Generating Quantum Matrix Geometry from Gauged Quantum Mechanics

TL;DR

This paper introduces a new quantum-oriented scheme to generate matrix geometries for coset spaces, revealing pure quantum Nambu geometries with unique non-commutative structures that lead to novel solutions in Yang-Mills matrix models.

Contribution

It proposes a novel non-commutative scheme based on quantum Nambu algebra for generating matrix geometries of coset spaces from gauged quantum mechanics.

Findings

Unveiled pure quantum Nambu geometries with nested fuzzy structures.

Demonstrated these geometries lead to new solutions in Yang-Mills matrix models.

Showed continuum limits form classical manifolds but fuzzification differs from original quantum geometry.

Abstract

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.