Inspired by the general relation between the boundary global symmetry and the bulk gauge symmetry in AdS/CFT, we reformulate the d + 1 dimensional AdS gravity theory as a SO (2, d ) gauge theory. In this formalism, the pull back of the bulk equation of motion onto a co-dimension one hypersurface Σ can be naturally related to the SO (2, d ) conservation law under a local energy scale in the dual CFT. The consistency of CFT requires that the SO (2, d ) covariant anomaly vanishes at arbitrary scale. After imposing a covariant area law as the renormalization condition, the vanishing of SO (2, d ) anomaly implies that the dual bulk geometries must satisfy the Einstein equation.
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Wang et al. (2021) studied this question.
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