Enriched string-net models and their excitations

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 A representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an A-enriched unitary fusion category X on a 2D boundary of the 3D Walker-Wang model associated to A. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer ZA(X) of A in Z(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 Z2(A), and we construct bulk-to-boundary hopping operators Z2(A)→ZA(X) reflecting how the UMTC of boundary excitations ZA(X) is symmetric-braided enriched in Z2(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.