1Ohio State University
2Vanderbilt University
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Abstract
Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC $mathcal{A}$ representing the Witt class of an anomaly, the article [10] gave a commuting projector model associated to an $mathcal{A}$-enriched unitary fusion category $mathcal{X}$ on a 2D boundary of the 3D Walker-Wang model associated to $mathcal{A}$. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer $Z^mathcal{A}(mathcal{X})$ of $mathcal{A}$ in $Z(mathcal{X})$.
In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center $Z_2(mathcal{A})$, and we construct bulk-to-boundary hopping operators $Z_2(mathcal{A})to Z^{mathcal{A}}(mathcal{X})$ reflecting how the UMTC of boundary excitations $Z^{mathcal{A}}(mathcal{X})$ is symmetric-braided enriched in $Z_2(mathcal{A})$.
This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal $6j$ symbol viewpoint.
Featured image: An $mathcal{A}$-enriched fusion category $mathcal{X}$ is the data necessary to attach the (2+1)D Levin-Wen string net model associated to the fusion category $mathcal{X}$ to the (3+1)D Walker-Wang string net model associated to the modular tensor category $mathcal{A}$.
► BibTeX gögn
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Vitnað af
[1] Corey Jones, Pieter Naaijkens, David Penneys, and Daniel Wallick, “Local topological order and boundary algebras”, arXiv: 2307.12552, (2023).
[2] Mario Tomba, Shuqi Wei, Brett Hungar, Daniel Wallick, Kyle Kawagoe, Chian Yeong Chuah, and David Penneys, “Boundary algebras of the Kitaev Quantum Double model”, arXiv: 2309.13440, (2023).
[3] Kyle Kawagoe, Corey Jones, Sean Sanford, David Green, and David Penneys, “Levin-Wen is a gauge theory: entanglement from topology”, arXiv: 2401.13838, (2024).
[4] Ying Chan, Tian Lan, and Linqian Wu, “Torus algebra and logical operators at low energy”, arXiv: 2403.01577, (2024).
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