  We present effective field theories for dipole symmetric topological matters
that can be described by the Chern-Simons theory. Unlike most studies using
higher-rank gauge theory, we develop a framework with both U(1) and dipole
gauge fields. As a result, only the highest multipole symmetry can support the
't Hooft anomaly. We show that with appropriate point group symmetries, the
dipolar Chern-Simons theory can exist in any dimension and, moreover, the
bulk-edge correspondence can depend on the boundary. As two applications, we
draw an analogy between the dipole anomaly and the torsional anomaly and
generalize particle-vortex duality to dipole phase transitions. All of the
above are in the flat spacetime limit, but our framework is able to
systematically couple dipole symmetry to curved spacetime. Based on that, we
give a proposal about anomalous dipole hydrodynamics. Moreover, we show that
the fracton-elasticity duality arises naturally from a non-abelian Chern-Simons
theory in 3D.
