Geometric actions and flat space holography

TL;DR

This work advances flat space holography in 2+1 dimensions by deriving a boundary theory from 3D Chern-Simons gravity under BMS3 boundary conditions, explicitly mapping bulk holonomies to BMS3 coadjoint orbits and obtaining a geometric action. It then uses this framework to compute one-loop torus partition functions, construct bilocal boundary operators from Wilson lines, and obtain entanglement entropy and BMS3 blocks, including quantum (1/c2) corrections. A key finding is that quantum effects do not renormalize Newton’s constant (c2 remains fixed) but induce a nonzero c1 shift, signaling a quantum gravitational anomaly; this also yields one-loop exact expressions for EE and blocks in various flat-space backgrounds, including Minkowski, null orbifold, and flat space cosmologies. The results solidify the connection between bulk flat space gravity and the coadjoint orbit quantization of BMS3, and they lay the groundwork for further explorations of flat-space holography, BMS3 quantum field theories, and potential extensions to supersymmetry and higher dimensions.

Abstract

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 $c_1$.