Enriched string-net models and their excitations

TL;DR

This paper rigorously analyzes enriched string-net models and their excitations in Walker-Wang models, using TQFT techniques to clarify boundary and bulk excitations and their algebraic structures.

Contribution

It provides a rigorous TQFT-based verification of boundary excitations as enriched centers and describes bulk excitations via the M"uger center, also reviewing Levin-Wen models from a tensor category perspective.

Findings

Boundary excitations are given by the enriched center Z^A(X).

Bulk point excitations correspond to the M"uger center Z_2(A).

Constructs bulk-to-boundary hopping operators reflecting symmetry-braided enrichment.

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 [arXiv:2208.14018] 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/M\"uger 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 M\"uger 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.