  We consider the topological abelian BF theory with radial boundary on a
generic 3D manifold. Our aim is to study if, where and how the boundary keeps
memory of the details of the background metric. We find that some features are
topologically protected and do not depend on the bulk metric. The 2D action
holographically induced on the boundary depends on two scalar fields, and can
be decoupled in two Luttinger actions describing two chiral bosons moving on
the edge of the 3D bulk. The outcome is that these edge excitations are
accelerated, as a direct consequence of the non-flat nature of the bulk
spacetime. The chiral velocities of the edge modes, indeed, acquire a local
dependence through the determinant of the induced metric on the boundary. We
find three possibilities for the motion of the edge quasiparticles: same
directions, opposite directions and a single-moving mode. But, requiring that
the Hamiltonian of the 2D theory is bounded by below, the case of edge modes
moving in the same direction is ruled out: systems involving parallel Hall
currents (for instance Fractional Quantum Hall Effect with $\nu=2/5$) cannot be
described by a BF theory with boundary, independently from the geometry of the
bulk spacetime, because of positive energy considerations. We are therefore
left with physical situations characterized by edge excitations moving with
opposite velocities (examples are FQHE with $\nu=1-1/n$, with $n$ positive
integer, and Helical Luttinger Liquids phenomena) or a single-moving mode
(Quantum Anomalous Hall). A strong restriction is obtained by requiring Time
Reversal symmetry, which uniquely identifies modes with equal and opposite
velocities, and we know that this is the case of Topological Insulators. The
novelty, with respect to the flat bulk background, is that the modes have local
velocities, which corresponds to Topological Insulators with accelerated edge
modes.
