Geometric actions and flat space holography

TL;DR

This paper explores the Hamiltonian reduction of 3D flat space gravity using Chern-Simons theory, deriving boundary actions, computing quantum corrections to partition functions and entanglement entropy, and revealing non-zero central charge contributions.

Contribution

It introduces a geometric boundary action for flat space gravity, computes quantum corrections to BMS$_3$ structures, and links Wilson line operators to entanglement entropy in BMS-invariant theories.

Findings

Quantum corrections induce a non-zero $c_1$ central charge.

One-loop contributions to the torus partition function are computed.

Wilson lines relate to bilocal operators and entanglement entropy.

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 $c_1$.