Quantum Gravity and Riemannian Geometry on the Fuzzy Sphere

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra $[x_i,x_j]=2imathlambda_p epsilon_{ijk}x_k$ 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}({rm Tr}(g^2)-frac{1}{2}{rm 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.