We propose a quantum mechanical theory of quantum spaces described by large $N$ noncommutative geometry as a model for quantum gravity. The theory admits fuzzy sphere and fuzzy ellipsoid as solution. We show that these solutions reproduces precisely the horizon radius of a Schwarzschild black hole and a Kerr black hole. Moreover our quantum mechanical description gives rise to a set of microstates over these geometries, which reproduces precisely the Bekenstein-Hawking entropy of black hole. These results provide strong support that our proposed theory of quantum spaces is a plausible candidate for the theory of quantum gravity.

References:
[1] 2406.01466 [hep-th]
A Matrix Model Proposal for Quantum Gravity and the Quantum Mechanics of Black Holes
[2] 2406.12704 [hep-th]
Quantum Kerr Black Hole from Matrix Theory of Quantum Gravity
[3] Recent unpublished results.