Interpolating Between Asymptotic and Near Horizon Symmetries

TL;DR

The paper addresses how to interpolate between two distinct boundary condition sets in three-dimensional AdS gravity by employing the Chern–Simons formulation with two boundaries. It develops a general construction using a Gauss-decomposed group element and an interpolating function $f(r)$ to glue a connection $A_\infty$ obeying one symmetry (e.g., Brown–Henneaux Virasoro) to a connection $A_0$ obeying another (e.g., Heisenberg $u(1)$ currents) while preserving flatness. It provides two explicit realizations: Brown–Henneaux at infinity to horizon Heisenberg, and Compère–Song–Strominger CSS at infinity to horizon Heisenberg, with holonomy matching tying certain zero modes ($V_0,V_\infty$, $K_0,K_\infty$) but leaving the full tower of horizon excitations unconstrained. As a result, BTZ-like black holes can be endowed with horizon soft hair invisible to asymptotic observers, with potential implications for causal patch holography and information flow, and the framework extends to Lorentzian signature, higher-spin theories, and higher dimensions.

Abstract

We develop basic tools and matching conditions to interpolate between asymptotic and near horizon symmetries. We focus on black holes in three dimensions. In particular, we match Brown--Henneaux boundary conditions at infinity, which yields two Virasoro algebras, to Heisenberg boundary conditions at the horizon yielding two u(1) current algebras. Our construction allows to equip BTZ black holes with soft hair excitations at the horizon invisible to the asymptotic observer.