Enriched string-net models and their excitations

David Green; Peter Huston; Kyle Kawagoe; David Penneys; Anup Poudel; Sean Sanford

doi:10.22331/q-2024-03-28-1301

Enriched string-net models and their excitations

David Green¹, Peter Huston², Kyle Kawagoe¹, David Penneys¹, Anup Poudel¹, and Sean Sanford¹

¹The Ohio State University
²Vanderbilt University

Published:	2024-03-28, volume 8, page 1301
Eprint:	arXiv:2305.14068v2
Doi:	https://doi.org/10.22331/q-2024-03-28-1301
Citation:	Quantum 8, 1301 (2024).

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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 data

@article{Green2024enrichedstringnet,
  doi = {10.22331/q-2024-03-28-1301},
  url = {https://doi.org/10.22331/q-2024-03-28-1301},
  title = {Enriched string-net models and their excitations},
  author = {Green, David and Huston, Peter and Kawagoe, Kyle and Penneys, David and Poudel, Anup and Sanford, Sean},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1301},
  month = mar,
  year = {2024}
}

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