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Advances in Mathematical Physics
Volume 2010 (2010), Article ID 671039, 18 pages
Research Article

Microscopic Description of 2D Topological Phases, Duality, and 3D State Sums

1Institute for Scientific Interchange Foundation, Villa Gualino, Viale Settimio Severo 75, 10131 Torino, Italy
2Dipartimento di Fisica Nucleare e Teorica, Istituto Nazionale di Fisica Nucleare, Universita degli Studi di Pavia, Sezione di Pavia, via A. Bassi 6, 27100 Pavia, Italy
3Dipartimento di Fisica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received 7 September 2009; Accepted 23 January 2010

Academic Editor: Debashish Goswami

Copyright © 2010 Zoltán Kádár et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Doubled topological phases introduced by Kitaev, Levin, and Wen supported on two-dimensional lattices are Hamiltonian versions of three-dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state is shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three-dimensional geometrical interpretation are given.