arXiv:1106.6033

String-net model of Turaev-Viro invariants

Alexander Kirillov

Published Jun 29, 2011• 86 citations

Abstract

In this paper, we describe the relation between the Turaev--Viro TQFT and the string-net space introduced in the papers of Levin and Wen. In particular, the case of surfaces with boundary is considered in detail.

DOI: 10.48550/arxiv.1106.6033

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arXiv:1106.6033

String-net model of Turaev-Viro invariants

Alexander Kirillov

Published Jun 29, 2011• 86 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.1106.6033

Related papers

Topological Flux Phases of Levin-Wen String-Net Models

Kaushal Patel, Parsa Bonderson, Kirill Shtengel +1 more

Bulletin of the American Physical SocietyMar 2, 20150 citations

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arXiv:2108.10798Annals of PhysicsMay 26, 20226 citations

Counterexamples in Levin-Wen string-net models, group categories, and Turaev unimodality

Spencer D. Stirling

We remark on the claim that the string-net model of Levin and Wen is a microscopic Hamiltonian formulation of the Turaev-Viro topological quantum field theory. Using simple counterexamples we indicate where interesting extra structure may be needed in the Levin-Wen model for this to hold (however we believe that some form of the correspondence is true). In order to be accessible to the condensed matter community we provide a brief and gentle introduction to the relevant concepts in category theory (relying heavily on analogy with ordinary group representation theory). Likewise, some physical ideas are briefly surveyed for the benefit of the more mathematical reader. The main feature of group categories under consideration is Turaev's unimodality. We pinpoint where unimodality should fit into the Levin-Wen construction, and show that the simplest example fails to be unimodal. Unimodality is straightforward to compute for group categories, and we provide a complete classification at the end of the paper.We remark on the claim that the string-net model of Levin and Wen is a microscopic Hamiltonian formulation of the Turaev-Viro topological quantum field theory. Using simple counterexamples we indicate where interesting extra structure may be needed in the Levin-Wen model for this to hold (however we believe that some form of the correspondence is true). In order to be accessible to the condensed matter community we provide a brief and gentle introduction to the relevant concepts in category theory (relying heavily on analogy with ordinary group representation theory). Likewise, some physical ideas are briefly surveyed for the benefit of the more mathematical reader. The main feature of group categories under consideration is Turaev's unimodality. We pinpoint where unimodality should fit into the Levin-Wen construction, and show that the simplest example fails to be unimodal. Unimodality is straightforward to compute for group categories, and we provide a complete classification at the end of the paper.

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We classify the irreducible anyon sectors of Levin-Wen models over an arbitrary unitary fusion category

\mathcal{C}

, showing that they are in one-to-one correspondence with equivalence classes of simple objects of the Drinfeld center

Z(\mathcal{C})

. We achieve this by making explicit how the Levin-Wen Hamiltonian stabilizes subspaces isomorphic to state spaces of the corresponding Turaev-Viro TQFT, and developing a detailed understanding of these state spaces on punctured disks. In particular, we construct Drinfeld insertion operators on such spaces which can move anyons between the punctures, and can change their fusion channels. Using these Drinfeld insertions, we construct explicit string operators that excite anyons above the ground state. The fusion and braiding properties of these anyons will be analysed in a companion paper.