Enriched string-net models and their excitations

Quantum 8, 1301 (2024).

https://doi.org/10.22331/q-2024-03-28-1301

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.