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Conformal Anomaly Of Submanifold Observables In AdS/CFT Correspondence

C. Robin Graham, Edward Witten

TL;DR

We consider submanifold observables in AdS/CFT associated with k-branes and show that renormalized volumes yield conformal invariants for odd k but acquire a local anomaly for even k. The paper develops a general setup with conformally compact Einstein metrics, derives the asymptotic expansions of the brane embedding and induced metric, and demonstrates that the renormalized area is anomaly-free for odd k while providing the explicit k=2 anomaly formula in terms of the boundary data (mean curvature, P-tensor, and conformal factor). The k=2 calculation connects to Willmore-type functionals in the conformally flat case and supplies new information about higher-dimensional CFTs arising in AdS/CFT contexts. The results offer a precise, local expression for the conformal anomaly of brane observables and a framework for computing similar anomalies in other dimensions and brane configurations.

Abstract

We analyze the conformal invariance of submanifold observables associated with $k$-branes in the AdS/CFT correspondence. For odd $k$, the resulting observables are conformally invariant, and for even $k$, they transform with a conformal anomaly that is given by a local expression which we analyze in detail for $k=2$

Conformal Anomaly Of Submanifold Observables In AdS/CFT Correspondence

TL;DR

We consider submanifold observables in AdS/CFT associated with k-branes and show that renormalized volumes yield conformal invariants for odd k but acquire a local anomaly for even k. The paper develops a general setup with conformally compact Einstein metrics, derives the asymptotic expansions of the brane embedding and induced metric, and demonstrates that the renormalized area is anomaly-free for odd k while providing the explicit k=2 anomaly formula in terms of the boundary data (mean curvature, P-tensor, and conformal factor). The k=2 calculation connects to Willmore-type functionals in the conformally flat case and supplies new information about higher-dimensional CFTs arising in AdS/CFT contexts. The results offer a precise, local expression for the conformal anomaly of brane observables and a framework for computing similar anomalies in other dimensions and brane configurations.

Abstract

We analyze the conformal invariance of submanifold observables associated with -branes in the AdS/CFT correspondence. For odd , the resulting observables are conformally invariant, and for even , they transform with a conformal anomaly that is given by a local expression which we analyze in detail for

Paper Structure

This paper contains 2 sections, 4 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. The Computation

Key Result

Lemma 2.1

A metric on $M$ in the conformal infinity of $g_+$ determines a unique defining function $r$ in a neighborhood of $M$ in $\overline{X}$ such that $\overline{g}|_{TM}$ is the prescribed boundary metric and such that $|dr|^2_{\overline{g}}=1$.

Theorems & Definitions (7)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4