Quantum gravity and Riemannian geometry on the fuzzy sphere

Abstract

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra [x_i,x_j]=2imath 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.