Quantum Gravity and Riemannian Geometry on the Fuzzy Sphere

2020/04/29 by Torres, Evelyn Lira, Majid, Shahn · 1 citation
#FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2004.14363

Abstract

We study the quantum geometry of the fuzzy sphere defined as the angular momentum algebra [x_i,x_j]=2\imathλ_p ε_ijkx_k modulo setting ∑_i x_i² to a constant, using a recently introduced 3D rotationally invariant differential structure. Metrics are given by symmetric 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 (1)/(2)(\rm Tr(g²)-(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.

Cited by

• Kaluza-Klein ansatz from Lorentzian quantum gravity on the fuzzy sphere

Related

• q-Fuzzy spheres and quantum differentials on B_q[SU₂] and U_q(su₂)
• Quantum gravity on finite spacetimes and dynamical mass
• Kaluza-Klein ansatz from Lorentzian quantum gravity on the fuzzy sphere
• Quantum Gravity: are we there yet?
• Hidden classical symmetry in quantum spaces at roots of unity : From q-sphere to fuzzy sphere