The asymptotic safety program builds on a high-energy completion of gravity based on the Reuter fixed point, a non-trivial fixed point of the gravitational renormalization group flow. At this fixed point the canonical mass-dimension of coupling constants is balanced by anomalous dimensions induced by quantum fluctuations such that the theory enjoys quantum scale invariance in the ultraviolet. The crucial role played by the quantum fluctuations suggests that the geometry associated with the fixed point exhibits non-manifold like properties. In this work, we continue the characterization of this geometry employing the composite operator formalism based on the effective average action. Explicitly, we give a relation between the anomalous dimensions of geometric operators on a background d-sphere and the stability matrix encoding the linearized renormalization group flow in the vicinity of the fixed point. The eigenvalue spectrum of the stability matrix is analyzed in detail and we identify a “perturbative regime” where the spectral properties are governed by canonical power counting. Our results recover the feature that quantum gravity fluctuations turn the (classically marginal) R²-operator into a relevant one. Moreover, we find strong indications that higher-order curvature terms present in the two-point function play a crucial role in guaranteeing the predictive power of the Reuter fixed point.

渐近安全程序基于Reuter固定点（重力重新归一化流的非平凡固定点）的高能量重力完成。在这个固定点上，耦合常数的规范质量维被量子涨落引起的反常尺寸所平衡，因此该理论在紫外线下享有量子尺度不变性。量子涨落起着至关重要的作用，表明与固定点相关的几何形状表现出非流形性质。在这项工作中，我们将基于有效平均作用，继续使用复合算子形式主义对该几何进行表征。明确地，我们给出了背景上几何算子的异常尺寸之间的关系d-球和编码线性化重归一化组的稳定性矩阵在不动点附近流动。对稳定性矩阵的特征值谱进行了详细分析，我们确定了一种“扰动状态”，其中的谱属性由规范的功率计数控制。我们的结果恢复了量子引力起伏使（通常是边际的）转向[R^2-运算符变成一个相关的运算符。此外，我们发现有力的迹象表明，存在于两点函数中的高阶曲率项在保证Reuter不动点的预测能力中起着至关重要的作用。