Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

TL;DR

This paper explores the classification and algebraic structure of string-like excitations in (3+1)d topological gauge models with gapped boundaries, revealing their relation to bicategories and pseudo-algebra objects.

Contribution

It introduces a classification of bulk dyonic string excitations and relates them to boundary point-like excitations via bicategory theory, advancing understanding of (3+1)d topological phases.

Findings

Classified string-like excitations using tube algebra

Established a bicategory encoding boundary excitations

Linked gapped boundaries to pseudo-algebra objects

Abstract

We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory.