Quantum gravity and Riemannian geometry on the fuzzy sphere
AbstractWe study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra $$[x_i,x_j]=2\imath \lambda _p \epsilon _{ijk}x_k$$ [ x i , x j ] = 2 ı λ p ϵ ijk x k modulo setting $$\sum _i x_i^2$$ ∑ i x i 2 to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric $$3 \times 3$$ 3 × 3 matrices g and we show that for each metric there is a unique quantum Levi-Civita connection with constant coefficients, with scalar curvature $$ \frac{1}{2}(\mathrm{Tr}(g^2)-\frac{1}{2}\mathrm{Tr}(g)^2)/\det (g)$$ 1 2 ( Tr ( g 2 ) - 1 2 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.