Generating Quantum Matrix Geometry from Gauged Quantum Mechanics

Quantum matrix geometry is the underlying geometry of M(atrix) theory. Expanding upon the idea of level projection, we propose a quantum-oriented non-commutative scheme for generating the matrix geometry of the coset space $G/H$. We employ this novel scheme to unveil unexplored matrix geometries by utilizing gauged quantum mechanics on higher dimensional spheres. The resultant matrix geometries manifest as $\it{pure}$ quantum Nambu geometries: Their non-commutative structures elude capture through the conventional commutator formalism of Lie algebra, necessitating the introduction of the quantum Nambu algebra. This matrix geometry embodies a one-dimension-lower quantum internal geometry featuring nested fuzzy structures. While the continuum limit of this quantum geometry is represented by overlapping classical manifolds, their fuzzification cannot reproduce the original quantum geometry. We demonstrate how these quantum Nambu geometries give rise to novel solutions in Yang-Mills matrix models, exhibiting distinct physical properties from the known fuzzy sphere solutions.