Braiding and gapped boundaries in fracton topological phases

We study gapped boundaries of Abelian type-I fracton systems in three spatial dimensions. Using the X-cube model as our motivating example, we give a conjecture, with partial proof, of the conditions for a boundary to be gapped. In order to state our conjecture, we use a precise definition of fracton braiding and show that bulk braiding of fractons has several features that make it insufficient to classify gapped boundaries. Most notable among these is that bulk braiding is sensitive to geometry and is ``nonreciprocal''; that is, braiding an excitation a around b need not yield the same phase as braiding b around a. Instead, we define fractonic ``boundary braiding, '' which resolves these difficulties in the presence of a boundary. We then conjecture that a boundary of an Abelian fracton system is gapped if and only if a ``boundary Lagrangian subgroup'' of excitations is condensed at the boundary; this is a generalization of the condition for a gapped boundary in two spatial dimensions, but it relies on boundary braiding instead of bulk braiding. We also discuss the distinctness of gapped boundaries and transitions between different topological orders on gapped boundaries.