We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra \([x_i,x_j]=2\imath \lambda _p \epsilon _{ijk}x_k\) modulo setting \(\sum _i x_i^2\) to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric \(3 \times 3\) matrices g and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature \( \frac{1}{2}(\mathrm{Tr}(g^2)-\frac{1}{2}\mathrm{Tr}(g)^2)/\det (g)\). As an application, we construct Euclidean quantum gravity on the fuzzy unit sphere. We also calculate the charge 1 monopole for the 3D differential structure.

我们研究定义为角动量代数\（[x_i，x_j] = 2 \ imath \ lambda _p \ epsilon _ {ijk} x_k \）模设置\（\ sum _i x_i ^ 2 \）的模糊球体的量子几何。使用最近推出的3D旋转不变差分结构将其设置为常数。度量由对称\（3 × 3 \）矩阵g给出，我们证明对于每个度量，都有一个具有常数系数且标量曲率为\（\ frac {1} {2}（\ mathrm {Tr}（g ^ 2）-\ frac {1} {2} \ mathrm {Tr}（g）^ 2）/ \ det（g）\）。作为应用，我们在模糊单位球面上构造了欧几里得量子引力。我们还为3D差分结构计算了电荷1单极子。