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.

受 AdS/CFT 中边界全局对称性和体规范对称性之间的一般关系的启发，我们将d + 1 维 AdS 引力理论重新表述为SO (2, d ) 规范理论。在这种形式主义中，将体运动方程拉回到同维一超曲面 Σ 上可以自然地与双 CFT 中局部能量标度下的SO (2, d ) 守恒定律相关。CFT 的一致性要求SO (2, d ) 协变异常在任意尺度上消失。将协变面积定律作为重整化条件后，SO (2, d) 异常意味着双体几何必须满足爱因斯坦方程。