  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.
