The Holographic Ward identity: Examples from 2+1 gravity

TL;DR

This work demonstrates that in $AdS_3/CFT_2$, the bulk Chern–Simons constraints encode the boundary Virasoro and super Virasoro Ward identities under Fefferman–Graham boundary conditions. By analyzing the two $SL(2,\mathbb{R})$ CS sectors and their boundary data, the authors derive the bosonic and supersymmetric Ward identities, identify the holographic 1-point functions $T,\overline{T},Q,\overline{Q}$ from a renormalised action $W[\mu,\bar{\mu},\epsilon,\bar{\epsilon}]$, and show how local improvements decouple left and right sectors. The paper also generalizes to $\mathcal{N}$-extended supersymmetry using $Osp(N|2)$ and discusses extensions to higher-dimensional AdS/CFT setups. These results provide a precise, bulk-based derivation of boundary conformal structures, reinforcing the holographic interpretation of Ward identities and suggesting avenues for full path-integral treatments and higher-dimensional generalisations. The approach highlights the central role of bulk constraints, boundary conditions, and holographic renormalization in connecting gravity to boundary CFT data.

Abstract

In the AdS/CFT correspondence the boundary Ward identities are encoded in the bulk constraints. We study the three-dimensional version of this result using the Chern-Simons formulation of gravity. Due the metric boundary conditions the conformal identities cannot be derived in a straightforward way from the chiral ones. We pay special attention to this case and find the necessary modifications to the chiral currents in order to find the two Virasoro operators. The supersymmetric Ward identities are studied as well.