Geometric actions and flat space holography

In this paper we perform the Hamiltonian reduction of the action for three-dimensional Einstein gravity with vanishing cosmological constant using the Chern-Simons formulation and Bondi-van der Burg-Metzner-Sachs (BMS) boundary conditions. An equivalent formulation of the boundary action is the geometric action on BMS$_3$ coadjoint orbits, where the orbit representative is identified as the bulk holonomy. We use this reduced action to compute one-loop contributions to the torus partition function of all BMS$_3$ descendants of Minkowski spacetime and cosmological solutions in flat space. We then consider Wilson lines in the ISO$(2,1)$ Chern-Simons theory with endpoints on the boundary, whose reduction to the boundary theory gives a bilocal operator. We use the expectation values and two-point correlation functions of these bilocal operators to compute quantum contributions to the entanglement entropy of a single interval for BMS$_3$ invariant field theories and BMS$_3$ blocks, respectively. While semi-classically the BMS$_3$ boundary theory has central charges $c_1 = 0$ and $c_2 = 3/G_N$, we find that quantum corrections in flat space do not renormalize $G_N$, but rather lead to a non-zero $c_1$.