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We consider the topological abelian BF theory with radial boundary on ageneric 3D manifold. Our aim is to study if, where and how the boundary keepsmemory of the details of the background metric. We find that some features aretopologically protected and do not depend on the bulk metric. The 2D actionholographically induced on the boundary depends on two scalar fields, and canbe decoupled in two Luttinger actions describing two chiral bosons moving onthe edge of the 3D bulk. The outcome is that these edge excitations areaccelerated, as a direct consequence of the non-flat nature of the bulkspacetime. The chiral velocities of the edge modes, indeed, acquire a localdependence through the determinant of the induced metric on the boundary. Wefind three possibilities for the motion of the edge quasiparticles: samedirections, opposite directions and a single-moving mode. But, requiring thatthe Hamiltonian of the 2D theory is bounded by below, the case of edge modesmoving in the same direction is ruled out: systems involving parallel Hallcurrents (for instance Fractional Quantum Hall Effect with $\nu=2/5$) cannot bedescribed by a BF theory with boundary, independently from the geometry of thebulk spacetime, because of positive energy considerations. We are thereforeleft with physical situations characterized by edge excitations moving withopposite velocities (examples are FQHE with $\nu=1-1/n$, with $n$ positiveinteger, and Helical Luttinger Liquids phenomena) or a single-moving mode(Quantum Anomalous Hall). A strong restriction is obtained by requiring TimeReversal symmetry, which uniquely identifies modes with equal and oppositevelocities, and we know that this is the case of Topological Insulators. Thenovelty, with respect to the flat bulk background, is that the modes have localvelocities, which corresponds to Topological Insulators with accelerated edgemodes.

Topological BF description of 2D accelerated chiral edge modes