A Note On Holographic Ward Identities

TL;DR

The paper demonstrates that Ward identities in a QFT dual to AdS/CFT arise directly from bulk gauge and diffeomorphism constraints when these constraints are recast in Hamilton-Jacobi form, yielding a generating functional for the identities. By applying the GKPW prescription, the authors show how functional derivatives of the on-shell supergravity action $S$ with respect to boundary sources $igra J^{ai} igra$, $igra O^I igra$, and $T^{ij}$ reproduce Ward identities for the dual currents, scalars, and the stress tensor. The vector-case analysis includes nonconservation on backgrounds that break global symmetries explicitly, and the Chern-Simons term reproduces the expected anomaly; tachyonic scalars are handled via holographic renormalization to obtain finite, renormalized Ward identities. Extending to diffeomorphisms, the framework yields conserved stress-energy Ward identities and their correlators with currents, reinforcing the deep link between bulk constraints and boundary symmetries. Overall, the work clarifies how bulk constraint equations encode QFT Ward identities and informs both checks of AdS/CFT and the inverse problem of bulk reconstruction from boundary data, with potential routes to dual RG equations via a Lagrangian perspective.

Abstract

In this note we show how Ward identities may be derived for a quantum field theory dual of a string theory using the AdS/CFT correspondence. In particular associated with any gauge symmetry of the bulk supergravity theory there is a corresponding constraint equation. Writing this constraint in Hamilton-Jacobi form gives a generating functional for Ward identities in the dual QFT. We illustrate the method by considering various examples.