Phases and propagation of closed p-brane

We study phases and propagation of closed $p$-brane within the framework of effective field theory with higher-form global symmetries, i.e., {\it brane-field theory}. We extend our previous studies by including the kinetic term of the center-of-mass motion as well as the kinetic term for the relative motions constructed by the area derivatives. This inclusion gives rise to another scalar Nambu-Goldstone mode in the broken phase, enriching the phase structures of $p$-brane. For example, when the higher-form global symmetries are discrete ones, we show that the low-energy effective theory in the broken phase is described by a topological field theory of the axion $\varphi(X)$ and $p$-form field $A_p^{}(X)$ with multiple (emergent) higher-form global symmetries. After the mean-field analysis, we investigate the propagation of $p$-brane in the present framework. We find the (functional) plane-wave solutions for the kinetic terms and derive a path-integral representation of the brane propagator. This representation motivates us to study the brane propagation within the Born-Oppenheimer approximation, where the volume of $p$-brane is treated as constant. In the volume-less limit (i.e. point-particle limit), the propagator reduces to the ordinary propagator of relativistic particle, whereas it describes the propagation of the area elements in the large-volume limit. Correspondingly, it is shown that the Hausdorff dimension of $p$-brane varies from $2$ to $2(p+1)$ as we increase the $p$-brane volume within the Born-Oppenheimer approximation. Although these results are quite intriguing, we also point out that the Born-Oppenheimer approximation is invalid in the point-particle limit, highlighting the quantum nature of $p$-brane as an extended object in spacetime.