Journal ArticleEnriched string-net models and their excitationsGreen, David; Huston, Peter; Kawagoe, Kyle; Penneys, David; Poudel, Anup; Sanford, Sean<p>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<math><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow></math>representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an<math><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow></math>-enriched unitary fusion category<math><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>X</mi></mrow></math>on a 2D boundary of the 3D Walker-Wang model associated to<math><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow></math>. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer<math><msup><mi>Z</mi><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow></msup><mo stretchy='false'>(</mo><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>X</mi></mrow><mo stretchy='false'>)</mo></math>of<math><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow></math>in<math><mi>Z</mi><mo stretchy='false'>(</mo><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>X</mi></mrow><mo stretchy='false'>)</mo></math>.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<math><msub><mi>Z</mi><mn>2</mn></msub><mo stretchy='false'>(</mo><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow><mo stretchy='false'>)</mo></math>, and we construct bulk-to-boundary hopping operators<math><msub><mi>Z</mi><mn>2</mn></msub><mo stretchy='false'>(</mo><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow><mo stretchy='false'>)</mo><mo stretchy='false'>&#x2192;</mo><msup><mi>Z</mi><mrow class='MJX-TeXAtom-ORD'><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow></mrow></msup><mo stretchy='false'>(</mo><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>X</mi></mrow><mo stretchy='false'>)</mo></math>reflecting how the UMTC of boundary excitations<math><msup><mi>Z</mi><mrow class='MJX-TeXAtom-ORD'><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow></mrow></msup><mo stretchy='false'>(</mo><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>X</mi></mrow><mo stretchy='false'>)</mo></math>is symmetric-braided enriched in<math><msub><mi>Z</mi><mn>2</mn></msub><mo stretchy='false'>(</mo><mrow class='MJX-TeXAtom-ORD'><mi class='MJX-tex-caligraphic' mathvariant='script'>A</mi></mrow><mo stretchy='false'>)</mo></math>.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<math><mn>6</mn><mi>j</mi></math>symbol viewpoint.</p>Quantum2024-03-2810535011Quantum813012521-327Xhttps://doi.org/10.22331/q-2024-03-28-13012154389; 1654159; 2011876National Science Foundation