions on rings, CBMS Reg. Conf. Ser. Math., 82, American Mathematical Society, Providence, RI, 1993 | DOI | MR | Zbl

[18] Naaijkens P., “Localized endomorphisms in Kitaev's toric code on the plane”, Rev. Math. Phys., 23 (2011), 347–373, arXiv: 1012.3857 | DOI | MR | Zbl

[19] Naaijkens P., “Kitaev's quantum double model from a local quantum physics point of view”, Advances in Algebraic Quantum Field Theory, Math. Phys. Stud., Springer, Cham, 2015, 365–395, arXiv: 1508.07170 | DOI | MR | Zbl

[20] Sweedler M.E., Hopf algebras, Math. Lecture Note Ser., W.A. Benjamin, Inc., New York, 1969 | MR | Zbl

[21] White A.T., Graphs of groups on surfaces. Interactions and models, North-Holland Math. Stud., 188, North-Holland Publishing Co., Amsterdam, 2001 | MR | Zbl

[22] Yan B., Chen P., Cui S.X., “Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras”, J. Phys. A, 55 (2022), 185201, 34 pp., arXiv: 2105.08202 | DOI | MR | Zbl