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 $[x_i,x_j]=2ı{\lambda }_p{\epsilon }_{ijk}{x}_k$ 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.

15 pages latex, 1 figure

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