Topological Phases for Extended Objects: Semiclassical Phase-Space Approach with Tensorial Coordinates

It has long been established that certain higher-dimensional topological phases of matter support extended objects like quasi-strings and quasi-membranes in their bulk states. In this study, we investigate the physics of these topological systems using a phase-space approach in the semiclassical regime, incorporating tensorial coordinates related to the extended objects. Specifically, we explore the semiclassical currents associated to open quasi-strings in topological phases in three spatial dimensions, considering the presence of both coordinate-space Kalb-Ramond fields and momentum-space tensor Berry connections. Our results show that the currents associated to the endpoints of open quasi-strings on the system's boundary exhibit a quantized Hall response. This formalism accurately reproduces the boundary response as predicted by an Abelian BF theory proving a novel way to study extended objects in topological matter.