Extrinsic Geometry and Gappable Edges in Rotation-Invariant Topological Phases

Recent work on Abelian topological phases with rotation symmetry has raised the question of whether rotation symmetry can protect gapless propagating edge modes. Here we address this issue by considering the coupling of topological phases to the extrinsic geometry of the background. First, we analyze an effective hydrodynamic theory for an Abelian topological phase with vanishing hall conductance. After integrating out the bulk hydrodynamic degrees of freedom, we identify charge neutral, rotation invariant mass terms by coupling the propagating boundary modes to the extrinsic geometry. This allows us to integrate out the edge modes and we find a gapped theory described by the Gromov-Jensen-Abanov Lagrangian, first introduced in [Phys. Rev. Lett. 116, 126802 (2016)], regardless of the shift. Finally, we apply these ideas to a microscopic theory and find the explicit bulk terms which respect gauge and rotation symmetry, that gap out the edge modes.