Polyakov formulas for GJMS operators from AdS/CFT

TL;DR

The paper presents a holographic derivation of Polyakov formulas for the GJMS conformally covariant Laplacians by relating bulk one-loop corrections in AdS to subleading large-$N$ boundary effects under double-trace deformations. Focusing on conformally flat boundaries, the calculation reduces to volume renormalization, linking the holographic Type A anomaly to the boundary Q-curvature and yielding a universal coefficient for all $P_{2k}$. It provides explicit Polyakov-type formulas for the full family of GJMS operators, including the conformal primitive, and compares these results with known zeta-regularized computations, highlighting a scheme-dependent separation between universal and local terms. The work reinforces the central role of Q-curvature in holographic anomalies and offers a compact, testable framework for the interplay between bulk geometry and boundary conformal data in the AdS/CFT correspondence.

Abstract

We argue that the AdS/CFT calculational prescription for double-trace deformations leads to a holographic derivation of the conformal anomaly, and its conformal primitive, associated to the whole family of conformally covariant powers of the Laplacian (GJMS operators) at the conformal boundary. The bulk side involves a quantum 1-loop correction to the SUGRA action and the boundary counterpart accounts for a sub-leading term in the large-N limit. The sequence of GJMS conformal Laplacians shows up in the two-point function of the CFT operator dual to a bulk scalar field at certain values of its scaling dimension. The restriction to conformally flat boundary metrics reduces the bulk computation to that of volume renormalization which renders the universal type A anomaly. In this way, we directly connect two chief roles of the Q-curvature: the main term in Polyakov formulas on one hand, and its relation to the Poincare metrics of the Fefferman-Graham construction, on the other hand. We find agreement with previously conjectured patterns including a generic and simple formula for the type A anomaly coefficient that matches all reported values in the literature concerning GJMS operators, to our knowledge.